A unification of various conjectures related to algebraic cycles, Weil cohomology theories, and dynamical degrees
aa r X i v : . [ m a t h . AG ] F e b A UNIFICATION OF VARIOUS CONJECTURES RELATED TO ALGEBRAICCYCLES, WEIL COHOMOLOGY THEORIES, AND DYNAMICAL DEGREES
FEI HU AND TUYEN TRUNG TRUONGA
BSTRACT . The second author stated the dynamical degree comparison (DDC) conjecture ondynamical correspondences (which are natural generalizations of dominant rational maps) ofsmooth projective varieties, relating the pullback actions of their dynamical iterates on étalecohomology groups and on cycle class groups. This conjecture contains as special cases boththe celebrated Weil’s Riemann hypothesis and the positive characteristic analog of Serre’s resulton polarized endomorphisms.In this paper, we explore certain motivic analogs of the above DDC conjecture and then pro-pose the spectral radius comparison (SRC) conjecture and the norm comparison (NC) conjecture.We show some subtleties regarding these motivic analogs, and that there are some interestinginterrelations between these conjectures and the well-known standard conjectures. As a toolto resolve these conjectures, we propose a quantitative version of the standard conjecture C ,named Conjecture G r , which suffices to obtain the DDC and SRC conjectures for all effectivefinite correspondences. Under an additional assumption of the standard conjecture D , we showthat the DDC and NC conjectures hold. Conjecture G r and the standard conjecture D , for ourpurpose, can be regarded as a replacement of the various positivity notions (forms and currents)in Kähler geometry which are currently not available in positive characteristic.To illustrate the main results, several examples - involving curves, Abelian varieties, surfaces,threefolds, algebraically stable correspondences, as well as the complex field case - are given.Along the way, we show that for surfaces and threefolds, cohomological dynamical degrees arebirational invariants, and the DDC conjecture descents to finite quotients. C ONTENTS
1. Introduction 21.1. A dynamical generalization of Weil’s Riemann hypothesis 31.2. New comparison conjectures 51.3. Main results 81.4. Applications 121.5. Other Weil cohomology theories 131.6. Plan and main idea of the paper 141.7. Acknowledgments 15
Mathematics Subject Classification.
Key words and phrases. positive characteristic, algebraic cycle, correspondence, standard conjectures, Weilcohomology theory, dynamical degree.The authors are supported by Young Research Talents grant
Notations and conventions 152. Preliminaries on correspondences 162.1. Correspondences 162.2. Motivic composition of correspondences 172.3. Operations of correspondences on cycle groups and cohomology 182.4. Degrees of algebraic cycles 212.5. Norms on numerical classes 212.6. Numerical invariants of correspondences 222.7. Boundedness of self-intersections of correspondences 253. Dynamical correspondences and effective finite correspondences 273.1. Dynamical correspondences 283.2. Effective finite correspondences 324. Proofs of main results 334.1. Some auxiliary results 334.2. Proof of Theorem 1.7 404.3. Proofs of Theorem 1.9 and Corollary 1.15 424.4. Proofs of Theorem 1.12 and Corollary 1.17 445. Applications 475.1. Curves 475.2. Abelian varieties 485.3. Descent properties 515.4. Complex projective manifolds 545.5. Algebraically stable correspondences 555.6. Invariance of χ under birational morphisms 555.7. Surfaces and threefolds 59Appendix A. A brief review of the standard conjectures 63References 671. I NTRODUCTION
This paper revolves around correspondences and their natural actions on Weil cohomologygroups and cycle class groups. We propose a unified framework to treat both the celebratedstandard conjectures by Bombieri and Grothendieck [Gro69] (see [Kle68,Kle94] and referencestherein) and the dynamical degree comparison (DDC) conjecture by the second author [Tru16].
TANDARD CONJECTURES AND DYNAMICAL DEGREES 3
A dynamical generalization of Weil’s Riemann hypothesis.
Throughout, X is a smoothprojective variety of dimension n over an algebraically closed field K of arbitrary characteris-tic. A correspondence f of X is an algebraic cycle of codimension n with rational coefficients,or its equivalence class, on X × X . For two correspondences f, g of X , we denote by g ◦ f their motivic composition. If moreover f and g are both dynamical correspondences (which arenatural generalizations of dominant rational maps), we can also define their dynamical compo-sition g ♦ f , similar to the composition of dominant rational maps. We denote by f ◦ m and f ♦ m the motivic and dynamical iterates of f , respectively. There are natural pullback and pushfor-ward actions of correspondences on Weil cohomology groups H • ( X ) and cycle class groups Z • ( X ) Q / ∼ modulo certain equivalence relation ∼ . It is worth mentioning that motivic compo-sition is compatible with pullback and pushforward actions but does not necessarily preservethe effectiveness of correspondences, while dynamical composition preserves the effectivenessof correspondences but may not be compatible with pullback nor pushforward actions. SeeSections 2.2 and 3.1 for details.For simplicity, we now choose étale cohomology as our Weil cohomology theory (but wenote that many results below apply to other Weil cohomology theories as well). In other words,we let H • ( X ) := H • ét ( X, Q ℓ ) with ℓ = char( K ) and refer to Milne’s lecture notes [Mil13] forétale cohomology theory. To a dynamical correspondence f of X , we can assign two invariantsas follows. We fix a field isomorphism ι : Q ℓ ≃ C so that we may speak of the complexabsolute value of an element of Q ℓ : for any α ∈ Q ℓ , | α | ι := | ι ( α ) | . We endow a norm k·k ι on the finite-dimensional Q ℓ -vector space H • ( X ) . For any ≤ i ≤ n ,the i -th cohomological dynamical degree χ i ( f ) ι of f (with respect to ι ) is defined by χ i ( f ) ι := lim sup m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H i ( X ) (cid:13)(cid:13) /mι . (1.1)On the other hand, for any ≤ k ≤ n , we define the k -th numerical dynamical degree λ k ( f ) of f as λ k ( f ) := lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m , (1.2)where we fix an arbitrary norm on the R -vector space N k ( X ) R := N k ( X ) ⊗ Z R .Note that in the above definition of λ k ( f ) , each limit exists and is actually equal to thefollowing limit lim m →∞ (( f ♦ m ) ∗ H kX · H n − kX ) /m = lim m →∞ ( f ♦ m · pr ∗ H n − kX · pr ∗ H kX ) /m , where H X is an (arbitrary) ample divisor on X ; this was proved in [Tru20, Theorem 1.1(1)].See also [Dan20, Theorems 1 and 2] for the case when f is the graph of a dominant rationalself-map of X . For complex projective varieties (or more generally, compact Kähler manifolds),this was established earlier by Dinh and Sibony [DS05, DS08]. The difficulty in showing theexistence of the limit lies in the aforementioned fact that the dynamical composition does notnecessarily commute with the pullback action on cohomology. FEI HU AND TUYEN TRUNG TRUONG
We also note that the λ k ( f ) are defined in an intrinsic way, while apparently, the χ i ( f ) ι maydepend on the field isomorphism ι . Another difference is worth mentioning: we now knowthat the λ k ( f ) < ∞ and is birational invariants (see [Tru20, Theorem 1.1(2)]), but it is stillunknown if the χ i ( f ) ι are always finite or birational invariants.Inspired by results in Complex Dynamics (see [DS17] for a survey), in Esnault–Srinivas[ES13], and Weil’s Riemann hypothesis (now Deligne’s theorem [Del74]), the second authorproposed the following dynamical degree comparison conjecture (cf. [Tru16, Question 2]). Conjecture 1.1 (Dynamical Degree Comparison Conjecture) . Let X be a smooth projectivevariety of dimension n over K . Let f be a dynamical correspondence of X . Then χ k ( f ) ι = λ k ( f ) for any ≤ k ≤ n . Note that the celebrated Weil’s Riemann hypothesis (solved by Deligne [Del74]) is then aspecial case of Conjecture 1.1 when X is actually defined over a finite field F q and f is the graphof the Frobenius endomorphism of X relative to F q . See [Tru16, Theorem 1.5]. Moreover, bythe same argument there, the following positive characteristic analog of Serre’s result [Ser60]on the eigenvalues of polarized endomorphisms of compact Kähler manifolds is also a specialcase of Conjecture 1.1. Conjecture 1.2 (Eigenvalues of polarized endomorphisms) . Let X be a smooth projective vari-ety of dimension n over K . Let f be a polarized endomorphism of X , i.e., f ∗ H X ∼ rat qH X foran ample divisor H X and a positive integer q ∈ Z > . Then for any ≤ i ≤ n , the eigenvaluesof f ∗ | H i ( X ) are q -Weil numbers of weight i , i.e., algebraic numbers α such that | σ ( α ) | = q i/ for every embedding σ : Q ֒ → C . The above Conjecture 1.2 follows from the standard conjectures (see [Kle68, §4]). However,as far as we know, in positive characteristic, it is still open even for surfaces (except for trivialcases), since the standard conjecture of Hodge type remains open for products of surfaces. Theproofs of Weil’s Riemann hypothesis due to Deligne [Del72,Del74] cannot be extended to polar-ized endomorphisms in a straightforward way, since either in the Kuga–Satake construction orin the hypersurface cutting the fact that Frobenius endomorphisms are canonical is essentiallyused. As an application of the machinery developed here, we show that Conjecture 1.2 holdsfor all surfaces dominated by birational models of Abelian surfaces (e.g., Kummer surfaces),for all ground fields and all Weil cohomology theories; see Proposition 5.17.Here is a summary of the status of Conjecture 1.1. The case where K = C can be obtainedfrom known results in Complex Dynamics, using that the de Rham cohomology groups aregenerated by smooth closed ( p, q ) -forms, that on complex projective manifolds the first Chernclass of an ample line bundle is a Kähler class, and a result in [Din05]. In Section 5.4, weprovide a slightly stronger result in this case. In short, the positivity notion in cohomologyplays a crucial role.There has been some progress for the case where f is a self-morphism of X (or rather, thegraph of a self-morphism of X ). The case when f is an automorphism of a smooth projective TANDARD CONJECTURES AND DYNAMICAL DEGREES 5 surface was solved in [ES13] using the Enriques–Bombieri–Mumford classification of surfacesin arbitrary characteristic, Tate’s conjecture for divisors on Abelian varieties, and certain liftingproperty of automorphisms of K surfaces. In any dimension, for an arbitrary self-morphism f ,using the standard conjecture C (which is a consequence of Weil’s Riemann hypothesis when K = F p ; see [KM74]), it has been established in [Tru16] that max i =0 ,..., n χ i ( f ) ι = max k =0 ,...,n λ k ( f ) . (1.3)See also [Shu19] for a different approach towards this equality using dynamical zeta functions.As a consequence, if f is a self-morphism of a surface with λ ( f ) ≥ λ ( f ) (this includes thecase when f is an automorphism of a surface), then Conjecture 1.1 is solved. When f is a self-morphism of an Abelian variety, Conjecture 1.1 is more recently solved in [Hu19a, Hu19b], byusing special arguments available only on Abelian varieties.For the case of dominant rational self-maps, or more general dynamical correspondences,in positive characteristic, less is known about Conjecture 1.1. If we assume further that thestandard conjecture D holds in middle degree on X × X , then eq. (1.3) is obtained in [Tru16].We have seen from above that the standard conjectures (in particular, Conjecture C ( X ) andConjecture D n ( X × X ) ) are very useful in studying Conjecture 1.1. The remaining of thispaper aims to demonstrate reciprocally that Conjecture 1.1 and our new comparison conjecturesbelow could also be helpful in studying the standard conjectures, which will be recalled inAppendix A.1.2. New comparison conjectures.
We start with a motivic analog of Conjecture 1.1, whichwe call the spectral radius comparison (SRC) conjecture. Note that in the definitions of theabove dynamical degrees χ i ( f ) ι and λ k ( f ) , if we use motivic iterates which are functorialinstead of dynamical iterates, then by the spectral radius formula in Linear Algebra, they arenothing but the spectral radii of the corresponding linear operators: lim m →∞ (cid:13)(cid:13) ( f ◦ m ) ∗ | H i ( X ) (cid:13)(cid:13) /mι = lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | H i ( X ) (cid:13)(cid:13) /mι = ρ ( f ∗ | H i ( X ) ) ι , and lim m →∞ (cid:13)(cid:13) ( f ◦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m = lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | N k ( X ) R (cid:13)(cid:13) /m = ρ ( f ∗ | N k ( X ) R ) . For any ≤ k ≤ n , let A k ( X ) denote the Q -vector space of algebraic cycles of codimension k on X modulo homological equivalence ∼ hom , i.e., A k ( X ) := Im(cl X : Z k ( X ) Q −→ H k ( X )) . Conjecture 1.3 (Spectral Radius Comparison Conjecture) . Let X be a smooth projective vari-ety of dimension n over K . For any ≤ k ≤ n , let f be a correspondence of X which is eithereffective or in A k ( X ) ⊗ A n − k ( X ) . Then we have ρ ( f ∗ | H k ( X ) ) ι = ρ ( f ∗ | N k ( X ) R ) . FEI HU AND TUYEN TRUNG TRUONG
We also consider the following norm comparison (NC) conjecture. Although it only concernspullback action of correspondences on even-degree cohomology groups, by a standard producttrick, one can also have an appropriate estimate of Dinh’s type for odd degrees (see Lemma 4.3).
Conjecture 1.4 (Norm Comparison Conjecture) . Let X be a smooth projective variety of di-mension n over K . Then there is a constant C > such that for every ≤ k ≤ n and for everycorrespondence f of X which is either effective or in A k ( X ) ⊗ A n − k ( X ) , we have (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) . Remark . It is tempting to state Conjectures 1.3 and 1.4 for all correspondences, given thatthe statements do not involve dynamical notions, not just for correspondences of X which areeffective or in A k ( X ) ⊗ A n − k ( X ) . However, as we shall see in Remark 1.8 later, such a general-ization does not hold. On the other hand, the main results in the current paper provide a supportthat these conjectures should hold for the classes of correspondences mentioned in these con-jectures. One reasonable explanation for why this should be the case is that for the mentionedclasses of correspondences, the refined k -th degree deg k ( f ) of f introduced in Section 2.6is equivalent to the norm k f ∗ | N k ( X ) R k , while if we use this new numerical invariant deg k ( f ) instead of k f ∗ | N k ( X ) R k we have certain nice estimates shown in Theorems 1.9 and 1.12.Nonetheless, it is easy to see that Conjecture 1.4 implies Conjecture 1.1, by consideringdynamical iterates f ♦ m . However, it is unknown whether Conjecture 1.4 implies Conjecture 1.3.The reason is that a typical argument would go through considering motivic iterates f ◦ m . Notethat even if f is effective, there is no guarantee that f ◦ m is effective. The previous paragraphlets one cautious when encountering this case.As far as we know, Conjectures 1.3 and 1.4 are new and open, even over C . Since Conjecture1.1 has been proved over C by using various familiar positivity notions (such as forms, currents)in Kähler geometry, a natural approach would be trying to develop these notions in positivecharacteristic. However, currently, there are no such analogs of forms or currents in positivecharacteristic, let alone their "positivity". In the remaining of this subsection, we proposeinstead a new Conjecture G r , which is a quantitative version of the standard conjecture C and sufficient to solve Conjecture 1.2 (see Theorem 1.9 and Corollary 1.10). On the other hand,together with the standard conjecture D , it can replace the above "positivity" notions in practice(see Theorem 1.12 and Corollary 1.17). In Section 5, we shall show that curves and Abelianvarieties indeed satisfy Conjecture G r .We first introduce the so-called Effective Property G r on X , where r ∈ Q > is a positiverational number. Note that there always exists a homological correspondence γ X,r of X , i.e., γ X,r ∈ H n ( X × X ) = n M i =0 H i ( X ) ⊗ H n − i ( X ) ≃ n M i =0 End Q ℓ ( H i ( X )) , so that its pullback γ ∗ X,r on H i ( X ) is the multiplication-by- r i map for each i . Note that γ X,r commutes with all homological correspondences of X . We also denote it by γ r if there is TANDARD CONJECTURES AND DYNAMICAL DEGREES 7 no confusion. If Conjecture C ( X ) holds, then γ r can be represented by a correspondence P ni =0 r i ∆ i , where ∆ i ∈ Z n ( X × X ) Q corresponds the i -th Künneth component π i of the diag-onal class cl X × X (∆ X ) . In fact, we shall see in Lemma 4.4 that Conjecture C ( X ) is equivalentto the algebraicity of γ r for every r ∈ Q > . Effective Property G r . Let X be a smooth projective variety of dimension n over K . Then forany r ∈ Q > , the above homological correspondence γ r of X is algebraic and represented bya rational algebraic n -cycle G r on X × X , i.e., γ r = cl X × X ( G r ) ; moreover, for any effectivecorrespondence f of X , the motivic composition G r ◦ f is also effective.In practice, a weaker version is enough for our purpose. Recall that for an algebraic k -cycle Z on X , its degree deg( Z ) (with respect to the fixed ample divisor H X on X ) is defined by Z · H kX . If Z is effective, then by the Nakai–Moishezon criterion, we know that deg( Z ) ≥ .Let H X × X := pr ∗ H X + pr ∗ H X be a fixed ample divisor on X × X . Then the degree deg( f ) of a correspondence f of X is equal to f · H nX × X = n X i =0 (cid:18) ni (cid:19) f · pr ∗ H n − iX · pr ∗ H iX = n X i =0 (cid:18) ni (cid:19) f ∗ H iX · H n − iX . In the summation, each f ∗ H iX · H n − iX is called the i -th degree of f , denoted by deg i ( f ) ; it playsa significant role in studying dynamics of a dynamical correspondence f . Conjecture G r . Let X be a smooth projective variety of dimension n over K . Then for any r ∈ Q > , the above homological correspondence γ r of X is algebraic and represented bya rational algebraic n -cycle G r on X × X , i.e., γ r = cl X × X ( G r ) ; moreover, there exists aconstant C > independent of r , so that for any effective correspondence f of X , we have k G r ◦ f k ≤ C deg( G r ◦ f ) , where k G r ◦ f k denotes any norm of G r ◦ f as an element in N n ( X × X ) R ; see Section 2.6. We note that for an effective correspondence f of X , albeit G r ◦ f may not be effectiveanymore, we always have deg( G r ◦ f ) = n X i =0 (cid:18) ni (cid:19) ( G r ◦ f ) ∗ H iX · H n − iX = n X i =0 (cid:18) ni (cid:19) r i f · pr ∗ H n − iX · pr ∗ H iX ≥ . Conjecture G r is indeed weaker than Effective Property G r by Remark 2.17. Also, Lemma 4.5demonstrates that Conjecture G r is not too outrageous: modulo the standard conjecture C (which, as mentioned, always holds over finite fields as a consequence of Weil’s Riemann hy-pothesis), it is equivalent to requiring a uniform bound on certain constants C r . In characteristiczero, this uniform boundedness of the constants C r holds for all dynamical correspondences (in-deed, there we can prove the following stronger estimate for norms of cohomology classes thanthe one required in Conjecture G r ; see Section 5.4). FEI HU AND TUYEN TRUNG TRUONG
Conjecture SG r . Let X be a smooth projective variety of dimension n over K . Then thereexists a constant C > such that for any r ∈ Q > and for any effective correspondence f of X , we have (cid:13)(cid:13) γ r ◦ cl X × X ( f ) (cid:13)(cid:13) ι ≤ C max k =0 ,...,n r k deg k ( f ) , where the left-hand side is any norm on the finite-dimensional Q ℓ -vector space H n ( X × X ) . In characteristic zero, this new conjecture can be more convenient to use than Conjecture G r ,given that the standard conjecture C is unknown in general (and hence, a priori, we do not knowif γ r ◦ f is algebraic). On the other hand, if Conjecture D n ( X × X ) holds, then the inequalitiesin Conjecture G r and Conjecture SG r are equivalent. Remark . It is worthy to mention that in order to solve Conjecture 1.2, we only need Con-jecture G r for graphs of all endomorphisms via the use of Theorem 1.9(3), where more gen-eral results hold for effective finite correspondences. Over finite fields, the standard conjec-ture C holds [KM74], so we only need to prove the inequality part of Conjecture G r . Oncethis has been done, we can show that Conjecture 1.3 holds for all endomorphisms over finitefields. Then using the well-known spreading out and specialization arguments, Conjecture 1.3holds for all endomorphisms over arbitrary fields and for all Weil cohomology theories; seeSection 1.5 for more details. This, in turn, proves Conjecture 1.2 for all fields and all Weilcohomology theories. Alternatively, one can proceed to prove the stronger inequality Conjec-ture SG r for graphs of polarized endomorphisms, without assuming the standard conjectures.In characteristic zero, as mentioned, Conjecture SG r holds for all dynamical correspondences(see Section 5.4).1.3. Main results.
Now we are ready to state the main results of this paper. The first resultshows some relations between our new conjectures and the standard conjectures from a dynam-ical viewpoint.
Theorem 1.7.
Let X be a smooth projective variety of dimension n over K . Fix any ≤ k ≤ n . (1) The following statements are equivalent: (a)
Conjecture D k ( X ) holds, i.e., homological equivalence and numerical equiva-lence coincide for algebraic cycles of codimension k on X . (b) Conjecture 1.4 holds for all correspondences f ∈ A k ( X ) ⊗ A n − k ( X ) . (2) Assume that Conjecture D k ( X ) holds. Then the following statements are equivalent: (a) H k ( X ) = A k ( X ) ⊗ Q Q ℓ = N k ( X ) ⊗ Z Q ℓ . In other words, H k ( X ) is generatedby algebraic cycle classes. (b) Conjecture 1.3 holds for all correspondences f ∈ A n ( X × X ) . (3) Assume that Conjecture C ( X ) holds. Then Conjecture 1.3 holds for all correspon-dences f ∈ A k ( X ) ⊗ A n − k ( X ) .Remark . From the above theorem, we see that Conjecture 1.4 is a unified framework to treatboth the standard conjecture D and the dynamical degree comparison (DDC) Conjecture 1.1. TANDARD CONJECTURES AND DYNAMICAL DEGREES 9
However, since the standard conjecture D holds in codimension one by Matsusaka [Mat57](i.e., Conjecture D ( X ) holds), if H ( X ) is not generated by divisor classes then we have anexample of a variety X where the inequalities in Conjectures 1.3 and 1.4 cannot hold for all correspondences. Here is an explicit way. Let α i be a basis for N ( X ) Q and β i a dual basis for N n − ( X ) Q . Consider f := ∆ X − P i α i ⊗ β i . Then f ∗ | N ( X ) Q = 0 , but will be nonzero on H ( X ) := H n − ( X ) ⊥ ; see eq. (A.5) for this notion. In fact, since ( α i ⊗ β i ) ∗ | H ( X ) = 0 , wecan see that f ∗ | H ( X ) is the identity map. Of course, we can check that ρ (∆ ∗ X | H ( X ) ) = ρ (∆ ∗ X | N ( X ) Q ) = 1 and ρ (cid:0) X i ( α i ⊗ β i ) ∗ | H ( X ) (cid:1) = ρ (cid:0) X i ( α i ⊗ β i ) ∗ | N ( X ) Q (cid:1) = 1 . In the remaining of this paper, we will mainly consider effective correspondences, whichhave significant dynamical meanings. The idea obtained could be helpful for studying the case f ∈ A k ( X ) ⊗ A n − k ( X ) , which as we have seen in Theorem 1.7(1), leads to a proof of thestandard conjecture D . Note that when K = C , Dinh’s inequality [Din05, Proposition 5.8],which infers an affirmative answer to Conjecture 1.1 (see [Tru16]), can also be used to provethe validity of Conjecture 1.4 and Conjecture G r for all dynamical correspondences in charac-teristic zero; see Section 5.4 for more details. On the other hand, Conjecture 1.3 is widely openeven in characteristic zero (except for algebraically stable correspondences).The refined k -th degree deg k ( f ) mentioned in Remark 1.5, which will be discussed in Sec-tion 2.6, is used in the next results. For the time being, it suffices to mention that deg k ( f ) ≥ for all correspondences f of X , and if f is effective then it is equivalent to k f ∗ | N k ( X ) R k (seeRemark 2.17). By using the projective cross norm on tensor products, we can also see that if f ∈ A k ( X ) ⊗ A n − k ( X ) , then deg k ( f ) is equivalent to k f ∗ | N k ( X ) R k (see Proposition 2.18). Theorem 1.9.
Let X be a smooth projective variety of dimension n over K such that Conjec-ture G r holds. Then there is a constant C > so that the following statements hold. (1) If f is a correspondence of X , then we have for any ≤ k ≤ n , (cid:12)(cid:12) Tr( f ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ C deg k ( f ) . Further, if f is effective, then we have for any ≤ k ≤ n − , (cid:12)(cid:12) Tr( f ∗ | H k +1 ( X ) ) (cid:12)(cid:12) ≤ C q deg k ( f ) deg k +1 ( f ) . (2) If f is an effective correspondence of X so that the degree sequence { deg i ( f ) } i islog-concave (for example, f is an irreducible correspondence), then we have for any ≤ k ≤ n , ρ ( f ∗ | H k ( X ) ) ≤ C deg k ( f ) , and for any ≤ k ≤ n − , ρ ( f ∗ | H k +1 ( X ) ) ≤ C q deg k ( f ) deg k +1 ( f ) . (3) If f is a correspondence of X such that the m -th motivic iterate f ◦ m is numericallyeffective for every m ∈ N , then we have for any ≤ k ≤ n , ρ ( f ∗ | H k ( X ) ) = ρ ( f ∗ | N k ( X ) R ) , and for any ≤ k ≤ n − , ρ ( f ∗ | H k +1 ( X ) ) ≤ q ρ ( f ∗ | N k ( X ) R ) ρ ( f ∗ | N k +1 ( X ) R ) . As an immediate application of Theorem 1.9(3) to effective finite correspondences (see Def-inition 3.13), we obtain the following corollary. Note that by Remarks 3.14 and 3.15, thesecorrespondences are algebraically stable and their iterates are always effective. Hence its proofis straightforward.
Corollary 1.10.
Let X be a smooth projective variety of dimension n over K such that Conjec-ture G r holds. If f is an effective finite correspondence of X , then we have any ≤ k ≤ n , χ k ( f ) = ρ ( f ∗ | H k ( X ) ) = ρ ( f ∗ | N k ( X ) R ) = λ k ( f ) , and for any ≤ k ≤ n − , χ k +1 ( f ) = ρ ( f ∗ | H k +1 ( X ) ) ≤ q ρ ( f ∗ | N k ( X ) R ) ρ ( f ∗ | N k +1 ( X ) R ) = p λ k ( f ) λ k +1 ( f ) . In particular, Conjecture 1.2 holds on X .Remark . Note that under the assumption of Theorem 1.9 or Corollary 1.10, Conjec-ture C ( X ) holds. Hence both Tr( f ∗ | H i ( X ) ) and ρ ( f ∗ | H i ( X ) ) are independent of the choiceof ι : Q ℓ ≃ C by the Lefschetz trace formula (see Proposition 2.8).If f is effective or in A k ( X ) ⊗ A n − k ( X ) , then deg k ( f ) is equivalent to k f ∗ | N k ( X ) R k . Hence,the conclusion of Theorem 1.9(1) reads as | Tr( f ∗ | H k ( X ) ) | ≤ C k f ∗ | N k ( X ) R k . However, byTheorem 1.7(2), this new statement cannot hold for general (non-effective) correspondences.Indeed, taking k = 1 for simplicity, assume that | Tr( f ∗ | H ( X ) ) | ≤ C k f ∗ | N ( X ) R k for all corre-spondences f . Since the standard conjecture D holds for divisors, the proof of Theorem 1.7(2)implies that H ( X ) must be generated by divisor classes, which cannot be always true.Also, for Theorem 1.9(2), as far as we know, it is not straightforward to reduce the generaleffective case to the irreducible case as we proceed in the proof of Theorem 1.9(1), since wedo not have an inequality like ρ ( P i φ i ) ≤ P i ρ ( φ i ) for linear maps φ i (i.e., the spectral radiusfunction ρ ( · ) is in general not subadditive). Theorem 1.12.
Let X be a smooth projective variety of dimension n over K . (1) The following two statements are equivalent: (a)
Conjecture D n ( X × X ) holds, i.e., homological equivalence and numerical equiv-alence coincide for algebraic cycles of codimension n on X × X . (b) There is a constant
C > so that for all correspondences f of X and for all ≤ i ≤ n , we have (cid:13)(cid:13) f ∗ | H i ( X ) (cid:13)(cid:13) ι ≤ C k f k . TANDARD CONJECTURES AND DYNAMICAL DEGREES 11
Here k f k denotes the norm of f as an element in N n ( X × X ) R ; see Section 2.6. (2) Suppose that Conjecture D n ( X × X ) and Conjecture G r hold on X . Then there is aconstant C > such that the following two inequalities hold: (a) For all correspondences f of X and for any ≤ k ≤ n , we have (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C deg k ( f ) . (1.4)(b) For all effective correspondences f of X and for any ≤ k ≤ n − , we have (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ C q deg k ( f )deg k +1 ( f ) . (1.5)(3) Suppose that Conjecture C ( X ) holds. If the inequalities (1.4) and (1.5) in the assertion(2) hold for all effective correspondences, then Conjecture G r holds on X .Remark . In the above assertion (2a), if f is an effective correspondence of X or in A k ( X ) ⊗ A n − k ( X ) , then deg k ( f ) gives a norm on f ∗ | N k ( X ) R (see Remark 2.17). Hence, in particular,Conjectures 1.1 and 1.4 hold. Then by Theorem 1.7(1), Conjecture D ( X ) follows from theinequality (1.4) in (2a). The inequalities (1.4) and (1.5) in the assertion (2) are first proved forgraphs of automorphisms in characteristic zero by [Din05] (see also Section 5.4).By Propositions A.1 and A.6, Conjecture D ( X × X ) implies Conjecture C ( X ) . Hence,from Theorem 1.12 and Remark 1.13, we obtain immediately the following corollary. Corollary 1.14.
The following two statements are equivalent: (1)
The standard conjecture D and Conjecture G r hold for all X . (2) For each X , there is a constant C X > so that both inequalities (1.4) and (1.5) inTheorem 1.12(2) hold. Now we state some dynamical consequences of Theorems 1.9 and 1.12 by applying them todynamical correspondences.
Corollary 1.15.
Let X be a smooth projective variety of dimension n over K such that Con-jecture G r holds. Let f be a dynamical correspondence of X so that the dynamical degreesequence { λ i ( f ) } i is log-concave. Then for any ≤ k ≤ n , we have lim sup m →∞ ρ (( f ♦ m ) ∗ | H k ( X ) ) /m ≤ λ k ( f ) . Remark . The relevance of Corollary 1.15 to Conjecture 1.1 is that there is a speculation inComplex Dynamics that we should have, at least for a dominant rational self-map f of X , lim m →∞ ρ (( f ♦ m ) ∗ | N k ( X ) R ) /m = λ k ( f ) . Though the degree sequences { deg i ( f ) } i are log-concave for irreducible correspondences (see Lemma 4.2),the log-concavity property fails for general reducible correspondences. A priori, we do not know if the f ♦ m areirreducible even for an irreducible f . See [Tru20, Remark 1.4(5)] for details. If we assume that the following cohomological analog holds: lim sup m →∞ ρ (( f ♦ m ) ∗ | H k ( X ) ) /m = χ k ( f ) ι , then we obtain right away from the above corollary and Lemma 4.6 that χ k ( f ) ι = λ k ( f ) . Perhaps the best evidence to support this speculation is the recent work by [DF20], who showthat for any dominant rational self-map f : X X of a normal projective variety X over aground field of characteristic zero, there is a Banach space ( N • BPF ( X ) , k · k BPF ) constructedusing base-point-free b -classes in a subtle way, and a bounded linear pullback action of f on it,so that λ k ( f ) = ρ ( f ∗ | N k BPF ( X ) ) for all k . Here X stands for the Riemann–Zariski space of X defined as the projective limit ofall birational models over X , and the spectral radius is defined by lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | N k BPF ( X ) (cid:13)(cid:13) /m BPF . To have such an equality, they prove in [DF20, Proposition 4.3] that the norm k f ∗ | N k BPF ( X ) k BPF is equivalent to deg k ( f ) . We refer the interested reader to their paper for details. Corollary 1.17.
Let X be a smooth projective variety of dimension n over K such that Conjec-ture D n ( X × X ) and Conjecture G r hold. Let f be a dynamical correspondence of X . Thenthe following statements hold. (1) Conjecture 1.1 holds. In particular, for all ≤ k ≤ n , we have χ k ( f ) ι = lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) /mι = λ k ( f ) . (2) For all ≤ k ≤ n − , we have χ k +1 ( f ) ι := lim sup m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H k +1 ( X ) (cid:13)(cid:13) /mι ≤ p λ k ( f ) λ k +1 ( f ) . Remark . Hence, if the standard conjecture D and Conjecture G r hold, then Corollary 1.17solves Conjecture 1.1 in the affirmative, and also gives further information concerning χ i ( f ) ι for odd values of i . The latter in turn answers in the affirmative another question in [Tru16],where the inequality in (2) is named Dinh’s inequality.1.4. Applications.
In Section 5, we will present several explicit applications of the above re-sults. As some of the most interesting among these, we show that • Conjecture G r holds on curves and Abelian varieties (see Lemmas 5.1 and 5.3); • Conjectures 1.1 and 1.4 holds for dynamical correspondences of Abelian varieties overan algebraic closure of a finite field (see Section 5.2); • Conjecture 1.4 holds for dynamical correspondences of smooth complex projective va-rieties (see Theorem 5.10);
TANDARD CONJECTURES AND DYNAMICAL DEGREES 13 • The inequality part of Conjecture G r holds for all dynamical correspondences of smoothcomplex projective varieties (see Theorem 5.10); • Conjecture 1.1 holds on various types of surfaces and threefolds (see Section 5.7); • Conjecture 1.2 holds for all surfaces dominated by birational models of Abelian sur-faces (e.g., Kummer surfaces), for all ground fields and all Weil cohomology theories(see Proposition 5.17). Moreover, these surfaces satisfy Conjecture G r for dynamicalcorrespondences.In order to deal with the case of surfaces and threefolds, an invariant property of χ underbirational morphisms in every dimension is established (see Theorem 5.14). As an interestingconsequence of the mentioned invariant property and another descent property Theorem 5.7, weobtain the following result, which is valid for surfaces and threefolds, by virtue of the existenceof resolution of singularities. Since numerical dynamical degrees λ k are finite invariant undergenerically finite dominant rational maps (see [Tru20, Theorem 1.1(3)]), the theorem below isa consequence of Conjecture 1.1. Theorem 1.19. (1)
Let π : X Y be a birational map of smooth projective varieties ofdimension at most over K . Let g be a dynamical correspondence of Y and let f bethe dynamical pullback ( π × π ) ★ ( g ) of g under π × π (see Definition 3.7). Then wehave χ ( f ) ι = χ ( g ) ι and χ ( f ) ι = χ ( g ) ι . (2) Let π : X Y be a dominant rational map of smooth projective varieties of the samedimension at most over K . If Conjecture 1.1 holds on X , then so does on Y . Other Weil cohomology theories.
Although we mainly focus on étale cohomology inthis paper (for instance, in the definition of cohomological dynamical degrees χ i ), many resultsstill hold for other Weil cohomology theories. Indeed, in positive characteristic, it is believed(at least for us) that we should have a universal motivic cohomology theory which specializesto those well-known Weil cohomology theories; in particular, once a result holds for one Weilcohomology theory then it holds for all (for example, the independence of ℓ conjecture).Here is a detailed explanation. We will consider only smooth projective varieties X of di-mension n over ground fields K of positive characteristic, which falls into the following twocases. Case : K is an algebraic closure of a finite field. Then as a consequence of Deligne [Del74],we know that for every correspondence f of X and every ≤ i ≤ n , the characteristic polyno-mial of f ∗ | H i ( X ) has rational coefficients and is independent of the choice of Weil cohomologytheory H • ( X ) (see [KM74]). The main reason is that the Künneth projectors are Q -linear com-binations of powers of the Frobenius endomorphism, independent of the theory. In particular,all conjectures and results involving only Tr( f ∗ | H i ( X ) ) , ρ ( f ∗ | H i ( X ) ) , and the numerical cyclegroup N • ( X ) are independent of Weil cohomology theory H • ( X ) : if their conclusions are validfor one such theory A, then their conclusions are valid for all other theories B (even if one doesnot know if the assumptions needed to derive at these conclusions are valid in theory A would still hold in theories B). In particular, this is the case for the conclusions in: Conjectures 1.2,1.3 and G r , Theorem 1.9, Corollaries 1.10 and 1.15. Case : K is a general algebraically closed field of positive characteristic. In this case,if the correspondences under consideration are endomorphisms, or more generally, a positivesum f = P i a i Γ i , where a i ∈ Q > and each Γ i is the graph of an endomorphism of X ,then we can use the well-known spreading out and specialization arguments to reduce to theabove finite field case. For instance, in [Tru16], this reduction was used to prove eq. (1.3) forendomorphisms. The point is that the action on cohomology groups of an endomorphism (andhence its trace) specializes, and the spectral radius ρ ( M ) of a square matrix M can be computedthrough | Tr( M m ) | or k M m k of self-products of M . The general case here, where f is a positivesum of graphs of endomorphisms, can also be dealt with similarly, by noting that f ◦ m is againa positive sum of graphs of endomorphisms, and both traces and norms satisfy the triangleinequality. For example, to show that ρ ( f ∗ | H k ( X ) ) = ρ ( f ∗ | N k ( X ) R ) for all endomorphisms f (or a positive sum of such), we reduce to showing a weaker assertion that there is a constant C > so that | Tr( f ∗ | H k ( X ) ) | ≤ C k f ∗ | N k ( X ) R k . The latter inequality specializes.1.6. Plan and main idea of the paper.
In Section 2, we will provide preliminaries on cor-respondences, and the classical way to compose them, and various estimates on intersectionnumbers and norms. A useful result on the boundedness of self-intersections of correspon-dences, Lemma 2.19, is given therein. Dynamical correspondences and their compositions,which are less familiar than their motivic counterparts, will be discussed in Section 3. In thesame section, we will explain how these carry out to effective finite correspondences, usingendomorphisms (not necessarily surjective) as a model. The section after is devoted to proofsof main results. The last section provides more explicit applications, examples, and some ad-ditional results, including proof of Theorem 1.19. In Appendix A, a summary of the standardconjectures and a partial list of their current status is given.The main theme of this paper is that there are deep interrelation and unification betweenthe standard conjectures and Dynamical Systems. Conjecture 1.4 (stemming from ComplexDynamics) generalizes both the standard conjecture D and Conjecture 1.1 (also coming fromComplex Dynamics). The latter is a dynamical generalization of Weil’s Riemann hypothesis(and more generally Conjecture 1.2), which in turn is meaningful in both Algebraic and Arith-metic Geometry. These conjectures all predict, on smooth projective varieties, surprising in-timate relationships between cohomological and numerical equivalences concerning algebraiccycles. The conjectures are also quite subtle. While Conjectures 1.3 and 1.4 can be naturallystated for all correspondences, not necessarily effective, these generalizations do not hold truefor all correspondences. Our results show that the right context for these conjectures is to re-strict to correspondences f for which deg k ( f ) and k f ∗ | N k ( X ) R k are equivalent. Among thosefor which the latter condition is satisfied, we have effective correspondences (which generate allcorrespondences and have dynamical applications) and correspondences in A k ( X ) ⊗ A n − k ( X ) (which have applications to the standard conjecture D ). TANDARD CONJECTURES AND DYNAMICAL DEGREES 15
The main technical tool introduced in this paper, Conjecture G r , is a quantitative version ofthe standard conjecture C . The combination of Conjecture G r with the standard conjecture D ,as well as a quantitative version of Chow’s moving lemma, can replace the positivity notionsin Kähler manifolds (such as positive closed smooth forms and currents) for many purposes, inparticular, provide affirmative answers to both Conjectures 1.1 and 1.4. We expect thereforethat Conjecture G r could later find more applications for analogs in positive characteristic ofresults currently known only in characteristic zero. Besides this, a well-known fact, that thedegree sequence { deg k ( f ) } k of an irreducible correspondence f is log-concave, also plays acrucial role. Yet another point is that the approach of considering general correspondences ishelpful even if one is interested only in endomorphisms. For example, in Section 5.7, endomor-phisms on a finite quotient of an Abelian surface or threefold (or their birational models) canbe studied by (dynamically) pulling back to the Abelian variety, on which they are no longerendomorphisms but just dynamical correspondences in general.The program by Bombieri and Grothendieck is to use the standard conjecture B and the stan-dard conjecture of Hodge type to solve Weil’s Riemann hypothesis and Conjecture 1.2. Theresults in this paper suggest an alternative route: using the standard conjecture D and Conjec-ture G r . Indeed, for the purpose of solving Conjecture 1.2, we only need Conjecture G r forgraphs of surjective endomorphisms (see Remark 1.6 and Corollary 1.10). Note that by Propo-sition A.6, the standard conjecture D follows from the combination of the standard conjecture B and the standard conjecture of Hodge type. Also, even for Abelian varieties, the standardconjecture of Hodge type is currently known only up to dimension four [Anc21], while ourConjecture G r holds in all dimensions (see Section 5.2). Yet another fact is that in charac-teristic zero, the proof of the standard conjecture of Hodge type is much more difficult thanthat of the inequality part of Conjecture G r (more generally Conjecture SG r ) for dynamicalcorrespondences (see Theorem 5.10). Moreover, with this alternative route, we obtain not onlyWeil’s Riemann hypothesis or Conjecture 1.2, but also various much stronger, applicable inmore general settings and new results, several of which are in the if and only if form. In char-acteristic zero, most of the mentioned results are known unconditionally. These allow a newcharacteristic-free perspective of algebraic dynamics.1.7. Acknowledgments.
We would like to heartily thank Bruno Kahn and Paul Arne Østværfor referring us to the notion of finite correspondence, which helps to strengthen some results.We would also like to thank Giuseppe Ancona, Nguyen-Bac Dang, Charles Favre, Claire Voisin,Duc-Viet Vu, Yuri Zarhin, and De-Qi Zhang for inspiring comments and discussions.N
OTATIONS AND CONVENTIONS
Unless otherwise stated, the following notations remain in force throughout. K is an algebraically closed field of arbitrary characteristic. X, Y are smooth projective variety of dimension n over K . f : X ⊢ Y is a correspondence from X to Y , i.e., a rational algebraic cycle of codimension n , or its equivalence class, on X × Y . Corr ( X, X ) or Corr ∼ ( X, X ) is the group or ring of self-correspondences of X . H X , H Y are fixed ample divisors on X and Y . H X × Y := pr ∗ H X + pr ∗ H Y is a fixed ample divisor on X × Y , where pr i is the naturalprojection from X × Y to the i -th factor.‘ · ’ denotes the intersection product of algebraic cycle classes, while ‘ ∪ ’ stands for the cupproduct of cohomology classes. deg k ( f ) := f · pr ∗ H n − kX · pr ∗ H kY = f ∗ H kX · H n − kY is the k -th degree of f . deg( f ) := f · H nX × Y = P nk =0 (cid:0) nk (cid:1) deg k ( f ) is the degree of f (with respect to H X × Y ). Z k ( X ) is the Abelian group generated by integral algebraic cycles of codimension k on X . CH • ( X ) := Z • ( X ) / ∼ rat is the Chow ring of X , i.e., the ring of integral algebraic cycles on X modulo rational equivalence ∼ rat . H • ( X ) is a fixed Weil cohomology theory of X , e.g., étale cohomology H • ét ( X, Q ℓ ) .For an arbitrary Abelian group A (viewed as a Z -module) and a field F of characteristic zero(e.g., Q , R , C , or Q ℓ ), A F stands for A ⊗ Z F . cl X : Z k ( X ) Q −→ H k ( X ) is the cycle class map, which factors through CH k ( X ) Q . Some-times, we also call cl X : CH k ( X ) Q −→ H k ( X ) the cycle class map. A k ( X ) := Z k ( X ) Q / ∼ hom = Im(cl X : Z k ( X ) Q −→ H k ( X )) denotes the Q -vector space ofalgebraic cycles of codimension k on X modulo homological equivalence ∼ hom . N k ( X ) := Z k ( X ) / ≡ is the Abelian group of integral algebraic cycles of codimension k on X modulo numerical equivalence ≡ , which is finitely generated (see [Kle68, Theorem 3.5]). g ◦ f denotes the motivic composition of two composable correspondences f, g (see Sec-tion 2.2). g ♦ f denotes the dynamical composition of two composable dynamical correspondences f, g (see Definition 3.2). ( π × π ) ★ ( g ) denotes the dynamical pullback of g under π × π (see Definition 3.7).2. P RELIMINARIES ON CORRESPONDENCES
Correspondences.
Let K be an algebraically closed field of arbitrary characteristic. All K -varieties are assumed to be irreducible, reduced, separated schemes of finite type over K .We work with the category SmProj ( K ) of smooth projective varieties over K and fix a Weilcohomology theory with a coefficient field F of characteristic zero, i.e., a functor H from SmProj ( K ) to the category of finite-dimensional graded F -vector spaces (see [MNP13, Defini-tion 1.2.13]). In particular, we have a cup product ∪ , Poincaré duality, the Künneth formula,the cycle class map cl , the weak Lefschetz theorem, and the hard Lefschetz theorem. We referto Fulton’s book [Ful98] for the intersection theory on SmProj ( K ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 17
Definition 2.1.
Let X and Y be smooth projective varieties of dimension n over K . Through-out, unless otherwise stated , a correspondence f from X to Y is a rational algebraic cycle ofcodimension n , or its equivalence class, on X × Y , i.e., f ∈ Z n ( X × Y ) Q or Z n ( X × Y ) Q / ∼ ,where ∼ is an adequate equivalence relation (e.g., rational equivalence ∼ rat , homological equiv-alence ∼ hom , or numerical equivalence ≡ ; see [MNP13, §1.2] for a discussion on various ade-quate equivalence relations). We shall write f : X ⊢ Y to denote a correspondence f from X to Y . We also use the following shorthand notation: Corr ( X, Y ) := Z n ( X × Y ) Q , Corr ∼ ( X, Y ) := Z n ( X × Y ) Q / ∼ . The canonical transpose f T : Y ⊢ X of a correspondence f : X ⊢ Y is defined as thepushforward of f under the involution map τ : X × Y → Y × X which interchanges thecoordinates.A correspondence f : X ⊢ Y is effective , if it can be represented by an effective rational n -cycle on X × Y . It is further called irreducible , if f is represented by an (irreducible) algebraicsubvariety of X × Y of codimension n . Example 2.2.
Let π : X → Y be a morphism of smooth projective varieties of dimension n over K . Then its graph Γ π , defined by the image of the diagonal ∆ X under the productmorphism id X × π : X × X → X × Y , is an irreducible algebraic cycle of codimension n on X × Y , hence an irreducible correspondence from X to Y . Its transpose is thus denoted by Γ T π (or π T ). Similarly, one can consider the graph Γ π of a rational map π : X Y , which is alsoan irreducible correspondence from X to Y .2.2. Motivic composition of correspondences.
Correspondences can be naturally composedin intersection theory (see, e.g., [Ful98, §16.1]). More precisely, given two arbitrary correspon-dences f : X ⊢ Y and g : Y ⊢ Z , the composite correspondence, denoted by g ◦ f , is definedby g ◦ f := pr , ∗ (pr ∗ f · pr ∗ g ) , (2.1)where the pr ij denote the natural projections from X × Y × Z to the appropriate factors, re-spectively. Note that the intersection is in Z n ( X × Y × Z ) Q / ∼ , and hence the compositecorrespondence g ◦ f ∈ Corr ∼ ( X, Z ) . In particular, this makes Corr ∼ ( X, X ) a (not necessarilycommutative) ring. Clearly, the identity is given by the class of the diagonal ∆ X . To distinguishthe so-called dynamical composition of dynamical correspondences that will be introduced inSection 3, we call g ◦ f the motivic composition of f followed by g .For a self-correspondence f : X ⊢ X and any m ∈ N , we denote by f ◦ m the m -th motiviciterate of f . Certainly, one may speak of correspondences in a more general sense. For instance, correspondences mayalso mean arbitrary cycle classes on X × Y , or even arbitrary cohomology classes in H • ( X × Y ) as in [Kle68]. Remark . It is worth mentioning that given two morphisms φ : X → Y and ψ : Y → Z ofsmooth projective varieties of dimension n over K , we always have that Γ ψ ◦ Γ φ = Γ ψ ◦ φ ascorrespondences from X to Z , where ψ ◦ φ denotes the standard composition of morphisms.However, if both φ and ψ are endomorphisms of an Abelian variety, then in general, we do not have that Γ φ + Γ ψ = Γ φ + ψ .2.3. Operations of correspondences on cycle groups and cohomology.
We first recall thedefinition of the natural pullback and pushforward actions of correspondences on the algebraiccycle class groups.
Definition 2.4.
Let f : X ⊢ Y be a correspondence from X to Y . We define the pullback f ∗ and pushforward f ∗ on the algebraic cycle class groups as follows: f ∗ : Z k ( X ) Q / ∼ −→ Z k ( Y ) Q / ∼ , α pr , ∗ ( f · pr ∗ α ) ,f ∗ : Z k ( Y ) Q / ∼ −→ Z k ( X ) Q / ∼ , β pr , ∗ ( f · pr ∗ β ) , where the pr i denote the natural projections from X × Y to X and Y , respectively.Similarly, if f ∈ Corr ∼ ( X, Y ) for an equivalence relation ∼ finer than, or equal to, homolog-ical equivalence relation ∼ hom , we can define natural pullback f ∗ and pushforward f ∗ on thecohomology groups H i ( X ) as follows: f ∗ : H i ( X ) −→ H i ( Y ) , α pr , ∗ (cl X × Y ( f ) ∪ pr ∗ α ) ,f ∗ : H i ( Y ) −→ H i ( X ) , β pr , ∗ (cl X × Y ( f ) ∪ pr ∗ β ) , where cl X × Y : Z k ( X × Y ) Q −→ H k ( X × Y ) is the cycle class map, which factors through Z k ( X × Y ) Q / ∼ by assumption. Remark . The correspondences with respect to rational equivalence operate on the rationalChow groups CH • ( − ) Q , on cohomology groups H • ( − ) , and on the numerical class groups N • ( − ) Q or N • ( − ) R . The correspondences with respect to homological equivalence operateon cohomology groups and on the numerical class groups. Note, however, that the correspon-dences with respect to numerical equivalence operate only on cohomology groups provided thestandard conjecture D holds.The motivic composition of correspondences is functorial by the assertion (1) of the follow-ing proposition. Proposition 2.6 (cf. [Ful98, Proposition 16.1.2]) . Let f : X ⊢ Y and g : Y ⊢ Z be correspon-dences. Then the following statements hold. (1) ( g ◦ f ) ∗ = g ∗ ◦ f ∗ and ( g ◦ f ) ∗ = f ∗ ◦ g ∗ . (2) ( f T ) ∗ = f ∗ and ( f T ) ∗ = f ∗ . (3) If f is the graph Γ π of a proper morphism π : X → Y , then f ∗ = π ∗ , the properpushforward; if π is flat, then f ∗ = π ∗ , the flat pullback. TANDARD CONJECTURES AND DYNAMICAL DEGREES 19
Remark . As we shall be always working on pullback actions of self-correspondences oncycle class groups and cohomology groups, we remark that the following homomorphisms
Corr ∼ ( X, X ) −→ n M k =0 End R ( N k ( X ) R ) , f n X k =0 f ∗ | N k ( X ) R , Corr ∼ ( X, X ) −→ n M i =0 End F ( H i ( X )) , f n X i =0 f ∗ | H i ( X ) . are anti-homomorphisms of rings.Let ∆ X ⊂ X × X be the diagonal. Then by the Künneth formula H n ( X × X ) = n M i =0 H i ( X ) ⊗ H n − i ( X ) , we have the following decomposition: cl X × X (∆ X ) = n X i =0 π i , where π i ∈ H i ( X ) ⊗ H n − i ( X ) for each i . Note that the pullback π ∗ i of the Künneth component π i is exactly the i -th projection operator π i : H • ( X ) → H i ( X ) , hence the name. The following trace formula is particularly useful in studying correspondences. For instance,Jannsen [Jan92] made clever use of it to prove the semisimplicity of the ring of correspon-dences modulo numerical equivalence (see also [MNP13, §3.2]). The formula also shows thatthe traces of correspondences acting on cohomology groups are actually numerical invariantsprovided the standard conjecture of Künneth type.
Proposition 2.8 (Lefschetz trace formula) . Let g : X ⊢ Y and h : Y ⊢ Z be correspondences.Let f : X ⊢ X be a self-correspondence of X . Then we have g · h T = n X i =0 ( − i Tr(( h ◦ g ) ∗ | H i ( Z ) ) ,f · ∆ X = n X i =0 ( − i Tr( f ∗ | H i ( X ) ) , cl X × X ( f ) ∪ π n − i = ( − i Tr( f ∗ | H i ( X ) ) . Suppose moreover that Conjecture C ( X ) holds. Then the last formula reads as f · ∆ n − i = ( − i Tr( f ∗ | H i ( X ) ) . (2.2)The above Lefschetz trace formula is also valid for general homological correspondences.In this paper, we only focus on algebraic correspondences in Corr ∼ ( X, X ) and refer the readerto [Kle68, Proposition 1.3.6(ii)] for the general case. However, in [Kle68, Kle94, MNP13], they are all using the pushforward of the Künneth components to givethe projections, which is merely a conventional discrepancy.
We also need a useful lemma due to Lieberman.
Lemma 2.9 (Lieberman’s Lemma, cf. [Kle72, p. 73], [MNP13, Lemma 2.1.3]) . Let φ : X ⊢ Y , ψ : X ′ ⊢ Y ′ , f : X ⊢ X ′ , and g : Y ⊢ Y ′ be correspondences shown as follows: X ✤ f ❴ φ X ′ ❴ ψ Y ✤ g Y ′ . Then we have ( φ × ψ ) ∗ ( f ) = ψ ◦ f ◦ φ T , ( φ × ψ ) ∗ ( g ) = ψ T ◦ g ◦ φ, where we think of φ × ψ ∈ Z n ( X × Y × X ′ × Y ′ ) Q / ∼ as a correspondence from X × X ′ to Y × Y ′ by interchanging the second and the third factors. In particular, we have the followingpullback/pushforward identities on cycle class groups and cohomology groups: (( φ × ψ ) ∗ ( f )) ∗ = ψ ∗ ◦ f ∗ ◦ φ ∗ , (( φ × ψ ) ∗ ( g )) ∗ = ψ ∗ ◦ g ∗ ◦ φ ∗ , (( φ × ψ ) ∗ ( f )) ∗ = φ ∗ ◦ f ∗ ◦ ψ ∗ , (( φ × ψ ) ∗ ( g )) ∗ = φ ∗ ◦ g ∗ ◦ ψ ∗ . The lemma below is now straightforward.
Lemma 2.10.
Suppose that we have the following diagram of correspondences: X ✤ f ❴ π X ✤ f ❴ π X ❴ π Y ✤ g Y ✤ g Y . (1) If π is a generically finite surjective morphism of degree d , then we have d ( π × π ) ∗ ( f ) ◦ g = ( π × π ) ∗ ( f ◦ ( π × π ) ∗ ( g )) . (2) If π is a generically finite surjective morphism of degree d , then we have d ( π × π ) ∗ ( g ◦ g ) = ( π × π ) ∗ ( g ) ◦ ( π × π ) ∗ ( g ) . (3) If π is a generically finite surjective morphism of degree d , then we have d g ◦ ( π × π ) ∗ ( f ) = ( π × π ) ∗ (( π × π ) ∗ ( g ) ◦ f ) . Proof.
Note that for a generically finite surjective morphism π : X → Y of degree d , we havethat Γ π ◦ Γ T π = d ∆ Y (see [MNP13, Example 2.3(vi)]); in particular, π ∗ π ∗ is the multiplication-by- d map on CH • ( Y ) Q or H • ( Y ) . We only verify the first equality since the others are similar.Using Lieberman’s Lemma 2.9, the left-hand side is d Γ π ◦ f ◦ Γ T π ◦ g , while the right-handside is Γ π ◦ f ◦ (Γ T π ◦ g ◦ Γ π ) ◦ Γ T π , hence the equality. (cid:3) TANDARD CONJECTURES AND DYNAMICAL DEGREES 21
Degrees of algebraic cycles.
Let X be a smooth projective variety of dimension n over K . Let N k ( X ) denote the group of algebraic cycles of codimension k on X modulo numer-ical equivalence ≡ , which turns out to be a finitely generated Abelian group (see [Kle68,Theorem 3.5]). For a field F of characteristic zero (e.g., Q , Q ℓ , R , or C ), let N k ( X ) F := N k ( X ) ⊗ Z F .Fix an ample divisor H X on X . For an algebraic cycle Z of codimension k on X , we call Z · H n − kX the degree of Z , denoted by deg H X ( Z ) (or simply, deg( Z ) as the ample divisor isclear from the context). By definition, the degree deg( Z ) of Z is a numerical invariant of Z ,i.e., depending only the numerical class [ Z ] ∈ N k ( X ) R of Z .A numerical class α ∈ N k ( X ) R is effective , it α = [ Z ] for some effective cycle Z on X . Note that by the Nakai–Moishezon criterion, the degree of an effective cycle (class) isalways nonnegative. The effective cone Eff k ( X ) of X denotes the convex cone in N k ( X ) R generated by all numerical classes of effective cycles. The closure of Eff k ( X ) in N k ( X ) R iscalled the pseudoeffective cone Eff k ( X ) of X . We refer to Fulger–Lehmann [FL17, §2] formore positivity notions on higher-codimensional cycles.The classical Bézout’s theorem asserts that for irreducible subvarieties Z , Z ⊂ P n K of codi-mension c and c , respectively, if they intersect properly (i.e., all irreducible components of theintersection Z ∩ Z have codimension c + c ), then deg( Z ∩ Z ) = deg( Z ) deg( Z ) . Usingthe cone construction, initially due to Severi, one can prove a quantitative version of Chow’smoving lemma (see, e.g., [Rob72] or [EH16, Appendix A]). Lemma 2.11 (cf. [Tru20, Lemma 2.4]) . Let X be a smooth projective variety of dimension n over K . Then for any two effective cycles Z , W on X , there is a cycle e Z on X rationallyequivalent to Z and intersecting W properly. Moreover, there is a constant C > dependingonly on n and deg( X ) , such that e Z can be expressed as e Z + − e Z − , where e Z ± are effective with deg( e Z ± ) ≤ C deg( Z ) ; in particular, we have | deg( Z · W ) | ≤ C deg( Z ) deg( W ) .Proof. Note that Chow’s moving lemma is valid for all (not necessarily effective) cycles. Usingthe cone construction, we can still inductively find a cycle e Z = m X i =1 ( − i − ( g C L i ( Z i − )) · X + ( − m Z m on X rationally equivalent to Z and meeting W properly. Here we use the fixed ample divisor H X to give a fixed embedding X ֒ → P N K and C L i ( Z i − ) denotes the cone over Z i − with vertex L i , a linear subspace of P N K . The rest goes verbatim as the proof of [Tru20, Lemma 2.4]. (cid:3) Norms on numerical classes. As N k ( X ) R is a finite-dimensional real vector space, thereare many ways to endow it norms, equivalent though. For instance, given any α ∈ N k ( X ) R ,we define k α k := inf n deg( α + + α − ) : α ≡ α + − α − , α ± ∈ Eff k ( X ) o . (2.3)This is indeed a norm on N k ( X ) R (see, e.g., [Tru20, §2.4]). Remark . We note that if α ∈ Eff k ( X ) , then k α k = deg( α ) . First, k α k ≤ deg( α ) is clear.If we write α ≡ α + − α − for some α ± ∈ Eff k ( X ) , then deg( α + + α − ) ≥ deg( α ) . Takinginfimum, we obtain the converse.On the other hand, because of the duality between N k ( X ) R and N n − k ( X ) R , we can speak ofthe operator norm on N k ( X ) R as linear operators on N n − k ( X ) R : for any α ∈ N k ( X ) R , k α k ′ := sup n | α · β | : β ∈ N n − k ( X ) R , k β k = 1 o . (2.4)For any f ∈ N k ( X ) R ⊗ N n − k ( X ) R , there are norms of f induced from the norms on thelarger space N n ( X × X ) R . However, one also has the following projective cross norm. Definition 2.13.
The projective cross norm on N k ( X ) R ⊗ N n − k ( X ) R is defined as follows: π ( f ) := inf m X i =1 k α i kk β i k : f ≡ m X i =1 α i ⊗ β i , α i ∈ N k ( X ) R , β i ∈ N n − k ( X ) R . (2.5) Proposition 2.14 (cf. [Rya02, Proposition 2.1]) . The above π is a norm on N k ( X ) R ⊗ N n − k ( X ) R .Moreover, for every α ∈ N k ( X ) R , β ∈ N n − k ( X ) R , we have π ( α ⊗ β ) = k α kk β k . Numerical invariants of correspondences.
Let f ∈ Z n ( X × X ) Q be a correspondencesof X . Let H X × X := pr ∗ H X + pr ∗ H X be a fixed ample divisor on X × X . The degree deg( f ) of f (as an algebraic n -cycle on X × X ) is deg( f ) := f · H nX × X = n X i =0 (cid:18) ni (cid:19) f · pr ∗ H n − iX · pr ∗ H iX . (2.6)We then call deg i ( f ) := f · pr ∗ H n − iX · pr ∗ H iX = f ∗ H iX · H n − iX (2.7)the i -th degree of f (with respect to H X ). It is also clear that deg i ( f ) ≥ for any effectivecorrespondence f of X . Note that the two commonly used degrees of f in the literature are deg ( f ) and deg n ( f ) which are particularly helpful when studying correspondences of curves.We can also define norms on N n ( X × X ) R as in Section 2.5. For this, it is more natural towork with real correspondences f ∈ Z n ( X × X ) R : k f k := inf (cid:8) deg( f + + f − ) : f ≡ f + − f − , f ± ∈ Eff n ( X × X ) (cid:9) , (2.8)and similarly for the dual norm k f k ′ as follows: k f k ′ := sup (cid:8) | f · g | : g ∈ N n ( X × X ) R , k g k = 1 (cid:9) . (2.9)Note that for effective correspondences f , we always have k f k = deg( f ) ; see Remark 2.12.On the other hand, for a non-effective correspondence f , its k -th degree deg k ( f ) may benegative. In practice, inspired by the definition of the norm k f k , it is useful to consider thefollowing refined k -th degree: deg k ( f ) := inf (cid:8) deg k ( f + + f − ) : f ≡ f + − f − , f ± ∈ Eff n ( X × X ) (cid:9) . (2.10) TANDARD CONJECTURES AND DYNAMICAL DEGREES 23
By this definition, deg k ( f ) is also a numerical invariant of f . Lemma 2.15.
Let X be a smooth projective variety of dimension n over K . There exists aconstant C > (depending only on X itself) such that for any correspondences f, g of X , thefollowing inequalities hold: (cid:12)(cid:12) deg( g ◦ f ) (cid:12)(cid:12) ≤ C k g kk f k , k g ◦ f k ≤ C k g kk f k . Proof.
We first consider the case when both f and g are effective correspondences; in this case,by Remark 2.12, the norm function k·k and the degree function deg( · ) coincide. Recall bydefinition that g ◦ f = pr , ∗ (pr ∗ f · pr ∗ g ) . We denote the effective n -cycles pr ∗ f and pr ∗ g by V and W , respectively. Then by Lemma 2.11, we can find effective n -cycles V ± on X × X × X such that V ∼ rat V + − V − , both V ± intersect W properly, and deg( V ± ) ≤ C deg( V ) , where C > is a constant depending only on n and deg( X ) . Note that both V ± · W are now effective. It follows that (cid:12)(cid:12) deg( g ◦ f ) (cid:12)(cid:12) = (cid:12)(cid:12) deg(pr , ∗ ( V · W )) (cid:12)(cid:12) = (cid:12)(cid:12) deg(pr , ∗ ( V + · W ) − pr , ∗ ( V − · W )) (cid:12)(cid:12) ≤ deg(pr , ∗ ( V + · W )) + deg(pr , ∗ ( V − · W )) ≤ deg( V + · W ) + deg( V − · W ) ≤ deg( V + ) deg( W ) + deg( V − ) deg( W ) ≤ C deg( V ) deg( W ) . On the other hand, by the definition of the degree function on X × X × X , we have that deg(pr ∗ f ) = pr ∗ f · (pr ∗ H X × X + pr ∗ H X ) n = pr ∗ f · n X i =0 (cid:18) ni (cid:19) pr ∗ H n − iX × X · pr ∗ H iX . Note that the only nonzero term is when i = n . Hence we obtain that deg(pr ∗ f ) = (cid:0) nn (cid:1) deg( X ) deg( f ) ,and similar for pr ∗ g . Putting all together, we prove that (cid:12)(cid:12) deg( g ◦ f ) (cid:12)(cid:12) ≤ C (cid:18) nn (cid:19) deg( X ) k g kk f k . Now, let us consider the general case. Let f ≡ f + − f − and g ≡ g + − g − be arbitrarydecompositions of f and g , where f ± and g ± are effective. It follows readily that (cid:12)(cid:12) deg( g ◦ f ) (cid:12)(cid:12) ≤ X (cid:12)(cid:12) deg( g ± ◦ f ± ) (cid:12)(cid:12) ≤ C (cid:18) nn (cid:19) deg( X ) X k g ± kk f ± k = 2 C (cid:18) nn (cid:19) deg( X ) ( k g + k + k g − k )( k f + k + k f − k )= 2 C (cid:18) nn (cid:19) deg( X ) deg( g + + g − ) deg( f + + f − ) . Taking infimum over all decompositions, we prove the first inequality. The proof of the secondinequality is almost the same as the first one, so omitted here. (cid:3)
We now introduce norms on the pullback f ∗ of correspondences f on N k ( X ) R . First, wehave the following natural operator norms on End R ( N k ( X ) R ) induced from the norm k·k andthe dual norm k·k ′ on N k ( X ) R : for any φ ∈ End R ( N k ( X ) R ) , k φ k op := sup n(cid:13)(cid:13) φ ( α ) (cid:13)(cid:13) : α ∈ N k ( X ) R , k α k = 1 o , (2.11)and k φ k ′ op := sup n(cid:13)(cid:13) φ ( α ) (cid:13)(cid:13) ′ : α ∈ N k ( X ) R , k α k ′ = 1 o = sup n(cid:12)(cid:12) φ ( α ) · β (cid:12)(cid:12) : α ∈ N k ( X ) R , β ∈ N n − k ( X ) R , k α k ′ = k β k = 1 o . (2.12)Secondly, for a fixed k with ≤ k ≤ n , we consider the following group homomorphism Φ k : Z n ( X × X ) R −→ End R ( N k ( X ) R ) f f ∗ | N k ( X ) R , which naturally factors through N n ( X × X ) R by definition. Also, Φ k is surjective since N n ( X × X ) R ⊃ N k ( X ) R ⊗ N n − k ( X ) R ≃ End R ( N k ( X ) R ) . We note that for any f ∈ Ker(Φ k ) , its k -th degree deg k ( f ) is zero as f ∗ H kX ≡ . Now,in a similar vein, we define an invariant k·k on End R ( N k ( X ) R ) as follows: for any (real)correspondence f of X , (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) := inf ( deg k ( f + + f − ) : Φ k ( f − f + + f − ) = 0 ,f ± ∈ Eff n ( X × X ) ) . (2.13) Lemma 2.16.
The above assignment (2.13) gives a norm on the R -vector space End R ( N k ( X ) R ) .Proof. First, it is easy to check that this assignment is nonnegative, homogeneous and satisfiesthe triangle inequality.We only need to check that for any real correspondence f of X this assignment attains zeroonly if f ∗ | N k ( X ) R is the zero map, i.e., f ∈ Ker(Φ k ) . To this end, by the definition of thepullback action of correspondences on N k ( X ) R , it suffices to show that if k f ∗ | N k ( X ) R k = 0 , TANDARD CONJECTURES AND DYNAMICAL DEGREES 25 then f · ( Z × Z ) = f · pr ∗ Z · pr ∗ Z = Z · f ∗ Z = 0 for any effective cycles Z and Z on X of codimension n − k and k , respectively. Denote W = Z × Z . For any decomposition f = f + − f − + g , where f ± ∈ Z n ( X × X ) R are effective and g ∈ Ker(Φ k ) , we have | f · W | ≤ (cid:12)(cid:12) f + · W (cid:12)(cid:12) + (cid:12)(cid:12) f − · W (cid:12)(cid:12) + | g · W | = (cid:12)(cid:12) f + · W (cid:12)(cid:12) + (cid:12)(cid:12) f − · W (cid:12)(cid:12) . Now, applying Lemma 2.19 (proved independently later) to f ± and W = Z × Z , we get that (cid:12)(cid:12) f + · W (cid:12)(cid:12) + (cid:12)(cid:12) f − · W (cid:12)(cid:12) ≤ C deg n − k ( W ) deg k ( f + + f − ) . Taking infimum over all decompositions of f , we see that | f · W | = 0 and hence the lemmafollows. (cid:3) Remark . (1) Note that for any effective correspondence f of X , we have (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) = deg k ( f ) = deg k ( f ) . First, it is easy to see that k f ∗ | N k ( X ) R k ≤ deg k ( f ) = deg k ( f ) . For any effective correspon-dences f ± such that Φ k ( f − f + + f − ) = 0 , or equivalently, f − f + + f − = g ∈ Ker(Φ k ) ,we have deg k ( f + + f − ) ≥ deg k ( f + ) = deg k ( f + + g ) = deg k ( f + f − ) ≥ deg k ( f ) . Taking infimum, we get that k f ∗ | N k ( X ) R k ≥ deg k ( f ) , hence equalities.(2) In particular, it is easy to see that Effective Property G r implies Conjecture G r .For the convenience of the reader, we summarize relations of these norms just introduced,which basically follow from the fact that all norms on finite-dimensional real vector spaces areequivalent. Proposition 2.18. (1)
The norms k·k and k·k ′ on N n ( X × X ) R defined by eqs. (2.8) and (2.9) are equivalent. (2) The norms k·k op , k·k ′ op , and k·k on End R ( N k ( X ) R ) defined by eqs. (2.11) to (2.13) areequivalent. (3) All above are equivalent to the projective cross norm π ( · ) on N k ( X ) R ⊗ N n − k ( X ) R defined by eq. (2.5) via the canonical isomorphism N k ( X ) R ⊗ N n − k ( X ) R ∼ −−→ End R ( N k ( X ) R ) , f f ∗ | N k ( X ) R and the canonical embedding N k ( X ) R ⊗ N n − k ( X ) R ֒ → N n ( X × X ) R , α ⊗ β pr ∗ α · pr ∗ β. Boundedness of self-intersections of correspondences.
In this subsection, we prove alemma bounding the intersection of two effective correspondences f, g of X in terms of all i -thdegrees of f and g (with respect to a fixed ample divisor H X ). The strict intersection analoghas been proved in [Tru20, Lemma 4.1]. Here we adopt a similar idea to the case of the usualintersection, which is easier to deal with. Lemma 2.19.
Let X be a smooth projective variety of dimension n over K . Then there existsa constant C > such that for any two effective correspondences f , g of X , we have | f · g | ≤ C n X i =0 deg i ( f ) deg n − i ( g ) . Proof.
Let f , g be effective correspondences of X . By reduction to the diagonal, the intersec-tion number f · g taken over X × X is equal to ( f × g ) · ∆ X × X , where ∆ X × X := { ( x , x , x , x ) : x , x ∈ X } ⊂ X × X × X × X is the diagonal of X × X . We define an isomorphism τ byinterchanging the second and the third factors of the fourfold product Y := X × X × X × X .Now clearly, τ (∆ X × X ) = ∆ X × ∆ X , where ∆ X := { ( x, x ) : x ∈ X } ⊂ X × X is the diagonalof X . Moreover, using the projection formula, we have ( f × g ) · ∆ X × X = ( f × g ) · τ ∗ (∆ X × ∆ X ) = τ ∗ ( f × g ) · (∆ X × ∆ X ) . For ≤ i = j ≤ , let p i and p ij denote the natural projections from X × X × X × X to theappropriate X and X × X , respectively. Then clearly, we have p − k ◦ τ = p k for k = 2 , , p k ◦ τ = p k for k = 1 , , p ◦ τ = p , and p ◦ τ = p .We still denote the natural projections from X × X to X by pr and pr , and let H X × X =pr ∗ H X + pr ∗ H X denote the ample divisor on X × X . In particular, we have p = pr ◦ p , p = pr ◦ p , and so on; on Y we have p ∗ ij H X × X = p ∗ i H X + p ∗ j H X . We claim that there existsa constant C > so that the numerical class of p ∗ ( C H nX × X ± ∆ X ) · p ∗ ( C H nX × X ± ∆ X ) is in the nef cone Nef n ( Y ) , which is defined as the closed convex cone in N n ( Y ) R dual to thepseudoeffective cone Eff n ( Y ) (see [FL17, §2.3]).Proof of the claim: Note that H X × X is ample on X × X . It thus follow from [FL17,Lemma 3.14] that H nX × X is in the interior of the so-called pliant cone PL n ( X × X ) ⊂ N n ( X × X ) R , which is a closed convex cone contained in the nef cone Nef n ( X × X ) but behaves better.We refer to [FL17, §3] for details. Thus, there is a constant C > such that C H nX × X ± ∆ X isstill in the interior of PL n ( X × X ) . Since pliant cones are preserved by pullbacks of morphismsand by intersection products (see [FL17, Remarks 3.4 and 3.6]), we get that the numerical classof p ∗ ( C H nX × X ± ∆ X ) · p ∗ ( C H nX × X ± ∆ X ) is in the pliant cone PL n ( Y ) of Y ; in particular,it is in the nef cone Nef n ( Y ) of Y . We hence prove the claim.Using the above claim, it is not hard to deduce that C p ∗ H nX × X · p ∗ H nX × X ± p ∗ ∆ X · p ∗ ∆ X ∈ Nef n ( Y ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 27
Now, as τ ∗ ( f × g ) is effective and nef cones are dual to pseudoeffective cones, we obtain that (cid:12)(cid:12) τ ∗ ( f × g ) · (∆ X × ∆ X ) (cid:12)(cid:12) ≤ C τ ∗ ( f × g ) · p ∗ H nX × X · p ∗ H nX × X = C ( f × g ) · τ ∗ p ∗ H nX × X · τ ∗ p ∗ H nX × X = C ( f × g ) · p ∗ H nX × X · p ∗ H nX × X = C p ∗ f · p ∗ g · ( p ∗ H X + p ∗ H X ) n · ( p ∗ H X + p ∗ H X ) n = C X ≤ i, j ≤ n (cid:18) ni (cid:19)(cid:18) nj (cid:19)(cid:0) p ∗ f · p ∗ H n − iX · p ∗ H jX (cid:1) · (cid:0) p ∗ g · p ∗ H iX · p ∗ H n − jX (cid:1) = C X ≤ i, j ≤ n (cid:18) ni (cid:19)(cid:18) nj (cid:19)(cid:0) f · pr ∗ H n − iX · pr ∗ H jX (cid:1) · (cid:0) g · pr ∗ H iX · pr ∗ H n − jX (cid:1) = C n X i =0 (cid:18) ni (cid:19) (cid:0) f · pr ∗ H n − iX · pr ∗ H iX (cid:1) · (cid:0) g · pr ∗ H iX · pr ∗ H n − iX (cid:1) ≤ C n X i =0 deg i ( f ) deg n − i ( g ) , where C = C (cid:0) n ⌊ n/ ⌋ (cid:1) , the first and the second last equalities follow from the projection for-mula, and the last equality follows from the fact that in order to make the intersection mean-ingful one has to require n − i + j ≤ n and n − j + i ≤ n , i.e., i = j . We hence proveLemma 2.19. (cid:3) Remark . (1) In the case K = C , a different proof, for the strict intersection of positiveclosed currents - using regularization of positive closed currents - was given in [DN11] forsmooth complex projective varieties and [DNT12] for compact Kähler manifolds.(2) When n = 1 and f = g is an effective ( n, n ′ ) -correspondence of the curve X , the aboveinequality reads as | f · f | ≤ Cnn ′ . On the other hand, by the Castelnuovo–Severi inequal-ity (see, e.g., [Ful98, Example 16.1.10]), we know that for any ( n, n ′ ) -correspondence f of a curve, f · f ≤ nn ′ . Whether the self-intersection f · f has a uniform lower boundis called the Bounded Negativity Conjecture (see, e.g., [BHK + YNAMICAL CORRESPONDENCES AND EFFECTIVE FINITE CORRESPONDENCES
In this section, we first introduce the category
DynCorr ( K ) of dynamical correspondencesover K , which contains the category of dominant rational maps as a subcategory. In particu-lar, we collect some basic materials on dynamical correspondences, including their definition,dynamical compositions, dynamical pullbacks under dominant rational maps, and a fundamen-tal result concerning the numerical dynamical degrees λ k of dynamical correspondences. Wethen discuss dynamical aspects of effective finite correspondences, which already have theirown significance in constructions of various triangulated categories DM eff ( K ) of (effective)motives over K by Suslin and Voevodsky. Dynamical correspondences.
Dynamical correspondences are natural generalizations ofdominant rational maps that have been extensively studying for decades. In this subsection, wediscuss their basic properties. We refer to [Tru20] and references therein for details.
Definition 3.1 (Dynamical correspondence) . Let X and Y be smooth projective varieties ofdimension n over K . A correspondence f ∈ Z n ( X × Y ) Q is dominant , if for each irreduciblecomponent f i of f , the natural restriction maps pr | f i : f i → X and pr | f i : f i → Y inducedfrom the projections pr : X × Y → X and pr : X × Y → Y , respectively, are both surjective(and hence generically finite). We say that f is a dynamical correspondence , if f is botheffective and dominant.Clearly, the graph Γ π of a dominant rational map π : X Y is a dynamical correspondence.We can compose dynamical correspondences just like how we compose dominant rational maps(see [DS17, §2] or [Tru20, §3.1]). Definition 3.2 (Dynamical composition) . Let f ∈ Z n ( X × Y ) Q and g ∈ Z n ( Y × Z ) Q be twodynamical correspondences. Assume that they are both irreducible for simplicity since the gen-eral case can be defined by linearity. By generic flatness (see, e.g., [Sta21, Proposition 052A]),there are nonempty Zariski open subsets U ⊂ X and V ⊂ Y such that the restriction maps pr | f : f → X and pr | g : g → Y are finite and flat over U and V , respectively. By shrinking U , we may assume that the strict image f ( U ) := pr ( f ∩ pr − ( U )) of U under f is contained in V . It follows that for any point x ∈ U , the strict image f ( x ) of x under f is a finite subset of V , whose strict image under g is still finite. The dynamicalcomposition g ♦ f is then defined as the closure { ( x, g ( f ( x ))) : x ∈ U } of the composite graph in X × Z . Alternatively, let f (resp. g ) be a nonempty Zariski opensubset of f (resp. g ) which is finite over some open U ⊂ X (resp. open V ⊂ Y ). Then g ♦ f is the pushforward of the scheme-theoretic closure of the scheme-theoretic intersection ( f × Z ) ∩ ( U × g ) ⊂ U × V × Z in X × Y × Z under pr : X × Y × Z → X × Z . Sincewe are concerning dynamical correspondences here, it can be checked that the result does notdepend on the choice of f and g .Intuitively, g ♦ f is the same as pr , ∗ (( f × Z ) ∩ ( X × g )) with components of dimensiongreater than n and components whose projections to the factors of X × Z are not surjectiveremoved. Here we use ∩ for the scheme-theoretic intersection.For a dynamical self-correspondence f of X and any m ∈ N , we denote by f ♦ m the m -th dynamical iterate of f , which is still a dynamical self-correspondence of X . Unlike themotivic composition, this dynamical composition is, in general, not functorial in the sense that ( g ♦ f ) ∗ = f ∗ ◦ g ∗ . For example, let f ([ x : y : z ]) = [ yz : xz : xy ] be the standard Cremona TANDARD CONJECTURES AND DYNAMICAL DEGREES 29 map of X = P K . Then f ∗ acts on H ( X ) as multiplication by , while f ♦ = id X . Hence id = ( f ♦ ) ∗ = ( f ∗ ) = × on H ( X ) . Remark . Motivic compositions of correspondences are functorial but do not necessarilypreserve the effectiveness. Dynamical compositions of dynamical correspondences preservethe effectiveness but may not be functorial, which leads to a notion called algebraic stabil-ity (see below). The non-functorial nature of dynamical compositions or iterates makes thecomputation of dynamical degrees very hard. For instance, the following natural question hasbeen well-known for decades: are dynamical degrees always algebraic numbers? Recently, anegative answer to this question is given by Bell–Diller–Jonsson [BDJ20].
Definition 3.4 (Algebraic stability) . Let f ∈ Z n ( X × X ) Q be a dynamical correspondence of X . Fix an integer ≤ k ≤ n . We say that f is algebraically k -stable , if for all m ∈ N , ( f ♦ m ) ∗ | H k ( X ) = ( f ∗ ) m | H k ( X ) = ( f ◦ m ) ∗ | H k ( X ) . It is called totally algebraically stable , if for all m ∈ N , ( f ♦ m ) ∗ | H • ( X ) = ( f ∗ ) m | H • ( X ) = ( f ◦ m ) ∗ | H • ( X ) . Clearly, if f is (the graph of) a self-morphism of X , then it is algebraically k -stable for all k .The study of algebraic stability has attracted a lot attention [DF01, JW11, Tru13, LW14, DL16,DF16]. Here we collect some examples. Example 3.5. (1) If f = P i a i Γ i , where the Γ i are (graphs of) self-morphisms of X and a i ∈ Q > , then it is also algebraically k -stable for all k . Indeed, one can check that the m -thdynamical iterates f ♦ m are the same as the m -th motivic iterates f ◦ m .(2) Another interesting example is due to Voisin [Voi04] and studied by Amerik [Ame09]later. For a smooth cubic V ⊂ P , let X = F ( V ) be the variety of lines on V . Thus X is asmooth four-dimensional subvariety of the Grassmannian G (1 , . There is a rational self-map f of X defined by sending a general line l ⊂ V to the line l ′ , where l ′ ∪ l = P l ∩ V and P l isthe unique plane in P tangent to V along l . Amerik showed that the above f is algebraically k -stable for all k .For the convenience of the reader, we recall a fundamental result on the existence of the nu-merical dynamical degrees λ k of dynamical correspondences due to the second author [Tru20]. Theorem 3.6 (cf. [Tru20, Theorem 1.1(1)]) . Let X be a smooth projective variety of dimension n over K . Let f be a dynamical correspondence of X . Then for any ≤ k ≤ n , the limit lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m defining the k -th numerical dynamical degree λ k ( f ) of f exists and is equal to lim m →∞ (deg k ( f ♦ m )) /m = lim m →∞ (( f ♦ m ) ∗ H kX · H n − kX ) /m , where H X is an (arbitrary) ample divisor on X . Proof.
The existence of these limits follows immediately from [Tru20, Theorem 1.1(1)] andRemark 2.17. Note that the last limit is indeed independent of the choice of H X since theample cone Amp ( X ) ⊂ N ( X ) R of ample divisor classes on X is open. (cid:3) We also recall dynamical pullbacks of dynamical correspondences by dominant rationalmaps, which will be used in Section 5.
Definition 3.7 (Dynamical pullback) . Let π i : X i Y i be dominant rational maps of smoothprojective varieties of dimension n over K for i = 1 , . Let g be a dynamical correspon-dence from Y to Y . Assume that g is also irreducible. By generic flatness (see, e.g., [Sta21,Proposition 052A]), there are nonempty Zariski open subsets U i ⊂ X i and V i ⊂ Y i such that U i = π − i ( V i ) and π i | U i : U i → V i are finite and flat morphisms. Then the dynamical pullback ( π × π ) ★ ( g ) of g under ( π , π ) (or π × π ) is defined by the closure ( π × π | U × U ) − ( g ∩ ( V × V )) of the scheme-theoretic inverse image ( π × π | U × U ) − ( g ∩ ( V × V )) in X × X . It is indeeda dynamical correspondence from X to X by the following commutative diagram: ( π × π ) ★ ( g ) U X g X U V Y Y V pr π | U π pr pr π π | U The reducible case can be defined linearly.
Remark . Consider the dominant rational map π × π : X × X Y × Y . Let ( π × π ) − ( g ) denote the total transform of g under π × π , i.e., ( π × π ) − ( g ) := pr (Γ π × π ∩ pr − ( g )) , where π ij denotes the projection from X × X × Y × Y to the product of the i -th and j -th factors.Clearly, the difference between the total transform ( π × π ) − ( g ) and the dynamical pullback ( π × π ) ★ ( g ) lies in the indeterminacy locus I π × π and the non-flat locus N F π × π ⊂ X × X of π × π .If we assume further that both π i : X i → Y i are morphisms, then one can check that theimage of ( π × π ) ★ ( g ) under π × π is exactly g . Indeed, by definition, one easily has ( π × π )(( π × π ) ★ ( g )) = g ∩ ( V × V ) = g. It follows from the definition of proper pushforward that ( π × π ) ∗ (( π × π ) ★ ( g )) = deg( π × π ) g. TANDARD CONJECTURES AND DYNAMICAL DEGREES 31
Example 3.9.
Let π : P C → P C be the morphism defined by π ([ z : z ]) = [ z d , z d ] (hence ofdegree d ). Let g be the graph of a polynomial map p : P C → P C . Then the dynamical pullback f := ( π × π ) ★ ( g ) is given by { ([ z , z ] , [ w , w ]) ∈ P C × P C : [ w d : w d ] = p ([ z d : z d ]) } , which is a -to- d correspondence (i.e., deg ( f ) = d ). Note that ( π × π ) ★ ( g ♦ m ) = { ([ z , z ] , [ w , w ]) ∈ P C × P C : [ w d : w d ] = p m ([ z d : z d ]) } is a -to- d correspondence and f ♦ m is a -to- d m correspondence. One can check that f ♦ m = d m − ( π × π ) ★ ( g ♦ m ) . Hence, since the degree of π × π is d , we have ( π × π ) ∗ ( f ♦ m ) = d m +1 g ♦ m .In general, similar to Lemma 2.10, we have the following lemma for dynamical pullbacks ofdynamical correspondences. Lemma 3.10.
Let π : X → Y be a generially finite surjective morphism of smooth projectivevarieties over K . Let g , g be dynamical correspondences of Y and f = ( π × π ) ★ ( g ) , f =( π × π ) ★ ( g ) the dynamical pullbacks of g , g , respectively. Then we have deg( π )( π × π ) ★ ( g ♦ g ) = f ♦ f , ( π × π ) ∗ ( f ♦ f ) = deg( π ) g ♦ g . Proof.
The second one follows immediately from the first one and Remark 3.8. For the firstone, note that by definition, there is an appropriate Zariski open subset U ⊂ X such that forany x ∈ U , we have ( π × π ) ★ ( g ♦ g )( x ) = π − ◦ g ◦ g ◦ π ( x ) , and ( f ♦ f )( x ) = π − ◦ g ◦ π ◦ π − ◦ g ◦ π ( x ) = π − ◦ g ◦ g ◦ π ( x ) , where the left-hand sides are strict images of x under correspondences. Clearly, as correspon-dences (or rather, effective algebraic n -cycles on X × X ), we have f ♦ f = c ( π × π ) ★ ( g ♦ g ) for some constant c > . Comparing deg of both sides, one can see that c = deg( π ) . (cid:3) The next lemma has been implicitly proved by the second author in the more general set-ting of semi-conjugate correspondences; see [Tru20, Lemma 5.4]. The proof there utilizedan effective version of Chow’s moving lemma to estimate certain strict intersection; as an im-portant consequence, it is deduced that (relative) dynamical degrees are birational invariants(see [Tru20, Lemma 5.5]). Here, we single it out for our purpose in Section 5.
Lemma 3.11.
Let π : X → Y be a generically finite surjective morphism of smooth projectivevarieties of dimension n over K . Let g be a dynamical correspondence of Y and f := ( π × π ) ★ ( g ) the dynamical pullback of g under π × π . Then for any ≤ k ≤ n , we have deg k ( f ) ∼ deg( π ) deg k ( g ) , i.e., there are constants C , C > , independent of g and f , such that C deg( π ) deg k ( g ) ≤ deg k ( f ) ≤ C deg( π ) deg k ( g ) . Remark . We note that in the above lemma, if g is merely an effective correspondence of Y and f := ( π × π ) ∗ ( g ) , the motivic pullback of g by π × π (viewed as a correspondence from X × X to Y × Y ), is also effective, then one can easily get the equivalence between deg k ( f ) and deg k ( g ) . In fact, by Remark 2.17, the k -th degree of g (resp. f ) is equivalent to the norm k g ∗ | N k ( Y ) R k (resp. k f ∗ | N k ( X ) R k ) on the k -th numerical cycle class group. Also, the properpushforward π ∗ : N k ( X ) R → N k ( X ) R is surjectve because π is surjective. It follows that N k ( X ) R ≃ N k ( Y ) R ⊕ Ker( π ∗ ) . Using this decomposition and the dual operator norm eq. (2.12), one can see that the two norms k g ∗ | N k ( Y ) R k ′ op and k f ∗ | N k ( X ) R k ′ op are equivalent, so are deg k ( g ) and deg k ( f ) .3.2. Effective finite correspondences.
As discussed in Section 3.1, typical examples of dy-namical correspondences are graphs of dominant rational maps (and surjective morphisms).The assumption that dynamical correspondences are dominant is used only to assure that com-position of two dynamical correspondences given in Definition 3.2 is always well-defined. Incase f is the graph of a (not necessarily surjective) morphism, then we do not need it to be sur-jective, since we can always compose them (Remark 2.3). Similarly, many results for dynami-cal correspondences can be adapted to a positive sum of graphs of (not necessarily surjective)morphisms. The key point is that the projection of the graph of an endomorphism to the firstfactor has all fibers finite (in fact, it is an isomorphism). This latter property is shared by a verylarge class of correspondences, which we recall as follows. Definition 3.13 (Finite correspondence) . Let
X, Y be smooth projective varieties of dimension n over K . Let f ∈ Z n ( X × Y ) Q be a correspondence. We say that f is a finite correspondence ,if for each irreducible component f i of f , the restriction map pr | f i : f i → X induced from theprojection pr : X × Y → X is finite and surjective. Remark . Finite correspondences were introduced by Suslin and Voevodsky in construc-tions of various triangulated categories DM eff ( K ) of (effective) motives over K (see [MVW06]).An advantage of finite correspondences is about their motivic compositions described as fol-lows. Let f ∈ Z n ( X × Y ) Q and g ∈ Z n ( Y × Z ) Q be two finite correspondences. Without lossof generality, we may assume that both f and g are irreducible. Then [MVW06, Lemma 1.7]asserts that pr ∗ f intersects pr ∗ g properly and each component of the pr -pushforward of pr ∗ f · pr ∗ g is finite and surjective over X . Hence g ◦ f ∈ Z n ( X × Z ) Q is also a finitecorrespondence (note, however, that in Section 2.2 one has to modulo at least rational equiva-lence). In particular, if f, g are effective finite correspondences, so is their composition g ◦ f .Therefore, our Theorem 1.9(3) applies to all effective finite correspondences.Arithmetic dynamics of effective finite correspondences f whose transposes f T are alsofinite correspondences has been carried out recently (see [Ing19] and references therein); seealso [DKW20] for certain equidistribution property of these correspondences in the special caseof curves. TANDARD CONJECTURES AND DYNAMICAL DEGREES 33
Remark . As mentioned earlier, one can consider dynamical compositions of non-dynamicalcorrespondences whose second projection may not be surjective (e.g., graphs of arbitrary mor-phisms). This is also the case for effective finite correspondences. More precisely, let f ∈ Z n ( X × Y ) Q and g ∈ Z n ( Y × Z ) Q be two effective finite correspondences. Then by Def-inition 3.2 and the above Remark 3.14, the dynamical composition g ♦ f coincides with themotivic composition g ◦ f , since the restriction maps pr | f : f → X and pr | g : g → Y arefinite over X and Y , respectively. In particular, all effective finite correspondences are totallyalgebraically stable (see Definition 3.4), and hence all of their dynamical degrees are nothingbut the spectral radii of the corresponding linear maps. Remark . It is worth mentioning that by [FV00, Theorem 7.1], any correspondence f : X ⊢ Y is rationally equivalent to some finite correspondence g : X ⊢ Y . However, g need not beeffective even if f is an effective correspondence. Lemma 3.17.
Let π : X → Y be a finite surjective morphism of smooth projective varieties X, Y of dimension n over K . Let g : Y ⊢ Y be an effective finite correspondence of Y . Thenthe dynamical pullback f := ( π × π ) ★ ( g ) of g under π × π , which coincides with the usual flatpullback ( π × π ) ∗ ( g ) , is an effective finite correspondence of X .Proof. It suffices to consider the case when g is an irreducible subvariety of Y × Y . Sincethe base change of a finite surjective morphism is still finite and surjective (see, e.g., [Sta21,Lemma 01S1 and Lemma 01WL]), it follows from the following diagram f h gX × X X × Y Y × YX Y id X × π pr pr π × id Y pr π that h → X is finite and surjective. Also, since π is finite and surjective, so is f → h . Thelemma thus follows from [Sta21, Lemma 01WK]. (cid:3)
4. P
ROOFS OF MAIN RESULTS
Some auxiliary results.
We first recall a generalization of Grothendieck’s Hodge indextheorem on surfaces to higher-dimensional varieties, which is also known as the Khovanskii–Teissier inequality. We refer to [Laz04, §1.6.A] for a proof. Note that the inequality alsofollows from the Hodge–Riemann bilinear relation for divisors (see, e.g., [Hu20, §2.5]).
Theorem 4.1 (Khovanskii–Teissier inequality) . Let Z be a projective variety of dimension n over K . Let α, β ∈ N ( Z ) R be nef Cartier divisor classes on Z . Then for all ≤ i ≤ n − ,we have ( α n − i · β i ) ≥ ( α n − i +1 · β i − )( α n − i − · β i +1 ) . From this one can easily deduce, as in the next lemma, that the degree sequences { deg i ( f ) } i of irreducible correspondences f are log-concave. Here a finite sequence { a i } ni =0 of positivereal numbers is said to be log-concave , if a i ≥ a i − a i +1 for all ≤ i ≤ n − . In particular,there are integers ≤ s ≤ t ≤ n such that < a < · · · < a s = · · · = a t > · · · > a n > . For meromorphic maps of compact Kähler manifolds, this log-concavity property is well-known in Complex Dynamics. The version for irreducible correspondences (possibly on asingular variety) can be found, e.g., in [Tru20]. In this paper, we only treat the smooth case.
Lemma 4.2.
Let X be a smooth projective variety of dimension n over K . Let H X be a fixedample divisor on X . Let f be an irreducible correspondence of X . Then the degree sequence { deg i ( f ) } i is log-concave, where deg i ( f ) := f ∗ H iX · H n − iX = f · pr ∗ H n − iX · pr ∗ H iX . Precisely, for all ≤ i ≤ n − , we have deg i ( f ) ≥ deg i − ( f ) deg i +1 ( f ) . Proof.
We let α = (pr ∗ H X ) | f (resp. β = (pr ∗ H X ) | f ) be the restriction of the nef divisor pr ∗ H X (resp. pr ∗ H X ) to f . Note that f · pr ∗ H n − iX · pr ∗ H iX = α n − i · β i . Then the log-concavityof the degree sequence { deg i ( f ) } i of f follows readily from Theorem 4.1. (cid:3) Using the Künneth formula, below is a standard trick by applying argument to product cor-respondences of product varieties. We include it here for the convenience of the reader.
Lemma 4.3.
Let X be a smooth projective variety of dimension n over K . Suppose that Con-jecture 1.4 holds on X × X . Then there is a constant C > such that for any ≤ k ≤ n − and for any effective correspondence f of X , we have (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ C q deg k ( f ) deg k +1 ( f ) . Proof.
Note that by the Cauchy–Schwarz inequality, the function p deg k ( · ) deg k +1 ( · ) definedover effective correspondences of X is superadditive. So it suffices to consider the case when f is irreducible. For any ≤ k ≤ n − , let us consider the pullback action of f × f on H k +2 ( X × X ) . Here, f × f is the product correspondence of X × X naturally identified as pr ∗ f · pr ∗ f . It follows from our assumption and the Künneth formula that (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ι = (cid:13)(cid:13) ( f × f ) ∗ | H k +1 ( X ) ⊗ H k +1 ( X ) (cid:13)(cid:13) ι ≤ (cid:13)(cid:13) ( f × f ) ∗ | H k +2 ( X × X ) (cid:13)(cid:13) ι ≤ C (cid:13)(cid:13) ( f × f ) ∗ | N k +1 ( X × X ) R (cid:13)(cid:13) = C deg k +1 ( f × f ) , TANDARD CONJECTURES AND DYNAMICAL DEGREES 35 where C > is a constant from Conjecture 1.4. On the other hand, by definition of deg k +1 (see eq. (2.7)), one has deg k +1 ( f × f ) = ( f × f ) · pr ∗ H n − k − X × X · pr ∗ H k +1 X × X = (pr ∗ f · pr ∗ f ) · (pr ∗ H X + pr ∗ H X ) n − k − · (pr ∗ H X + pr ∗ H X ) k +1 = min { k +1 ,n } X s =max { , k +1 − n } c s,k pr ∗ f · pr ∗ f · pr ∗ H n − sX · pr ∗ H n − k − sX · pr ∗ H sX · pr ∗ H k +1 − sX = min { k +1 ,n } X s =max { , k +1 − n } c s,k deg s ( f ) deg k +1 − s ( f ) , where the pr ij (resp. pr i ) denote appropriate projections from the fourfold product X × X × X × X to X × X (resp. X ), the third equality holds since pr ∗ f · pr ∗ H rX · pr ∗ H sX = 0 unless r = n − s , and c s,k := (cid:0) n − k − n − s (cid:1)(cid:0) k +1 s (cid:1) ≤ (cid:0) nn (cid:1) . By Lemma 4.2, the degree sequence { deg s ( f ) } s is log-concave. Then we have that deg s ( f ) deg k +1 − s ( f ) ≤ deg k ( f ) deg k +1 ( f ) forall possible s . This shows that deg k +1 ( f × f ) ≤ n (cid:18) nn (cid:19) deg k ( f ) deg k +1 ( f ) . Putting the above together, Lemma 4.3 follows. (cid:3)
Recall that the homological correspondence γ r of X , introduced before Effective Prop-erty G r , is an element in H n ( X × X ) ≃ ⊕ i End Q ℓ ( H i ( X )) so that its pullback γ ∗ r on H i ( X ) is the multiplication-by- r i map for each i . The lemma below says that Conjecture C ( X ) isequivalent to the algebraicity of γ r for every r ∈ Q > . In particular, Conjecture G r and henceEffective Property G r imply Conjecture C ( X ) . Lemma 4.4.
The following statements are equivalent: (1)
Conjecture C ( X ) holds. (2) The γ r is algebraic for every r ∈ Q > .Proof. Suppose that Conjecture C ( X ) holds. Then there are correspondences ∆ i ∈ Z n ( X × X ) Q such that cl X × X (∆ i ) corresponds the projector π i : H • ( X ) → H i ( X ) . It is then easy tosee that γ r can be represented by P ni =0 r i ∆ i , so is algebraic.Conversely, assuming that γ r is represented by a rational algebraic n -cycle G r on X × X forany r ∈ Q > , we shall prove Conjecture C ( X ) using a similar trick in [Kle68, Theorem 2A11]due to Lieberman. Recall that γ r acts on each H i ( X ) by the multiplication by r i and ∆ X is theidentity in the ring of correspondences. It thus follows from the functoriality of the cycle classmap that γ r = cl X × X ( G r ◦ ∆ X ) = γ r ◦ cl X × X (∆ X ) = n X i =0 γ r ◦ π i = n X i =0 r i π i . Therefore, if we choose n + 1 different rational numbers for r , we can express π i as rationallinear combinations of the γ r . In particular, the π i are represented by rational algebraic n -cycleson X × X , which concludes the proof of Conjecture C ( X ) . (cid:3) Lemma 4.5.
Fix a positive rational number r ∈ Q > . Suppose that the homological corre-spondence γ r is represented by G r ∈ Z n ( X × X ) Q . Then there is a constant C r > so thatfor any effective correspondence f of X , we have k G r ◦ f k ≤ C r deg( G r ◦ f ) . Proof.
Let α i be a basis for N n ( X × X ) Q with k α i k = 1 . Then as discussed in Section 2.5, k G r ◦ f k is equivalent to max i | ( G r ◦ f ) · α i | . By Lemma 2.11, we have max i | ( G r ◦ f ) · α i | ≤ C max { , r n } deg( f ) ≤ C max { , r n } max { , r − n } deg( G r ◦ f ) , where C > is a constant independent of both f and r . Hence, C r := C max { , r n } max { , r − n } satisfies the requirement. (cid:3) The lemma below should be standard. Using this, many of our problems can be reduced toconsider the other (nontrivial) direction.
Lemma 4.6.
Let f ∈ Z n ( X × X ) Q be an arbitrary correspondence of X . Then we have ρ ( f ∗ | N k ( X ) R ) ≤ ρ ( f ∗ | H k ( X ) ) ι , (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) . (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι , k f k . k cl X × X ( f ) k ι , where k f k denotes the norm of the numerical class of f in N n ( X × X ) R and k cl X × X ( f ) k ι denotes the norm of cl X × X ( f ) ∈ H n ( X × X ) .Proof. Note that we have the following diagram of finite-dimensional Q ℓ -vector spaces (re-specting the pullback action f ∗ by the functoriality and the multiplicativity of the cycle map): H k alg ( X ) := A k ( X ) ⊗ Q Q ℓ H k ( X ) N k ( X ) Q ℓ := N k ( X ) ⊗ Z Q ℓ . (4.1)Then it follows readily that ρ ( f ∗ | N k ( X ) R ) = ρ ( f ∗ | N k ( X ) Q ℓ ) ≤ ρ ( f ∗ | H k ( X ) ) ι . The second andthe third follow similarly by considering the quotient norm inherited from norms on H k ( X ) and H n ( X × X ) , respectively. (cid:3) The following two lemmas are used in the proof of Theorem 1.7.
TANDARD CONJECTURES AND DYNAMICAL DEGREES 37
Lemma 4.7.
Suppose that Conjecture 1.3 holds for all correspondences of X . Then for everytwo correspondences f , g of X such that f ∗ | N k ( X ) R = g ∗ | N k ( X ) R , we have Tr( f ∗ | H k ( X ) ) =Tr( g ∗ | H k ( X ) ) .Proof. Since both the assumption and conclusion are linear in f and g , we may assume that g ∗ | N k ( X ) R = 0 . Then | Tr( f ∗ | H k ( X ) ) | ≤ b k ρ ( f ∗ | H k ( X ) ) ι = b k ρ ( f ∗ | N k ( X ) R ) = 0 , where b k := dim Q ℓ H k ( X ) is the k -th ℓ -adic Betti number of X . (cid:3) Lemma 4.8.
Suppose that Conjecture C ( X ) holds. Then there exists a constant C > so thatfor any f ∈ A k ( X ) ⊗ A n − k ( X ) , we have | Tr( f ∗ | H k ( X ) ) | ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) . Proof.
First, the Lefschetz trace formula asserts that
Tr( f ∗ | H k ( X ) ) = f · ∆ n − k , where ∆ n − k ∈ Z n ( X × X ) Q corresponds to the projection H • ( X ) → H n − k ( X ) (seeProposition 2.8). Therefore, both sides of the inequality are numerically invariant so that wecan only consider the numerical class of f in N k ( X ) R ⊗ N n − k ( X ) R .Let f ≡ P i α i ⊗ β i be an arbitrary numerical representation of f , where α i ∈ N k ( X ) R and β i ∈ N n − k ( X ) R . Then we have | Tr( f ∗ | H k ( X ) ) | = | f · ∆ n − k |≤ X i | ( α i ⊗ β i ) · ∆ n − k |≤ X i k ∆ n − k kk α i ⊗ β i k ′ , where k·k ′ is the dual norm on N n ( X × X ) R defined by eq. (2.9). On the other hand, we havea so-called projective cross norm π ( · ) on N k ( X ) R ⊗ N n − k ( X ) R (see Definition 2.13). Clearly,they are equivalent as N k ( X ) R ⊗ N n − k ( X ) R is a finite-dimensional R -vector space. We thusobtain that k α i ⊗ β i k ′ ≤ C π ( α i ⊗ β i ) = C k α i kk β i k , where C > is a constant and the equality follows from Proposition 2.14. Combining theabove together gives us that | Tr( f ∗ | H k ( X ) ) | ≤ C k ∆ n − k k X i k α i kk β i k . By Definition 2.13, taking infimum over all numerical representations of f , we get that | Tr( f ∗ | H k ( X ) ) | ≤ C k ∆ n − k k π ( f ) . Note that π ( · ) is a norm on N k ( X ) R ⊗ N n − k ( X ) R (see Proposition 2.14), and we have thefollowing canonical isomorphism N k ( X ) R ⊗ N n − k ( X ) R ∼ −−→ End R ( N k ( X ) R ) , f f ∗ | N k ( X ) R . Hence we can deduce that π ( f ) is equivalent to k f ∗ | N k ( X ) R k (see also Proposition 2.18). Lemma 4.8thus follows. (cid:3) Using the fact that any log-concave sequence is stable under scaling by definition, we obtaina technical lemma as follows, which is particularly useful in practice by reducing the upperbound from the maximum of a log-concave sequence to a prescribed index k . This is also amotivation for us to introduce Conjecture G r and Effective Property G r . Lemma 4.9.
Let { a i } ni =0 be a log-concave sequence of positive real numbers, i.e., a i ≥ a i − a i +1 for all ≤ i ≤ n − . Let { b j } nj =0 be a finite sequence of nonnegative real num-bers such that for any positive rational number r ∈ Q > and any ≤ j ≤ n , r j b j ≤ max ≤ i ≤ n r i a i . (4.2) Then we have for any ≤ k ≤ n , b k ≤ a k , (4.3) and for any ≤ k ≤ n − , b k +1 ≤ √ a k a k +1 . (4.4) Proof.
By continuity, eq. (4.2) holds true for any positive real number r ∈ R > . Let us firstconsider the case when ≤ k ≤ n − . For this fixed k , we choose r k := a k /a k +1 ∈ R > .Clearly, the new finite sequence { c k,i := r ik a i } ni =0 of positive real numbers is log-concave sinceso is { a i } ni =0 by assumption. In particular, one has · · · ≤ c k,k − c k,k ≤ c k,k c k,k +1 = a k r k a k +1 = 1 ≤ c k,k +1 c k,k +2 ≤ · · · . It follows immediately that c k,k − ≤ c k,k = c k,k +1 ≥ c k,k +2 . The log-concavity property infersthat the maximum of all c k,i = r ik a i can be achieved at c k,k (or c k,k +1 ). Substituting r k into theinequality (4.2), we obtain that for any ≤ j ≤ n , r jk b j ≤ max ≤ i ≤ n r ik a i = r kk a k . In particular, for j = 2 k , we have the inequality (4.3); if j = 2 k + 1 , the inequality (4.4)follows.The remaining case k = n has been covered by the case k = n − . Indeed, it follows fromthe above proof that r nn − b n ≤ max ≤ i ≤ n r in − a i = c n − ,n − = c n − ,n = r nn − a n . (cid:3) The next result has been stated as [Hru04, Proposition 12.11] in a slightly different way,whose proof uses indeed the full strength of Weil’s Riemann hypothesis. It is also stated asa claim inside the proof of [Tru16, Theorem 1.2] with a condensed proof. For the sake ofcompleteness, here we provide a more detailed proof.
TANDARD CONJECTURES AND DYNAMICAL DEGREES 39
Lemma 4.10.
Let X be a smooth projective variety of dimension n over K satisfying Conjec-ture C ( X ) . Then there is a positive constant C > such that for any correspondence f of X ,we have for any ≤ i ≤ n , ρ ( f ∗ | H i ( X ) ) ≤ C k f k . In particular, if f is effective, we can replace k f k by deg( f ) .Proof. By the Lefschetz trace formula (see Proposition 2.8), we have
Tr(( f ◦ m ) ∗ | H i ( X ) ) = ( − i (cl X × X ( f ◦ m ) ∪ π n − i ) = ( − i ( f ◦ m · ∆ n − i ) ∈ Q . It follows from the functoriality of motivic composition that ρ ( f ∗ | H i ( X ) ) = lim m →∞ (cid:12)(cid:12) Tr(( f ∗ ) m | H i ( X ) ) (cid:12)(cid:12) /m = lim m →∞ (cid:12)(cid:12) Tr(( f ◦ m ) ∗ | H i ( X ) ) (cid:12)(cid:12) /m . On the other hand, using the dual norm k · k ′ of correspondences (see eq. (2.9)), we have (cid:12)(cid:12) f ◦ m · ∆ n − i (cid:12)(cid:12) ≤ (cid:13)(cid:13) f ◦ m (cid:13)(cid:13) ′ (cid:13)(cid:13) ∆ n − i (cid:13)(cid:13) . Thanks to Lemma 2.15, there is a constant
C > such that (cid:13)(cid:13) f ◦ m (cid:13)(cid:13) ′ ≤ C m − ( k f k ′ ) m . Note that all norms on N n ( X × X ) R are equivalent. Putting the above all together, Lemma 4.10follows. (cid:3) In a similar vein, we can prove a stronger lemma below with the help of Conjecture G r andLemma 4.9. It will be used in the proof of Theorem 1.9(1). Lemma 4.11.
Let X be a smooth projective variety of dimension n over K satisfying Conjec-ture G r . Then there is a constant C > such that for any irreducible correspondence f of X ,we have for any ≤ k ≤ n , (cid:12)(cid:12) Tr( f ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ C deg k ( f ) , and for any ≤ k ≤ n − , (cid:12)(cid:12) Tr( f ∗ | H k +1 ( X ) ) (cid:12)(cid:12) ≤ C q deg k ( f ) deg k +1 ( f ) . Proof.
We first note that by Lemma 4.4, Conjecture G r implies Conjecture C ( X ) . Hence bythe same argument in the proof of Lemma 4.10, namely, applying the Lefschetz trace formulato G r ◦ f and using the dual norm k · k ′ on N n ( X × X ) R , we have for each ≤ i ≤ n , (cid:12)(cid:12) Tr(( G r ◦ f ) ∗ | H i ( X ) ) (cid:12)(cid:12) = (cid:12)(cid:12) ( G r ◦ f ) · ∆ n − i (cid:12)(cid:12) ≤ k ∆ n − i kk G r ◦ f k ′ . According to Conjecture G r , there is a constant C > so that for any r ∈ Q > , k G r ◦ f k ≤ C deg( G r ◦ f ) . Using the definition of γ r = cl X × X ( G r ) and combining the last three displayed equations, weobtain that r i (cid:12)(cid:12) Tr( f ∗ | H i ( X ) ) (cid:12)(cid:12) ≤ C k ∆ n − i k deg( G r ◦ f )= C k ∆ n − i k n X j =0 (cid:18) nj (cid:19) deg j ( G r ◦ f ) . Note that deg j ( G r ◦ f ) = ( G r ◦ f ) ∗ H jX · H n − jX = ( f ∗ G ∗ r H jX ) · H n − jX = r j f ∗ H jX · H n − jX = r j deg j ( f ) , and (cid:0) nj (cid:1) ≤ (cid:0) n ⌊ n/ ⌋ (cid:1) . It follows that r i (cid:12)(cid:12) Tr( f ∗ | H i ( X ) ) (cid:12)(cid:12) ≤ C max ≤ j ≤ n r j deg j ( f ) , where the constant C > is independent of r ∈ Q > and f . Note that by Lemma 4.2, thesequence { deg j ( f ) } j is log-concave. Hence Lemma 4.11 follows directly from Lemma 4.9with b i := | Tr( f ∗ | H i ( X ) ) | and a j := C deg j ( f ) . (cid:3) Proof of Theorem 1.7.
Proof of Theorem 1.7 (1).
Suppose that Conjecture D k ( X ) holds. Then by Lemma A.4, wehave a direct sum decomposition H k ( X ) = H k alg ( X ) ⊕ H n − k tr ( X ) . Let f ∈ A k ( X ) ⊗ A n − k ( X ) .Recall by definition that if we write f = P i,j α i ⊗ β j for α i ∈ A k ( X ) and β j ∈ A n − k ( X ) , then f ∗ α = P i,j ( α ∪ β j ) α i for any α ∈ A k ( X ) . It is now easy to verify that f ∗ ( H k alg ( X )) ⊆ H k alg ( X ) and f ∗ ( H k tr ( X )) = 0 . In particular, we have that f ∗ | H k ( X ) = f ∗ | H k alg ( X ) = f ∗ | A k ( X ) ⊗ Q Q ℓ = f ∗ | N k ( X ) Q ⊗ Q Q ℓ . We thus prove that k f ∗ | H k ( X ) k ι = k f ∗ | N k ( X ) Q k = k f ∗ | N k ( X ) R k .Conversely, suppose that Conjecture 1.4 holds for all correspondences f ∈ A k ( X ) ⊗ A n − k ( X ) .To prove Conjecture D k ( X ) , by Proposition A.3, it is equivalent to showing that if β ∈ A n − k ( X ) such that α ∪ β = 0 for all α ∈ A k ( X ) , then β = 0 . Now let us pick up a nonzero α ∈ A k ( X ) and consider the correspondence f := α ⊗ β ∈ A k ( X ) ⊗ A n − k ( X ) . Then, clearly,one has f ∗ | A k ( X ) = 0 and hence f ∗ | N k ( X ) Q = 0 . Now it follows from the norm comparison inConjecture 1.4 that k f ∗ | H k ( X ) k ι ≤ C k f ∗ | N k ( X ) R k = 0 , which implies that f ∗ | H k ( X ) = 0 . Inother words, for any γ ∈ H k ( X ) , we have ( γ ∪ β ) α = f ∗ γ = 0 . This is only possible when γ ∪ β = 0 since α ∈ A k ( X ) is nonzero. By Poincaré duality, we prove that β = 0 . (cid:3) Proof of Theorem 1.7 (2).
Suppose that Conjecture D k ( X ) holds. If H k ( X ) is generated byalgebraic cycle classes, i.e., H k ( X ) = H k alg ( X ) = A k ( X ) ⊗ Q Q ℓ = N k ( X ) ⊗ Q Q ℓ , then wehave f ∗ | H k ( X ) = f ∗ | N k ( X ) Q ⊗ Q Q ℓ , TANDARD CONJECTURES AND DYNAMICAL DEGREES 41 for any correspondence f ∈ A n ( X × X ) . This infers that ρ ( f ∗ | H k ( X ) ) ι = ρ ( f ∗ | N k ( X ) Q ) = ρ ( f ∗ | N k ( X ) R ) . Hence Conjecture 1.3 holds for all correspondences f ∈ A n ( X × X ) .Conversely, suppose that Conjecture D k ( X ) holds and Conjecture 1.3 holds for all corre-spondences of X . We shall prove that H k ( X ) = N k ( X ) ⊗ Z Q ℓ . By Proposition A.3, it isequivalent to showing that dim Q ℓ H k ( X ) = dim Q N k ( X ) Q . First, it follows from Lemma A.4that there is a direct sum decomposition H k ( X ) = H k alg ( X ) ⊕ H n − k tr ( X ) . As in the proof ofLemma A.4, we denote by α i and β j the bases of A k ( X ) and A n − k ( X ) , respectively, which aredual to each other, i.e., α i ∪ β j = δ ij . Now let f ∈ A n ( X × X ) be an arbitrary correspondenceof X . Consider the following correspondence g := X j f ∗ α j ⊗ β j ∈ A k ( X ) ⊗ A n − k ( X ) . It is easy to verify that g ∗ α i = P j δ ij f ∗ α j = f ∗ α i for all i . Therefore, the pullback actions of f and g coincide on A k ( X ) and hence on N k ( X ) Q (as by assumption Conjecture D k ( X ) holds).In other words, we have that f ∗ | N k ( X ) Q = f ∗ | A k ( X ) Q = g ∗ | A k ( X ) Q = g ∗ | N k ( X ) Q . Thanks to Lemma 4.7, we obtain an equality of traces:
Tr( f ∗ | H k ( X ) ) = Tr( g ∗ | H k ( X ) ) . On the other hand, as we have seen in the proof of Theorem 1.7(1) that g ∗ ( H k tr ( X )) = 0 whichimplies that Tr( g ∗ | H k ( X ) ) = Tr( g ∗ | H k alg ( X ) ) = Tr( g ∗ | A k ( X ) Q ) . Now, combining the last three displayed equations, we get that
Tr( f ∗ | H k ( X ) ) = Tr( f ∗ | N k ( X ) Q ) . In particular, applying this to f = ∆ X , the identity in the ring of correspondences, we see that dim Q ℓ H k ( X ) = dim Q N k ( X ) Q . (cid:3) Proof of Theorem 1.7 (3).
We first note that A k ( X ) ⊗ A n − k ( X ) is a two-sided ideal in the ring Corr ∼ hom ( X, X ) = A n ( X × X ) of correspondences (modulo homological equivalence ). Infact, one can verify that for any α ⊗ β ∈ A k ( X ) ⊗ A n − k ( X ) and g ∈ A n ( X × X ) , the motiviccompositions g ◦ ( α ⊗ β ) = α ⊗ g ∗ β and ( α ⊗ β ) ◦ g = g ∗ α ⊗ β are both in A k ( X ) ⊗ A n − k ( X ) . Inparticular, for our f ∈ A k ( X ) ⊗ A n − k ( X ) , its m -th motivic iterates f ◦ m ∈ A k ( X ) ⊗ A n − k ( X ) .Applying Lemma 4.8 to f ◦ m , we get that (cid:12)(cid:12) Tr(( f ◦ m ) ∗ (cid:12)(cid:12) H k ( X ) ) | ≤ C (cid:13)(cid:13) ( f ◦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) . Though the same are also true modulo rational equivalence and numerical equivalence, here we do not needthem.
Since the motivic composition is functorial, i.e., ( f ◦ m ) ∗ = ( f ∗ ) m , one readily deduces that ρ ( f ∗ | H k ( X ) ) = lim m →∞ (cid:12)(cid:12) Tr(( f ∗ ) m | H k ( X ) ) (cid:12)(cid:12) /m = lim m →∞ (cid:12)(cid:12) Tr(( f ◦ m ) ∗ (cid:12)(cid:12) H k ( X ) ) | /m ≤ lim m →∞ (cid:0) C (cid:13)(cid:13) ( f ◦ m ) ∗ | N k ( X ) R (cid:13)(cid:13)(cid:1) /m ≤ lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | N k ( X ) R (cid:13)(cid:13) /m = ρ ( f ∗ | N k ( X ) R ) , where the first equality is a fact in Linear Algebra that ρ ( M ) = lim m →∞ | Tr( M m ) | /m and thelast equality follows from the spectral radius formula. Note that in our situation, the spectralradius is independent of ι because by the Lefschetz trace formula these traces are rationalnumbers. The other direction has been proved for general correspondences in Lemma 4.6. Wethus prove the assertion (3) and conclude the proof of Theorem 1.7. (cid:3) Proofs of Theorem 1.9 and Corollary 1.15.
Proof of Theorem 1.9 (1).
Note that the irreducible case has been proved in Lemma 4.11. Letus consider the case when f is effective. In this case, the invariant deg k ( f ) is the same as theusual k -th degree deg k ( f ) (see Remark 2.17), which is linear with respect to effective corre-spondences. Write f = P i a i f i , where a i ∈ Q > and the f i are irreducible correspondences of X . It is thus straightforward to see that (cid:12)(cid:12) Tr( f ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ X i a i (cid:12)(cid:12) Tr( f ∗ i | H k ( X ) ) (cid:12)(cid:12) ≤ X i C deg k ( a i f i ) = C deg k ( f ) , where the second inequality follows from Lemma 4.11. On the other hand, we note that by theCauchy–Schwarz inequality, the function p deg k ( · ) deg k +1 ( · ) defined over effective correspon-dences of X is superadditive. Precisely, for any two effective correspondences g, h of X , wealways have q deg k ( g ) deg k +1 ( g ) + q deg k ( h ) deg k +1 ( h ) ≤ q deg k ( g + h ) deg k +1 ( g + h ) . In particular, one has X i q deg k ( a i f i ) deg k +1 ( a i f i ) ≤ q deg k ( f ) deg k +1 ( f ) . So similarly, applying Lemma 4.11 for each f i , we have (cid:12)(cid:12) Tr( f ∗ | H k +1 ( X ) ) (cid:12)(cid:12) ≤ X i a i (cid:12)(cid:12) Tr( f ∗ i | H k +1 ( X ) ) (cid:12)(cid:12) ≤ C q deg k ( f ) deg k +1 ( f ) . This proves the second half of Theorem 1.9(1). We note that by continuity this inequality holdsfor any effective correspondence of X with real coefficients not just rational coefficients.We then consider an arbitrary correspondence f of X . Write f ≡ f + − f − , where f ± are(real) effective correspondences of X . Note that by Lemma 4.4, Conjecture C ( X ) holds. It TANDARD CONJECTURES AND DYNAMICAL DEGREES 43 thus follows from the Lefschetz trace formula (see Proposition 2.8), that
Tr( f ∗ | H k ( X ) ) is anumerical invariant of f and depends only the numerical equivalence class of f . Hence byapplying the previous discussion on the effective correspondence case to f ± , we have (cid:12)(cid:12) Tr( f ∗ | H k ( X ) ) (cid:12)(cid:12) = (cid:12)(cid:12) Tr(( f + − f − ) ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) Tr(( f + ) ∗ | H k ( X ) ) (cid:12)(cid:12) + (cid:12)(cid:12) Tr(( f − ) ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ C deg k ( f + + f − ) . By taking infimum over all numerical decompositions of f , the first half of Theorem 1.9(1)follows directly from the definition of deg k ; see eq. (2.10). (cid:3) Proof of Theorem 1.9 (2).
Let f be an effective correspondence of X . Let r ∈ Q > be anarbitrary positive rational number. Let g denote the motivic composition G r ◦ f . Note again thatby Lemma 4.4, Conjecture C ( X ) holds under our assumption. It thus follows from Lemma 4.10that there is a constant C > such that for each ≤ i ≤ n , ρ ( g ∗ | H i ( X ) ) ≤ C k g k . Now by assumption that Conjecture G r holds, there is a constant C > (independent of r )such that k g k = k G r ◦ f k ≤ C deg( G r ◦ f ) . Combining them together, we show that r i ρ ( f ∗ | H i ( X ) ) ≤ C C deg( G r ◦ f ) ≤ C max ≤ j ≤ n r j deg j ( f ) , (4.5)where C := ( n + 1) (cid:0) n ⌊ n/ ⌋ (cid:1) C C . See also proof of Lemma 4.11. Note that by assumptionthe degree sequence { deg j ( f ) } j is log-concave. Hence Theorem 1.9(2) follows readily fromLemma 4.9 with b i := ρ ( f ∗ | H i ( X ) ) and a j := C deg j ( f ) . (cid:3) Proof of Theorem 1.9 (3).
Since f ◦ m is numerically effective, it follows from Remark 2.17 that deg k ( f ◦ m ) = deg k ( f ◦ m ) = (cid:13)(cid:13) ( f ◦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) . Applying Theorem 1.9(1) to f ◦ m for all m , we get that (cid:12)(cid:12) Tr(( f ◦ m ) ∗ | H k ( X ) ) (cid:12)(cid:12) ≤ C (cid:13)(cid:13) ( f ◦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) , which, by functoriality, is equivalent to the following (cid:12)(cid:12) Tr(( f ∗ ) m | H k ( X ) ) (cid:12)(cid:12) ≤ C (cid:13)(cid:13) ( f ∗ ) m | N k ( X ) R (cid:13)(cid:13) . Taking limits of the m -th roots of both sides yields that ρ ( f ∗ | H k ( X ) ) ≤ ρ ( f ∗ | N k ( X ) R ) . Notethat the converse direction has been proved in Lemma 4.6 so that we actually have the equality.Using the same argument, the second half of Theorem 1.9(3) follows similarly from the secondhalf of Theorem 1.9(1). We hence conclude the proof of Theorem 1.9. (cid:3) Proof of Corollary 1.15.
We follow the proof of Theorem 1.9(2). In particular, for any effectivecorrespondence h of X and any r ∈ Q > , by considering G r ◦ h , we have shown that r k ρ ( h ∗ | H k ( X ) ) ≤ C max ≤ i ≤ n r i deg i ( h ) , for some constant C > independent of h and r ; see eq. (4.5).Now, consider a dynamical correspondence f of X so that the dynamical degree sequence { λ i ( f ) } i is log-concave. Note that for any m ∈ N , the m -th dynamical iterate f ♦ m of f is stilla dynamical correspondence and hence effective. So applying the same argument to G r m ◦ f ♦ m yields that r km ρ (( f ♦ m ) ∗ | H k ( X ) ) ≤ C max ≤ i ≤ n r im deg i ( f ♦ m ) . Taking the limsups of the m -th roots of both sides, we obtain that r k lim sup m →∞ ρ (( f ♦ m ) ∗ | H k ( X ) ) /m ≤ max ≤ i ≤ n r i lim sup m →∞ (deg i ( f ♦ m )) /m = max ≤ i ≤ n r i λ i ( f ) , where the equality is guaranteed by Theorem 3.6. Hence, similar with the proof of Theo-rem 1.9(2), our Corollary 1.15 follows readily from Lemma 4.9. (cid:3) Proofs of Theorem 1.12 and Corollary 1.17.
Proof of Theorem 1.12 (1). (1a) ⇒ (1b): This is essentially contained in the proof of [Tru16,Theorem 1(1)]. The main idea is as follows. Since by assumption Conjecture D n ( X × X ) holds, we then have an orthogonal decomposition H n ( X × X ) = H n alg ( X × X ) M H n tr ( X × X ) , where H n tr ( X × X ) := H n alg ( X ) ⊥ := { α ∈ H n ( X ) : α ∪ β = 0 , for all β ∈ H n alg ( X ) } . See Lemma A.4. Let τ denote the orthogonal projection from H n ( X × X ) to H n alg ( X × X ) .Let f be an arbitrary correspondence of X ; in particular, cl X × X ( f ) ∈ H n alg ( X × X ) . It followsthat for any α ∈ H i ( X ) and β ∈ H n − i ( X ) , we have | f ∗ α ∪ β | ι = | cl X × X ( f ) ∪ pr ∗ α ∪ pr ∗ β | ι = | f · τ (pr ∗ α ∪ pr ∗ β ) |≤ k f k ′ k τ (pr ∗ α ∪ pr ∗ β ) k , where we are using the dual norm k · k ′ of correspondences; see eq. (2.9).On the other hand, we adopt the following operator norm of f ∗ | H i ( X ) : (cid:13)(cid:13) f ∗ | H i ( X ) (cid:13)(cid:13) ι := sup {k f ∗ α k ι : α ∈ H i ( X ) , k α k ι = 1 } = sup ((cid:12)(cid:12) f ∗ α ∪ β (cid:12)(cid:12) ι : α ∈ H i ( X ) , β ∈ H n − i ( X ) , k α k ι = k β k ι = 1 ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 45
It follows that k f ∗ | H i ( X ) k ι . k f k ′ , which is equivalent to saying that k f ∗ | H i ( X ) k ι . k f k .(1b) ⇒ (1a): Suppose that the conclusion of (1b) holds. In particular, for any correspondence f ∈ Z n ( X × X ) Q and any ≤ i ≤ n , we have k f ∗ | H i ( X ) k ι ≤ C k f k . Now, if f is numericallytrivial, then clearly k f k = 0 and hence k f ∗ | H i ( X ) k ι = 0 for all i . By Poincaré duality and theKünneth formula, we know that H n ( X × X ) = n M i =0 H i ( X ) ⊗ H n − i ( X ) ≃ n M i =0 End Q ℓ ( H i ( X )) . It follows that cl X × X ( f ) = 0 in H n ( X × X ) , i.e., Conjecture D n ( X × X ) holds. We thusprove Theorem 1.12(1). (cid:3) Proof of Theorem 1.12 (2).
As in the proof of Theorem 1.9(1), it suffices to consider the casewhen f is an irreducible correspondence of X . In particular, the degree sequence { deg i ( f ) } i islog-concave by Lemma 4.2. Since Conjecture G r holds, we can consider g := G r ◦ f . By thedefinition of G r , we immediately have for any ≤ i ≤ n , (cid:13)(cid:13) g ∗ | H i ( X ) (cid:13)(cid:13) ι = r i (cid:13)(cid:13) f ∗ | H i ( X ) (cid:13)(cid:13) ι . Under our assumption, Theorem 1.12(1b) holds. Applying it to g yields that (cid:13)(cid:13) g ∗ | H i ( X ) (cid:13)(cid:13) ι ≤ C k g k , where C > is a constant. On the other hand, Conjecture G r asserts that for any r ∈ Q > ,there is a constant C > such that k g k = k G r ◦ f k ≤ C deg( G r ◦ f ) . Putting the above all together, we show that for any r ∈ Q > , r i (cid:13)(cid:13) f ∗ | H i ( X ) (cid:13)(cid:13) ι ≤ C max ≤ j ≤ n r j deg j ( f ) , where C > is a constant independent of f and r . Then Theorem 1.12(2) follows fromLemma 4.9 with b i := k f ∗ | H i ( X ) k ι and a j := C deg j ( f ) . (cid:3) Proof of Theorem 1.12 (3).
Note first that by assumption and Lemma 4.4, the homological cor-respondence γ r ∈ H n ( X × X ) is algebraic for all r ∈ Q > . Denote γ r = cl X × X ( G r ) as usual.It remains to show that for any effective correspondence f of X , there is a constant C > ,independent of f and r , such that k G r ◦ f k ≤ C deg( G r ◦ f ) . (4.6)Denote G r ◦ f by g for simplicity. Then by Lemma 4.6, one can easily have k g k . k cl X × X ( g ) k . On the other hand, by the Künneth formula and Poincaré duality, the norm from the right-handside is equivalent to max ≤ i ≤ n (cid:13)(cid:13) g ∗ | H i ( X ) (cid:13)(cid:13) ι . Note that g ∗ | H i ( X ) = r i f ∗ | H i ( X ) and deg( g ) = g · H nX × X = n X k =0 (cid:18) nk (cid:19) deg k ( g ) = n X k =0 (cid:18) nk (cid:19) r k deg k ( f ) by the definition of G r .It is now clear that eq. (4.6) follows from the inequalities (1.4) and (1.5) in Theorem 1.12(2)for effective correspondences. In fact, if i = 2 k is even, then the first inequality (1.4) assertsthat there is a constant C (independent of f ) such that (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C deg k ( f ) = C deg k ( f ) . Here by Remark 2.17, we have equalities deg k ( f ) = deg k ( f ) since f is assumed to be effective.In particular, we obtain that (cid:13)(cid:13) g ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ Cr k deg k ( f ) . Similarly, if i = 2 k + 1 is odd, then the second inequality (1.5) yields that (cid:13)(cid:13) g ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ Cr k +1 q deg k ( f ) deg k +1 ( f ) ≤ C/ r k deg k ( f ) + r k +2 deg k +1 ( f )) ≤ C max { r k deg k ( f ) , r k +2 deg k +1 ( f ) } . Then putting the above all together, we have shown that k g k . max ≤ k ≤ n r k deg k ( f ) = max ≤ k ≤ n deg k ( g ) , (4.7)in particular, k g k . deg( g ) . This concludes the proof of Theorem 1.12(3). (cid:3) Proof of Corollary 1.17. (1) Under our assumption, by Theorem 1.12(2a) and Remark 2.17, weknow that Conjecture 1.4 holds for all effective correspondences of X . Note in particular thatall dynamical iterates f ♦ m are effective. Hence we obtain that for any m ∈ N , (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) , for some constant C > independent of f and m . Taking the limsups of the m -th roots of bothsides yields that lim sup m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) /mι ≤ lim sup m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m = lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m , where the equality again follows from Theorem 3.6.On the other hand, by Lemma 4.6, it is easy to have lim inf m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) /mι ≥ lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | N k ( X ) R (cid:13)(cid:13) /m = λ k ( f ) . Combining the above together, we show that χ k ( f ) ι = lim m →∞ (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) /mι = λ k ( f ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 47 (2) By applying Theorem 1.12(2b) to the m -th dynamical iterate f ♦ m of f , we have (cid:13)(cid:13) ( f ♦ m ) ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ C q deg k ( f ♦ m ) deg k +1 ( f ♦ m ) , where C > is a constant independent of f and m . We then take the limsups of the m -throots of both sides. Now the desired inequality follows since the right-hand side converges to p λ k ( f ) λ k +1 ( f ) by Theorem 3.6. We hence complete the proof of Corollary 1.17. (cid:3)
5. A
PPLICATIONS
In this section, we apply the main results of the paper to various cases: complex projectivemanifolds, Abelian varieties, surfaces, threefolds, and algebraically stable correspondences.Along the way, we also prove several additional results.5.1.
Curves.
We emphasize that our Conjecture G r , a quantitative version of the standardconjecture C , plays an important role in Theorems 1.9 and 1.12. In this subsection, we firstshow that it holds on curves. We then explain why the inequalities (1.4) and (1.5) in Theo-rem 1.12(2) can be regarded as certain higher-dimensional generalizations of (a weaker versionof) the Castelnuovo–Severi inequality. It follows from this that on a curve Conjecture G r isequivalent to the mentioned weaker version of Castelnuovo–Severi inequality. Lemma 5.1.
Conjecture G r holds on curves.Proof. Let X be a smooth projective curve over K . It is well-known that the standard conjecture C holds on curves (in fact, one can choose ∆ = X × { pt } , ∆ = { pt } × X , and ∆ =∆ X − ∆ − ∆ ). Hence the algebraicity of the homological correspondence γ r ∈ H ( X × X ) follows from Lemma 4.4. Say, γ r is represented by G r as usual.Let f be an effective correspondence of X . Using the dual norm k · k ′ of correspondences(see eq. (2.9)), it suffices to show that for an arbitrary (test) correspondence g of X , there is aconstant C > , independent of f and r , such that | ( G r ◦ f ) · g | ≤ C deg( G r ◦ f ) . (5.1)Note that ( G T r ) ∗ = ( G r ) ∗ . Hence by the Lefschetz trace formula (see Proposition 2.8), onecan easily check that the self-intersection ( G r ◦ f ) · ( G r ◦ f ) equals r ( f · f ) . Recall that bydefinition, deg( G r ◦ f ) = deg ( f ) + r deg ( f ) . To simplify notation, we denote n := deg ( f ) , n ′ := deg ( f ) , m := deg ( g ) , and m ′ := deg ( g ) . Now, by the Castelnuovo–Severi inequality(see, e.g., [Ful98, Example 16.1.10]), we obtain that (cid:12)(cid:12) ( G r ◦ f ) · g − mn ′ − m ′ nr (cid:12)(cid:12) ≤ p r (2 nn ′ − f · f )(2 mm ′ − g · g ) . Further, since our f is assumed to be effective, by Lemma 2.19, we have | f · f | ≤ C nn ′ forsome constant C > depending only on X . It follows readily that (cid:12)(cid:12) ( G r ◦ f ) · g (cid:12)(cid:12) ≤ ( mn ′ + m ′ nr ) + C √ r nn ′ , where C := p ( C + 2)(2 mm ′ − g · g ) . Note that √ r nn ′ ≤ r n + n ′ . Thus eq. (5.1)is verified with C := max { m, m ′ } + C / independent of f and r . We hence prove thatConjecture G r holds on curves. (cid:3) Remark . In the above proof, we used only the following weaker version of Castelnuovo–Severi inequality: Let X be a smooth projective curve. Then, for each effective correspondence g , there is a constant C g > (depending on g ) so that (cid:12)(cid:12) ( G r ◦ f ) · g (cid:12)(cid:12) ≤ C g ( n ′ + nr + √ r nn ′ ) (5.2)for all effective correspondences f and for all r > . Here n = deg ( f ) , n ′ = deg ( f ) , m = deg ( g ) , and m ′ = deg ( g ) .This version readily follows from the inequalities (1.4) and (1.5) in Theorem 1.12(2) as weshall explain. We decompose cl X × X ( f ) = φ + φ + φ and cl X × X ( g ) = ψ + ψ + ψ , where φ , ψ ∈ H ( X ) ⊗ H ( X ) , φ , ψ ∈ H ( X ) ⊗ H ( X ) and φ , ψ ∈ H ( X ) ⊗ H ( X ) . Then cl X × X ( G r ◦ f ) = φ + rφ + r φ . Then, by degree consideration, we have ( G r ◦ f ) · g = φ ∪ ψ + rφ ∪ ψ + r φ ∪ ψ . The proof of Theorem 1.12(3) shows that | φ ∪ ψ | ≤ C k φ kk ψ k ∼ C k f ∗ | H ( X ) kk g ∗ | H ( X ) k = Cmn ′ . Similarly, one also has r | φ ∪ ψ | ≤ Cr m ′ n and r | φ ∪ ψ | ≤ Cr k f ∗ | H ( X ) kk g ∗ | H ( X ) k ≤ Cr √ nn ′ √ mm ′ , by the inequality in Theorem 1.12(2b). Hence we obtain the mentioned weaker version ofCastelnuovo–Severi inequality eq. (5.2).5.2. Abelian varieties.
In this subsection, we show that Abelian varieties satisfy EffectiveProperty G r and hence Conjecture G r holds on them. As a consequence, Conjecture 1.4 holdsfor Abelian varieties (in the case of positive characteristics, we require additionally that Abelianvarieties are defined over finite fields and the coefficient field Q ℓ is appropriately chosen; for de-tails see Remark 5.5). As far as we know, the latter result is new and extends [Hu19a, Hu19b].Results more special to Abelian surfaces in positive characteristic will be presented in Sec-tion 5.7. Lemma 5.3.
Abelian varieties satisfy Effective Property G r .Proof. Let X be an Abelian variety of dimension n over K . It is well-known that the standardconjecture C holds on Abelian varieties by Lieberman (see, e.g., [Kle68, Theorem 2A11]).Hence the algebraicity of γ r for every r ∈ Q > follows from Lemma 4.4. Say, γ r = cl X × X ( G r ) for some rational algebraic n -cycle G r on X × X . Let f be an effective correspondence of X . For an arbitrary r ∈ Q > , we write r = n /n ∈ Q , where n , n ∈ N are two coprimepositive integers. Define a new correspondence g of X by g := 1 n n ([ n ] × [ n ]) ∗ f, TANDARD CONJECTURES AND DYNAMICAL DEGREES 49 where [ n i ] is the multiplication-by- n i map of X and [ n ] × [ n ] is the product endomorphismof X × X . Clearly, g is effective since effective cycles are preserved by proper pushforward ofmorphisms. Also, thanks to Lieberman’s Lemma 2.9, for any ≤ i ≤ n and any cohomologyclass α ∈ H i ( X ) , we have g ∗ ( α ) = 1 n n (([ n ] × [ n ]) ∗ f ) ∗ ( α ) = 1 n n ([ n ] ∗ ◦ f ∗ ◦ [ n ] ∗ )( α )= n − i f ∗ ( n i α ) = r i f ∗ ( α ) = ( G r ◦ f ) ∗ ( α ) , where a fact that [ n ] ∗ α = n i α and [ n ] ∗ β = n n − i β for any α, β ∈ H i ( X ) is used. In otherwords, we show that g ∗ | H i ( X ) = r i f ∗ | H i ( X ) for all i . By Poincaré duality and the Künnethformula, this implies that G r ◦ f is homologically equivalent to the effective correspondence g .Thus X satisfies Effective Property G r . (cid:3) Now we state some consequences for Abelian varieties.
Remark . It is well-known that the standard conjecture of Hodgetype holds in characteristic zero by the Hodge–Riemann bilinear relations (see, e.g., Remark A.7(1)).Also, the standard conjecture B holds on Abelian varieties (see, e.g., [Kle68, Theorem 2A11]).Therefore, the standard conjecture D holds on all Abelian varieties in characteristic zero (seeProposition A.6(2)). Hence, as a direct corollary of Theorem 1.12(2a) and the above Lemma 5.3,Conjecture 1.4 holds. Precisely, if X is an Abelian variety of dimension n over an algebraicallyclosed field K of characteristic zero, there is a constant C > so that for any effective corre-spondence f of X and any ≤ k ≤ n , we have (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) , and (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ C q(cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13)(cid:13)(cid:13) f ∗ | N k +1 (cid:13)(cid:13) . As far as we know, this result is new (see Theorem 5.10 for a comparison, which is valid for dy-namical correspondences). As mentioned in Section 5.4, Conjecture 1.1 holds in characteristiczero, but here we obtain a new proof for this specific X .Now, let f be an arbitrary correspondence of X over K so that the m -th motivic iterate f ◦ m is numerically effective for any m ∈ N . From Theorem 1.9(3), then ρ ( f ∗ | H k ( X ) ) = ρ ( f ∗ | N k ( X ) R ) for all ≤ k ≤ n . If f is a totally algebraically stable dynamical correspondence(see Definition 3.4), then f ◦ m is homologically equivalent to f ♦ m which is always effective sothat the above condition is satisfied; moreover, one also has that χ k ( f ) = ρ ( f ∗ | H k ( X ) ) = ρ ( f ∗ | N k ( X ) R ) = λ k ( f ) , and χ k +1 ( f ) = ρ ( f ∗ | H k +1 ( X ) ) ≤ q ρ ( f ∗ | N k ( X ) R ) ρ ( f ∗ | N k +1 ) = p λ k ( f ) λ k +1 ( f ) . This is in particular the case when f is an effective finite correspondence. Remark . Let X be an Abelian variety over an algebraicallyclosed field K of positive characteristic.(1) Assume that K = F p , an algebraic closure of a finite field. It is known that the standardconjecture D holds for infinitely many primes ℓ = p by Clozel [Clo99]. We fix such a prime ℓ . Then all the results discussed in Remark 5.4 hold. In particular, there is a constant C > sothat for any effective correspondence f of X and any ≤ k ≤ n , we have (cid:13)(cid:13) f ∗ | H k ( X, Q ℓ ) (cid:13)(cid:13) ι ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) , and (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ι ≤ C q(cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13)(cid:13)(cid:13) f ∗ | N k +1 (cid:13)(cid:13) . Also, Conjecture 1.1 holds on X .(2) In general, we still have by Theorem 1.9(3) that for every correspondence f of X suchthat f ◦ m is effective for all m , Conjecture 1.3 holds for f . This is in particular the case when f is an effective finite correspondence. We also obtain the same inequalities as in part (1). Notethat the spreading out and specialization arguments (see Section 1.5), combined with part (1),also implies this conclusion for the more special case when f is a positive sum of graphs ofendomorphisms. This special case generalizes the results in [Hu19a, Hu19b], where the case ofendomorphisms is solved. Remark . Let X be an Abelian variety of dimension n over K . To each correspondence f of X , we can assign a morphism α f given by the formula: α f ( x ) := X f ( x ) , where f ( x ) is the direct image of x under the correspondence f (counting with multiplicity)and the sum is taken using the group structure of X .It can be checked that f ∗ | H ( X ) = α ∗ f | H ( X ) . However, for i > we do not have f ∗ | H i ( X ) = α ∗ f | H i ( X ) in general. Hence, this assignment does not help in understanding the pullback ofcorrespondences on H • ( X ) . In particular it does not help with Conjectures 1.1, 1.3 and 1.4.One reason is that since α f is a self-morphism, we have α ∗ f ( β ∪ γ ) = α ∗ f ( β ) ∪ α ∗ f ( γ ) for all β, γ ∈ H • ( X ) . On the other hand, for a general correspondence we do not have this property.Another, related reason is that, α f , being an effective correspondence, satisfies Conjectures 1.3and 1.4. On the other hand, on Abelian varieties X where H k alg ( X ) = H k ( X ) , Conjectures 1.3and 1.4 cannot hold for all correspondences, because of Remark 1.8.Here is an explicit example. Let φ be a surjective endomorphism of X . Fix an integer m andlet mφ denote the composition of φ with the multiplication by m on X . If f = Γ φ + Γ mφ , then α f = (1 + m ) φ . Hence, for all i , we have f ∗ | H i ( X ) = (Γ ∗ φ + Γ ∗ mφ ) | H i ( X ) = (1 + m i ) φ ∗ | H i ( X ) , while α ∗ f | H i ( X ) = (1 + m ) i φ ∗ | H i ( X ) . With an appropriate choice of m , we have that f ∗ | H i ( X ) = α ∗ f | H i ( X ) for all < i ≤ n . TANDARD CONJECTURES AND DYNAMICAL DEGREES 51
Descent properties.
This subsection concerns certain descent properties about EffectiveProperty G r , Conjecture G r , Conjecture 1.4, and Conjecture 1.1, for dynamical correspon-dences under generically finite surjective morphisms. Theorem 5.7.
Let π : X → Y be a generically finite surjective morphism of smooth projectivevarieties of dimension n over K . Then the following statements hold. (1) Suppose that Effective Property G r holds for all dynamical correspondences of X . Thatis, for any r ∈ Q > , the homological correspondence γ X,r = cl X × X ( G X,r ) for some G X,r ∈ Z n ( X × X ) Q , and for any dynamical correspondence f of X , the composite G X,r ◦ f is effective. Then the same holds on Y . (2) Suppose that Conjecture G r holds for all dynamical correspondences of X . That is, forany r ∈ Q > , the homological correspondence γ X,r = cl X × X ( G X,r ) for some G X,r ∈ Z n ( X × X ) Q , and for any dynamical correspondence f of X , we have k G X,r ◦ f k ≤ C X deg( G X,r ◦ f ) for some constant C X > independent of f and r . Then the sameholds on Y . (3) If Conjecture 1.4 holds for all dynamical correspondences of X , then so does for alldynamical correspondences of Y . (4) If Conjecture 1.1 holds on X , then so does on Y .Proof. Let d ∈ Z > denote the degree of the generically finite morphism π .(1) We first prove that G Y,r := d ( π × π ) ∗ ( G X,r ) can represent γ Y,r , which is by definition a ho-mological correspondence of Y such that its pullback γ ∗ Y,r on each H i ( Y ) is the multiplication-by- r i map. In fact, it follows easily from Lieberman’s Lemma 2.9 that for any α ∈ H i ( Y ) , wehave G ∗ Y,r ( α ) = 1 d (( π × π ) ∗ ( G X,r )) ∗ ( α ) = 1 d π ∗ ◦ G ∗ X,r ◦ π ∗ ( α )= 1 d π ∗ ◦ γ ∗ X,r ◦ π ∗ ( α ) = 1 d π ∗ ( r i π ∗ ( α )) = r i d π ∗ π ∗ α = r i α. Hence cl Y × Y ( G Y,r ) = γ Y,r by the Künneth formula and Poincaré duality.Secondly, let g be an arbitrary dynamical correspondence of Y . Let f := ( π × π ) ★ ( g ) bethe dynamical pullback of g , which is automatically effective; see Definition 3.7. Hence byassumption, G X,r ◦ f is effective. Note that by definition, the pullback of G − ,r on each H i ( − ) is given by the multiplication-by- r i map. It thus follows from Lemma 2.9 and Remark 3.8 that ( π × π ) ∗ ( G X,r ◦ f ) = Γ π ◦ G X,r ◦ f ◦ Γ T π = G Y,r ◦ Γ π ◦ f ◦ Γ T π = G Y,r ◦ ( π × π ) ∗ ( f ) = d G Y,r ◦ g. Therefore, G Y,r ◦ g = d ( π × π ) ∗ ( G X,r ◦ f ) is effective, as desired.(2) Suppose that Conjecture G r holds for all dynamical correspondences of X . First, wehave proved the algebraicity of γ Y,r in the assertion (1); in particular, G Y,r := d ( π × π ) ∗ ( G X,r ) . Note that since π is a projective morphism, thanks to Proposition 2.6, there is no difference between the properpushforward ( π × π ) ∗ and the pushforward of π × π as a correspondence (i.e., Definition 2.4). We just need to show that for any dynamical correspondence g of Y , we have k G Y,r ◦ g k ≤ C Y deg( G Y,r ◦ g ) for some constant C Y > independent of g and r .Let f be the dynamical pullback ( π × π ) ★ ( g ) of g . Then as in the proof of the assertion (1),we have G Y,r ◦ g = d ( π × π ) ∗ ( G X,r ◦ f ) . It thus follows that (cid:13)(cid:13) G Y,r ◦ g (cid:13)(cid:13) = 1 d (cid:13)(cid:13) ( π × π ) ∗ ( G X,r ◦ f ) (cid:13)(cid:13) ≤ d (cid:13)(cid:13) G X,r ◦ f (cid:13)(cid:13) ≤ C X d deg( G X,r ◦ f )= C X d n X i =0 (cid:18) ni (cid:19) deg i ( G X,r ◦ f ) . It remains to estimate deg i ( G X,r ◦ f ) = r i deg i ( f ) . Thanks to Lemma 3.11, there is a constant C > , independent of g and f , such that deg i ( f ) ≤ C deg i ( g ) . Now, putting all together, wehave actually shown that (cid:13)(cid:13) G Y,r ◦ g (cid:13)(cid:13) ≤ C Y deg( G Y,r ◦ g ) , where C Y := C C X /d . Namely, Conjecture G r holds for all dynamical correspondences of Y .(3) Suppose that Conjecture 1.4 holds for all dynamical correspondences of X . In otherwords, there is a constant C ′ X > such that for any ≤ k ≤ n and any dynamical correspon-dence f of X (which is in particular effective), we have (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≤ C ′ X (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) . (5.3)Now let g be an arbitrary dynamical correspondence of Y . As in the proof of the assertion (2),we let f := ( π × π ) ★ ( g ) be the dynamical pullback of g .Note that by assumption π ∗ : H k ( Y ) → H k ( X ) is an injection. Without loss of generality,we identify H k ( Y ) with the subspace π ∗ H k ( Y ) of H k ( X ) and assume that the norm k·k ι on H k ( Y ) is induced from that on H k ( X ) . In other words, these two norms are compatible withthe inclusion π ∗ . We shall use the following operator norms: (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι := sup {k f ∗ α X k ι : α X ∈ H k ( X ) , k α X k ι = 1 } = sup ((cid:12)(cid:12) f ∗ α X ∪ β X (cid:12)(cid:12) ι : α X ∈ H k ( X ) , β X ∈ H n − k ( X ) , k α X k ι = k β X k ι = 1 ) , (cid:13)(cid:13) g ∗ | H k ( Y ) (cid:13)(cid:13) ι := sup {k g ∗ α Y k ι : α Y ∈ H k ( Y ) , k α Y k ι = 1 } = sup ((cid:12)(cid:12) g ∗ α Y ∪ β Y (cid:12)(cid:12) ι : α Y ∈ H k ( Y ) , β Y ∈ H n − k ( Y ) , k α Y k ι = k β Y k ι = 1 ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 53
Now, consider α X := π ∗ α Y and β X := π ∗ β Y with α Y ∈ H k ( Y ) , β Y ∈ H n − k ( Y ) , and k α Y k ι = k β Y k ι = 1 . Then by Remark 3.8 and Lemma 2.9, we obtain that d g ∗ α Y ∪ β Y = (( π × π ) ∗ f ) ∗ α X ∪ β X = π ∗ f ∗ π ∗ α Y ∪ β Y = f ∗ π ∗ α Y ∪ π ∗ β Y = f ∗ α X ∪ β X . It follows readily that (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ι ≥ d (cid:13)(cid:13) g ∗ | H k ( Y ) (cid:13)(cid:13) ι . (5.4)On the other hand, by Remark 2.17 and Lemma 3.11, there is some constant C > so that (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) ≤ C (cid:13)(cid:13) g ∗ | N k ( Y ) R (cid:13)(cid:13) . (5.5)Putting eqs. (5.3) to (5.5) together, we prove that (cid:13)(cid:13) g ∗ | H k ( Y ) (cid:13)(cid:13) ι ≤ C Y (cid:13)(cid:13) g ∗ | N k ( Y ) R (cid:13)(cid:13) , where C Y := C C ′ X /d . Hence Conjecture 1.4 holds for all dynamical correspondences of Y .(4) Suppose that Conjecture 1.1 holds on X . That is, for any dynamical correspondence f of X , we have χ k ( f ) ι = λ k ( f ) . Let g be an arbitrary dynamical correspondence of Y and f itsdynamical pullback ( π × π ) ★ ( g ) as before. An iterated use of Lemma 3.10 yields that f ♦ m = d m − ( π × π ) ★ ( g ♦ m ) , for any m ∈ N > . Now, applying eq. (5.4) to g ♦ m and its dynamical pullback, we have (cid:13)(cid:13) (( π × π ) ★ ( g ♦ m )) ∗ | H k ( X ) (cid:13)(cid:13) ι ≥ d (cid:13)(cid:13) ( g ♦ m ) ∗ | H k ( Y ) (cid:13)(cid:13) ι . It thus follows that (cid:13)(cid:13) ( f ♦ m ) ∗ | H k ( X ) (cid:13)(cid:13) ι ≥ d m +1 (cid:13)(cid:13) ( g ♦ m ) ∗ | H k ( Y ) (cid:13)(cid:13) ι , which implies that χ k ( f ) ι ≥ d χ k ( g ) ι by definition.Similarly, applying Lemma 3.11 to g ♦ m and its dynamical pullback, we also have the follow-ing equivalence deg k ( f ♦ m ) ∼ d m − deg k ( g ♦ m ) , which yields that λ k ( f ) = d λ k ( g ) . On the other hand, it is known that χ k ( f ) ι = λ k ( f ) by assumption. Putting all together, we show that χ k ( g ) ι ≤ λ k ( g ) . The other direction iseasy; see, e.g., Lemma 4.6. Hence Conjecture 1.1 holds on Y . We thus complete the proof ofTheorem 5.7. (cid:3) Remark . Indeed, the proof of Theorem 5.7 works for more general correspondences. Weonly need the correspondence g on Y to be effective and have no irreducible components be-longing to the image of the non-flat locus N F π × π of π × π so that its pullback f = ( π × π ) ★ ( g ) ,defined in the way as dynamical pullback, is a nonzero effective correspondence of X , and that X satisfies Effective Property G r , Conjecture G r , or Conjecture 1.4 for this f . In particular, thefollowing descent theorem for finite surjective morphisms holds, whose proof is similar to thatof Theorem 5.7 and left to the reader to verify. Theorem 5.9.
Let π : X → Y be a finite surjective morphism between smooth projective vari-eties of dimension n over K . Then the following statements hold. (1) If Effective Property G r holds on X , then so does on Y . (2) If Conjecture G r holds on X , then so does on Y . (3) If Conjecture 1.4 holds on X , then so does on Y . Complex projective manifolds.
In the case when K = C , we obtain an affirmative an-swer to Conjecture 1.4 for dynamical correspondences (and hence to Conjecture 1.1 too). Wewill make use of an inequality, valid on compact Kähler manifolds, due to Dinh (see [Din05,Proposition 5.8]). While in [Din05] only the case of automorphisms is considered, the proofcan be adapted to dynamical correspondences (see also [Tru16], [Hu19b, Lemma 6.1]). Ageneralization to the case of compact complex manifolds is also available [Vu19, Lemma 4.1].Additionally, we remark that the inequality part of Conjecture G r holds for all dynamical corre-spondences of smooth complex projective varieties. We provide here a proof for completeness. Theorem 5.10.
Let X be a smooth complex projective variety of dimension n . Then there is aconstant C > so that for all dynamical correspondences f of X , we have for any ≤ k ≤ n , (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) , and for any ≤ k ≤ n − , (cid:13)(cid:13) f ∗ | H k +1 ( X ) (cid:13)(cid:13) ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13)(cid:13)(cid:13) f ∗ | N k +1 ( X ) R (cid:13)(cid:13) , where H • ( X ) is chosen to be the classical de Rham cohomology H • ( X, C ) .As a consequence, Conjecture SG r holds for all dynamical correspondences.Proof. Since f is effective, by Remark 2.17, k f ∗ | N k ( X ) R k is equivalent to deg k ( f ) which islinear in f . Thus it suffices to consider the case when f is an irreducible dynamical correspon-dence of X . In particular, by Lemma 4.2, the degree sequence { deg i ( f ) } i is log-concave.We present the proof for the estimate of k f ∗ | H k ( X ) k only, the one for k f ∗ | H k +1 ( X ) k is similar.We note that since f is a dynamical correspondence, outside of a small analytic proper subset,it is locally a finite sum of holomorphic maps. Using the Hodge decomposition H k ( X ) = M p + q =2 k H p,q ( X ) , it suffices to estimate the size of f ∗ ( α ) for α in a finite set of smooth closed ( p, q ) -forms whichgenerate H p,q ( X ) . By using the product trick of looking at F = ( f, f ) and α ∧ α , we obtainas in the proof of [Din05, Proposition 5.8] - which is local in nature - that there is a constant C > , independent of f , so that k f ∗ ( α ) k ≤ C deg p ( f ) deg q ( f ) . Taking supremum on the finite set of such forms α , as mentioned above, we obtain that (cid:13)(cid:13) f ∗ | H p,q ( X ) (cid:13)(cid:13) ≤ C deg p ( f ) deg q ( f ) . TANDARD CONJECTURES AND DYNAMICAL DEGREES 55
Since f is assumed to be irreducible, the degree sequence j deg j ( f ) is log-concave byLemma 4.2. Therefore, one can easily check that deg p ( f ) deg k − p ( f ) ≤ deg k ( f ) . Takingmaximum over all ≤ p ≤ n and using that deg k ( f ) ∼ k f ∗ | N k ( X ) R k when f is effective, weobtain that (cid:13)(cid:13) f ∗ | H k ( X ) (cid:13)(cid:13) ≤ C (cid:13)(cid:13) f ∗ | N k ( X ) R (cid:13)(cid:13) , as desired.The assertion about the validity of Conjecture SG r follows as in the proof of Theorem 1.12(3);see eq. (4.7). (cid:3) Remark . As a consequence of Theorem 5.10 and Theorem 1.12(3), in the complex case,our Conjecture G r for dynamical correspondences follows from the standard conjecture C .5.5. Algebraically stable correspondences.
An immediate corollary of Theorem 1.9 is fordynamical correspondences that are algebraically k -stable (see Definition 3.4). Corollary 5.12.
Let X be a smooth projective variety of dimension n over K such that Con-jecture G r holds. Fix an integer ≤ k ≤ n . Let f be an algebraically k -stable dynamicalcorrespondence of X . Then we have χ k ( f ) ι = λ k ( f ) .Proof. By applying Theorem 1.9(1) to the m -th dynamical iterates f ♦ m of f , we have (cid:12)(cid:12) Tr(( f ♦ m ) ∗ | H k ( X ) ) (cid:12)(cid:12) . deg k ( f ♦ m ) . Since f is algebraically k -stable, it follows that (cid:12)(cid:12) Tr(( f ∗ ) m | H k ( X ) ) (cid:12)(cid:12) . deg k ( f ♦ m ) . Taking the limits of m -th roots of both sides, we obtain that ρ ( f ∗ | H k ( X ) ) ≤ λ k ( f ) . On theother hand, by the spectral radius formula and the definition of algebraic stability, we also have χ k ( f ) ι = lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | H k ( X ) (cid:13)(cid:13) /mι = ρ ( f ∗ | H k ( X ) ) ,λ k ( f ) = lim m →∞ (cid:13)(cid:13) ( f ∗ ) m | N k ( X ) R (cid:13)(cid:13) /m = ρ ( f ∗ | N k ( X ) R ) . Putting all together, we thus prove that χ k ( f ) ι = ρ ( f ∗ | H k ( X ) ) ≤ ρ ( f ∗ | N k ( X ) R ) = λ k ( f ) . The converse inequality is easy by Lemma 4.6. So Corollary 5.12 follows. (cid:3)
Invariance of χ under birational morphisms. In Theorem 5.14, we show that the sec-ond cohomological dynamical degree χ ( · ) ι is invariant under birational morphisms. This willbe used to prove Theorem 1.19 subsequently. Lemma 5.13.
Let π : X → Y be a birational morphism of smmoth projective varieties over K .Then we have H ( X, Q ℓ ) = π ∗ H ( Y, Q ℓ ) ⊕ h cl X ( E ) , cl X ( E ) , . . . i , where h cl X ( E ) , cl X ( E ) , . . . i is the Q ℓ -span of exceptional divisor classes in H ( X, Q ℓ ) . Proof.
Note that in our situation the exceptional locus E of π has pure codimension one in X ;in other words, E = ∪ k E k . Also, since Y is a smooth K -variety, the indeterminacy locus I π − of π − has codimension ≥ in Y . Let Z := I π − , U := X r E , and V := Y r Z . Denote theclosed immersion E ֒ → X by i and the open immersion U ֒ → X by j . When restricting π to U , we have an isomorphism π | U : U → V . Indeed, since here π is a morphism, the graph Γ π isisomorphic to X . Now, outside the indeterminacy locus of π − , the graph Γ π − is isomorphicto Y . Since π is a birational map, we have Γ π − = Γ T π . Thanks to [Mil13, Theorem 9.4], wehave the following commutative diagram of long exact sequences: H ( V, Q ℓ ) H Z ( Y, Q ℓ ) H ( Y, Q ℓ ) H ( V, Q ℓ ) H Z ( Y, Q ℓ ) H ( U, Q ℓ ) H E ( X, Q ℓ ) H ( X, Q ℓ ) H ( U, Q ℓ ) H E ( X, Q ℓ ) . ≃ ≃ π ∗ ≃ i ∗ j ∗ Since codim Y ( Z ) ≥ , by the semi-purity Lemma 23.1 in [Mil13], we have H Z ( Y, Q ℓ ) = H Z ( Y, Q ℓ ) = 0 . Then a standard diagram chasing yields that H ( Y, Q ℓ ) ≃ H ( V, Q ℓ ) , i ∗ isinjective, and j ∗ is surjective.Note that by the definition of the cycle class map cl X (see [Mil13, §23, after Lemma 23.1]), i ∗ H E ( X, Q ℓ ) is exactly generated by the cl X ( E k ) . We claim that H ( X, Q ℓ ) = π ∗ H ( Y, Q ℓ ) ⊕ i ∗ H E ( X, Q ℓ ) . First, the right-hand side is a direct sum contained in H ( X, Q ℓ ) . Indeed, any nontrivial linearcombination of exceptional divisors E k cannot be a pullback of a divisor on Y under π , sincethe pushforward by π of such one is . Then it is easy to have the equality by the dimensionreason. We thus prove Lemma 5.13. (cid:3) Theorem 5.14.
Let π : X → Y be a birational morphism of smooth projective varieties ofdimension n over K . Let g be a dynamical correspondence of Y . Let f be the dynamicalpullback ( π × π ) ★ ( g ) of g under π × π (see Definition 3.7). Then we have χ ( f ) ι = χ ( g ) ι .Proof. Since the definitions of dynamical composition and dynamical pullback are generic, itis easy to verify that f ♦ m = ( π × π ) ★ ( g ♦ m ) for all m (see also Lemma 3.10). Note that ( π × π ) ∗ ( f ♦ m ) = g ♦ m by Remark 3.8. Then, as in the proof of Theorem 5.7(3), one can easilydeduce that (cid:13)(cid:13) ( g ♦ m ) ∗ | H ( Y ) (cid:13)(cid:13) ι ≤ (cid:13)(cid:13) (( π × π ) ★ ( g ♦ m )) ∗ | H ( X ) (cid:13)(cid:13) ι = (cid:13)(cid:13) ( f ♦ m ) ∗ | H ( X ) (cid:13)(cid:13) ι . Taking limsups of m -th roots of both sides yields that χ ( g ) ι ≤ χ ( f ) ι . It remains to prove the reverse inequality χ ( f ) ι ≤ χ ( g ) ι . First, by Lemma 5.13, we have H ( X ) = π ∗ H ( Y ) ⊕ ρ ′ M k =1 Q ℓ e k , In fact, this type of inequality holds for all degrees, i.e., χ i ( g ) ι ≤ χ i ( f ) ι for all i . TANDARD CONJECTURES AND DYNAMICAL DEGREES 57 where ρ ′ := ρ ( X/Y ) is the relative Picard number of X/Y , e , . . . , e ρ ′ are (linearly indepen-dent) π -exceptional divisor classes. Note the standard conjecture D holds for divisors by Mat-susaka [Mat57]. Hence we can identify N ( · ) Q ℓ with the subgroup H ( · ) of H ( · ) generatedby divisor classes and have N ( X ) Q ℓ = π ∗ N ( Y ) Q ℓ ⊕ ρ ′ M k =1 Q ℓ e k . By duality, the above also holds for one cycle classes so that we have similar decompositionsfor H n − ( X ) and N n − ( X ) Q ℓ ; we can also identify N n − ( · ) Q ℓ with the subgroup H n − ( · ) of H n − ( · ) generated by one cycle classes.Recall that the "transcendental" part H ( · ) of H ( · ) is defined by H ( · ) := H n − ( · ) ⊥ := { α ∈ H ( · ) : α ∪ β = 0 , for all β ∈ H n − ( · ) } . Then by Lemma A.4, we have the following direct sum decompositions H ( X ) = H ( X ) ⊕ H ( X ) = N ( X ) Q ℓ ⊕ H ( X ) , and H ( Y ) = H ( Y ) ⊕ H ( Y ) = N ( Y ) Q ℓ ⊕ H ( Y ) . In summary, we show that there is a Q -basis { α , . . . , α ρ } of N ( Y ) Q , which extends to afull Q ℓ -basis of H ( Y ) by adding a Q ℓ -basis { α ρ +1 , . . . , α b } of H ( Y ) ; moreover, { π ∗ α , . . . , π ∗ α ρ , e , . . . , e ρ ′ } is a Q -basis of N ( X ) Q , and { π ∗ α , . . . , π ∗ α ρ , π ∗ α ρ +1 , . . . , π ∗ α b , e , . . . , e ρ ′ } forms a Q ℓ -basis of H ( X ) . Similarly, by duality, we can find an appropriate dual basis { π ∗ β , . . . , π ∗ β ρ , π ∗ β ρ +1 , . . . , π ∗ β b , c , . . . , c ρ ′ } of H n − ( X ) . In fact, we let { β , . . . , β ρ } be the dual Q -basis of N n − ( Y ) R , let { β , . . . , β b } be the dual Q ℓ -basis of H n − ( Y ) , and let { c , . . . , c ρ ′ } be the relative -cycle classes dual tothe e k . Note that e k ∪ π ∗ β i = π ∗ e k ∪ β i = 0 and π ∗ α Y ∪ c k = α Y ∪ π ∗ c k = 0 for all i, k bythe projection formula. By Poincaré duality, one can check that the following assignments giverise to norms k·k ′ ι on H ( Y ) and H ( X ) : k α Y k ′ ι := b X i =1 (cid:12)(cid:12) α Y ∪ β i (cid:12)(cid:12) ι , k α X k ′ ι := b X i =1 (cid:12)(cid:12) α X ∪ π ∗ β i (cid:12)(cid:12) ι + ρ ′ X k =1 (cid:12)(cid:12) α X ∪ c k (cid:12)(cid:12) ι , for any α Y ∈ H ( Y ) and α X ∈ H ( X ) , respectively; see eq. (2.4) for a comparison. Here, ρ := ρ ( Y ) denotes the Picard number of Y and b := b ( Y ) is the second ℓ -adic Betti number of Y . We are now ready to give an appropriate estimate of the operator norm of f ∗ | H ( X ) inducedfrom the above k · k ′ ι . Let α X be an arbitrary element in H ( X ) with k α X k ′ ι = 1 , written as α X = b X i =1 a i π ∗ α i + ρ ′ X k =1 a b + k e k , for some a i , a b + k ∈ Q ℓ . Set α Y := P b i =1 a i α i ∈ H ( Y ) so that α X = π ∗ α Y + P ρ ′ k =1 a b + k e k .Then by definition, one can easily verify that k α X k ′ ι = b + ρ ′ X i =1 | a i | ι = k α Y k ′ ι + ρ ′ X k =1 | a b + k | ι . Note that ( π × π ) ∗ ( f ) = g and hence g ∗ = π ∗ f ∗ π ∗ by Lieberman’s Lemma 2.9. It follows that g ∗ α Y ∪ β i = π ∗ f ∗ π ∗ α Y ∪ β i = f ∗ π ∗ α Y ∪ π ∗ β i for any ≤ i ≤ b . In particular, one has (cid:13)(cid:13) g ∗ α Y (cid:13)(cid:13) ′ ι = b X i =1 (cid:12)(cid:12) g ∗ α Y ∪ β i (cid:12)(cid:12) ι = b X i =1 (cid:12)(cid:12) f ∗ π ∗ α Y ∪ π ∗ β i (cid:12)(cid:12) ι . We then deduce that (cid:13)(cid:13) f ∗ α X (cid:13)(cid:13) ′ ι = b X i =1 (cid:12)(cid:12) f ∗ α X ∪ π ∗ β i (cid:12)(cid:12) ι + ρ ′ X k =1 (cid:12)(cid:12) f ∗ α X ∪ c k (cid:12)(cid:12) ι ≤ b X i =1 (cid:12)(cid:12) f ∗ π ∗ α Y ∪ π ∗ β i (cid:12)(cid:12) ι + b X i =1 ρ ′ X k =1 (cid:12)(cid:12) a b + k f ∗ e k ∪ π ∗ β i (cid:12)(cid:12) ι + b X i =1 ρ ′ X k =1 (cid:12)(cid:12) a i f ∗ π ∗ α i ∪ c k (cid:12)(cid:12) ι + ρ ′ X j =1 ρ ′ X k =1 (cid:12)(cid:12) a b + k f ∗ e k · c j (cid:12)(cid:12) ι ≤ (cid:13)(cid:13) g ∗ α Y (cid:13)(cid:13) ′ ι + b X i =1 ρ ′ X k =1 (cid:12)(cid:12) f ∗ e k · π ∗ β i (cid:12)(cid:12) + b X i =1 ρ ′ X k =1 (cid:12)(cid:12) f ∗ π ∗ α i · c k (cid:12)(cid:12) + ρ ′ X j =1 ρ ′ X k =1 (cid:12)(cid:12) f ∗ e k · c j (cid:12)(cid:12) ≤ (cid:13)(cid:13) g ∗ α Y (cid:13)(cid:13) ′ ι + C (2 b ρ ′ + ρ ′ ) (cid:13)(cid:13) f ∗ | N ( X ) R (cid:13)(cid:13) ′ op , where the first inequality follows directly from the explicit expansion of f ∗ α X using α X = π ∗ α Y + P ρ ′ k =1 a b + k e k , in the second inequality the vanishing of f ∗ e k ∪ π ∗ β i and f ∗ π ∗ α i ∪ c k for ρ < i ≤ b follows from the definition of H ( X ) (see eq. (A.5); note also that | a i | ι , | a b + k | ι ≤ ), while the last inequality follows from the definition of the dual operator norm k·k ′ op ofcorrespondences (see eq. (2.12)), and C > is a constant depending only on the well-chosenbases of H ( X ) and H n − ( X ) . Now, one can easily see from the definition of the operatornorm in Linear Algebra that (cid:13)(cid:13) f ∗ | H ( X ) (cid:13)(cid:13) ′ op ,ι ≤ (cid:13)(cid:13) g ∗ | H ( Y ) (cid:13)(cid:13) ′ op ,ι + C (2 b ρ ′ + ρ ′ ) (cid:13)(cid:13) f ∗ | N ( X ) R (cid:13)(cid:13) ′ op . TANDARD CONJECTURES AND DYNAMICAL DEGREES 59
Applying the above estimate to the m -th dynamical iterates f ♦ m = ( π × π ) ★ ( g ♦ m ) yields that (cid:13)(cid:13) ( f ♦ m ) ∗ | H ( X ) (cid:13)(cid:13) ′ op ,ι ≤ (cid:13)(cid:13) ( g ♦ m ) ∗ | H ( Y ) (cid:13)(cid:13) ′ op ,ι + C (2 b ρ ′ + ρ ′ ) (cid:13)(cid:13) ( f ♦ m ) ∗ | N ( X ) R (cid:13)(cid:13) ′ op . By taking the limsups of the m -th roots as usual, we obtain that χ ( f ) ι ≤ max { χ ( g ) ι , λ ( f ) } .Since the numerical dynamical degrees λ k are birational invariants (see, e.g., Lemma 3.11), wehave that λ ( f ) = λ ( g ) ≤ χ ( g ) ι . Combining the above two together, we finally prove that χ ( f ) ι ≤ χ ( g ) ι and hence conclude the proof of Theorem 5.14. (cid:3) Remark . If resolution of singularities exists, then Theorem 5.14 can be extended to showthat χ is a birational invariant (i.e., invariant under any birational map). This is the casein characteristic zero, thanks to Hironaka’s resolution of singularities [Hir64]. In positivecharacteristic, resolution of singularities exists only up to dimension three by many people[Zar39, Abh56, Cut09, CP08, CP09]; it is still an open question in dimension ≥ . See Sec-tion 5.7 for a detailed discussion in the case of surfaces and threefolds.5.7. Surfaces and threefolds.
We first prove Theorem 1.19, which applies to surfaces andthreefolds. Then we use it to give a list of surfaces that can be shown to satisfy Conjecture 1.1or the stronger Conjecture 1.4. At the end, we show that Conjecture 1.2 holds on Kummersurfaces.
Proof of Theorem 1.19. (1) It is known that resolutions of singularities exists for surfaces andthreefolds in any characteristic (see [Zar39, Abh56, Hir64, Cut09, CP08, CP09]). Hence, byresolving singularities of the graph Γ π there is a smooth projective variety W over K , togetherwith two birational morphisms p : W → X , p : W → Y . Let h be the dynamical pullback of f under p × p , and let h be the dynamical pullback of g under p × p . Then it is easy to checkthat h = h . Note that both p and p are birational morphisms. Therefore, by Theorem 5.14,we obtain that χ ( f ) ι = χ ( h ) ι = χ ( h ) ι = χ ( g ) ι . If X and Y are surfaces, then χ ( f ) = χ ( g ) are topological degrees. If X and Y arethreefolds, we can see that χ ( f ) ι = χ ( g ) ι by considering their canonical transposes. Indeed,let g T and f T denote the transpose of g and f , respectively. Then f T is exactly the dynamicalpullback of g T under π × π . From what we just proved, one has χ ( f T ) ι = χ ( g T ) ι . On the otherhand, by duality, we have χ ( f ) ι = χ ( f T ) ι and χ ( g ) ι = χ ( g T ) ι . Hence χ ( f ) ι = χ ( g ) ι follows.(2) By resolution of singularities (valid for surfaces and threefolds), there is a smooth pro-jective variety W and a birational morphism W → X so that the composition W → Y is asurjective morphism. By the assertion (1) and the fact that λ ( f ) and λ ( f ) are birational invari-ants (see, e.g., Lemma 3.11), it follows that Conjecture 1.1 holds on W . Then Theorem 5.7(4)asserts that Conjecture 1.1 holds on Y as well, which concludes the proof of Theorem 1.19. (cid:3) Remark . We now deduce more explicit results on surfaces. The characteristic zero caseis handled by Theorem 5.10. Note that by a result of Shioda [Shi74], uniruled surfaces are
Shioda-supersingular (i.e., the Picard number and the second Betti number coincide), so thatour Conjectures 1.1, 1.3 and 1.4 hold automatically. We then consider surfaces with nonneg-ative Kodaira dimension in positive characteristic (see [Lie13] for a survey on algebraic sur-faces). All Enriques surfaces (including nonclassical ones), quasi-elliptic surfaces, and (quasi-)hyperelliptic surfaces are also Shioda-supersingular (see [Lie13, §5 and §7]), so that the samehold similar with the uniruled case.We next consider an Abelian surface A over K and choose a prime ℓ = p := char( K ) suchthat Conjecture D ( A × A ) holds by [Clo99, Anc21]. It follows from Lemma 5.3 and Corol-lary 1.17 that Conjecture 1.1 holds on A . Moreover, the inequalities in Theorem 1.12(2) alsohold. In particular, Conjecture 1.1 also holds on the associated Kummer surface S of the aboveAbelian surface A (for p = 2 ). Indeed, by the construction, S is the minimal resolution of thequotient A/ h− i of A by the sign involution. We thus have a dominant rational map A S .The claim then follows from Theorem 1.19(2). The remaining cases are elliptic surfaces ofKodaira dimension , surfaces of general type, and (general) K surfaces.As the last application of our Conjecture G r , we show, without using the standard conjecture D for Abelian fourfolds (which is known on ℓ -adic étale cohomology for infinitely many primes ℓ by [Clo99, Anc21]), that Conjecture 1.2 holds for all surfaces dominated by birational modelsof Abelian surfaces. Proposition 5.17.
Let A S be a dominant rational map from an Abelian surface A to asmooth projective surface S over K . Fix a Weil cohomology theory H • ( − ) . Then the followingstatements hold. (1) There is a constant
C > such that for any dynamical correspondence f of S , we have (cid:12)(cid:12) Tr( f ∗ | H k ( S ) ) (cid:12)(cid:12) ≤ C deg k ( f ) , (cid:12)(cid:12) Tr( f ∗ | H k +1 ( S ) ) (cid:12)(cid:12) ≤ C q deg k ( f ) deg k +1 ( f ) . (5.6)(2) Conjecture 1.1 holds on S for dynamical correspondences that are totally algebraicallystable. (3) Conjecture 1.2 holds on S (for polarized endomorphisms). (4) Conjecture G r holds on S for dynamical correspondences.Proof. (1) By resolution of singularities, there exist a birational morphism τ : e A → A from asmooth projective surface e A to A so that the composite π : e A → S is a morphism. Let f bea dynamical correspondence of S . Then we denote the dynamical pullback ( π × π ) ★ ( f ) of f under π × π by g , which is a dynamical correspondence of e A by definition. We further denotethe proper pushforward ( τ × τ ) ∗ ( g ) of g under τ × τ by h . It is easy to check that h is a TANDARD CONJECTURES AND DYNAMICAL DEGREES 61 dynamical correspondence of A . We thus have the following diagram A ❴ h e A τ o o π / / ❴ g S ❴ f A e A τ o o π / / S. First, it is well-known that the standard conjecture C holds on surfaces (see, e.g., [Kle68,Corollary 2A10]). Note by Remark 3.8 that ( π × π ) ∗ ( g ) = d f , where d := deg( π ) . So by theLefschetz trace formula (see Proposition 2.8), we have d (cid:12)(cid:12) Tr( f ∗ | H i ( S ) ) (cid:12)(cid:12) = d (cid:12)(cid:12) f · ∆ − i (cid:12)(cid:12) = (cid:12)(cid:12) ( π × π ) ∗ ( g ) · ∆ − i (cid:12)(cid:12) = (cid:12)(cid:12) g · ( π × π ) ∗ ∆ − i (cid:12)(cid:12) . We note that the dynamical pullback ( τ × τ ) ★ ( h ) coincides with the strict transform ( τ × τ ) − ∗ ( h ) of h under ( τ × τ ) − by definition and hence both are equal to g since g is a dynamicalcorrespondence. Moreover, the total inverse image ( τ × τ ) − ( h ) of h under the birationalmorphism τ × τ has dimension . Indeed, the difference ( τ × τ ) − ( h ) − ( τ × τ ) ★ ( h ) issupported inside the exceptional divisors E × e A or e A × E for some exceptional curve E of τ .As h is a dynamical correspondence of A , any irreducible component of the difference cannotbe an exceptional divisor of τ × τ , so has dimension . Therefore, the pullback ( τ × τ ) ∗ ( h ) of h under τ × τ is supported on ( τ × τ ) − ( h ) and the difference e := ( τ × τ ) ∗ ( h ) − g is an effective correspondence of e A . It follows that d (cid:12)(cid:12) Tr( f ∗ | H i ( S ) ) (cid:12)(cid:12) = (cid:12)(cid:12) (( τ × τ ) ∗ ( h ) − e ) · ( π × π ) ∗ ∆ − i (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( τ × τ ) ∗ ( h ) · ( π × π ) ∗ ∆ − i (cid:12)(cid:12) + (cid:12)(cid:12) e · ( π × π ) ∗ ∆ − i (cid:12)(cid:12) = (cid:12)(cid:12) h · ( τ × τ ) ∗ ( π × π ) ∗ ∆ − i (cid:12)(cid:12) + (cid:12)(cid:12) e · ( π × π ) ∗ ∆ − i (cid:12)(cid:12) . It remains to estimate the above two summands. Note that ( τ × τ ) ∗ ( π × π ) ∗ ∆ − i ∈ H i ( A ) ⊗ H − i ( A ) and h is a dynamical correspondence of A . Hence thanks to Lemma 5.3, we have r i (cid:12)(cid:12) h · ( τ × τ ) ∗ ( π × π ) ∗ ∆ − i (cid:12)(cid:12) = (cid:12)(cid:12) ( G r ◦ h ) · ( τ × τ ) ∗ ( π × π ) ∗ ∆ − i (cid:12)(cid:12) . (cid:13)(cid:13) G r ◦ h (cid:13)(cid:13) . deg( G r ◦ h ) ∼ max ≤ j ≤ r j deg j ( h ) ∼ max ≤ j ≤ r j d deg j ( f ) , where deg j ( h ) ∼ deg j ( g ) ∼ d deg j ( f ) follows from Lemma 3.11. Secondly, as our e issupported inside the exceptional divisors E × e A or e A × E for some exceptional curve E of τ , thecohomology class of e belongs to H ( e A ) ⊗ H ( e A ) . Hence, if i = 2 , then | e · ( π × π ) ∗ ∆ − i | = 0 ,while for i = 2 we have by Lemma 2.19 that (cid:12)(cid:12) e · ( π × π ) ∗ ∆ (cid:12)(cid:12) . deg ( e ) , which is bounded above by deg (( τ × τ ) ∗ ( h )) = ( τ × τ ) ∗ ( h ) · pr ∗ H e A · pr ∗ H e A = h · ( τ × τ ) ∗ (pr ∗ H e A · pr ∗ H e A ) . Note that ( τ × τ ) ∗ (pr ∗ H e A · pr ∗ H e A ) ∈ N ( A ) R ⊗ N ( A ) R . Hence using the dual operator norm(2.12) and Remark 2.17, one has h · ( τ × τ ) ∗ (pr ∗ H e A · pr ∗ H e A ) . (cid:13)(cid:13) h ∗ | N ( A ) R (cid:13)(cid:13) ′ op ∼ deg ( h ) ∼ d deg ( f ) . Now, putting the above together, Lemma 4.9 yields eq. (5.6).(2) Let f be a dynamical correspondence of S that is totally algebraically stable. Underthe same notation as in (1), we note that ( τ × τ ) ★ ( h ♦ m ) = g ♦ = d m − ( π × π ) ★ ( f ♦ m ) (seeLemma 3.10). Then by applying the above argument to f ♦ m and taking limits of m -th roots,we have that χ k ( f ) ι ≤ λ k ( f ) . The other direction is easy by Lemma 4.6. Hence Conjecture 1.1holds for this f .(3) Let f be a polarized endomorphism of S , i.e., f ∗ H S ∼ rat qH S for an ample divisor H S and a positive integer q . To prove Conjecture 1.2 holds for S , it suffices to show that theeigenvalues of f ∗ | H i ( X, Q ℓ ) all have absolute value q i/ for any field isomorphism ι : Q ℓ ≃ C ,equivalently, χ i ( f ) ι ≤ q i/ for any ι by Poincaré duality. This has just been proved in (2) forall dynamical correspondences of S that are totally algebraically stable.(4) The proof is similar to the proof of (1). Let f be a dynamical correspondence of S . Sincethe standard conjecture C holds on surfaces, we just need to show that for any r ∈ Q > , (cid:13)(cid:13) G r ◦ f (cid:13)(cid:13) . deg( G r ◦ f ) ∼ max ≤ j ≤ r j deg j ( f ) . Without loss of generality, we can use the dual norm (2.9) for G r ◦ f . Choose a basis β t for N ( S × S ) R . It suffices to show that for any β t , (cid:12)(cid:12) ( G r ◦ f ) · β t (cid:12)(cid:12) . max ≤ j ≤ r j deg j ( f ) . Using the Künneth projectors ∆ i , we decompose β t as P i =0 β t,i , where β t,i := ∆ i ◦ β t whosecohomology class is in H i ( S ) ⊗ H − i ( S ) . The same argument in (1) with ∆ i replaced by β t,i shows that for even i = 2 k , we have | f · β t, − i | . deg k ( f ) ; for odd i = 2 k + 1 , we have r k +1 | f · β t, − i | . r k +1 p deg k ( f ) deg k +1 ( f ) ≤ max j = k,k +1 r j deg j ( f ) . It thus follows that (cid:12)(cid:12) ( G r ◦ f ) · β t (cid:12)(cid:12) ≤ X i =0 (cid:12)(cid:12) ( G r ◦ f ) · β t, − i (cid:12)(cid:12) ≤ X i =0 r i (cid:12)(cid:12) f · β t, − i (cid:12)(cid:12) . max ≤ j ≤ r j deg j ( f ) . This shows that Conjecture G r holds on S for dynamical correspondences and hence concludesthe proof of Proposition 5.17. (cid:3) Remark . Let A be an Abelian threefold, and G a finite group acting faithfully on A . Let X be a resolution of singularities of A/G . Then we obtain a finite degree dominant rational map A X . An interesting example is when E is an elliptic curve, A = E , and G ≤ Aut( E ) acts on A diagonally. See [Cam11, COT14, OT15, CT15] for more on this kind of constructionswith applications in primitive automorphisms of positive entropy on rational threefolds. Untilnow, there are only two rational threefolds shown to have such automorphisms, both of themare of the mentioned type. When A is defined over some finite field, we fix a prime number ℓ TANDARD CONJECTURES AND DYNAMICAL DEGREES 63 for which Conjecture D ( A × A ) holds [Clo99]. Then Conjecture 1.1 holds for A . Hence, itfollows from Theorem 1.19(2) that Conjecture 1.1 holds for the above threefold X as well.A PPENDIX
A. A
BRIEF REVIEW OF THE STANDARD CONJECTURES
In this section, we give a brief review of the standard conjectures and their current status.We refer the reader to Kleiman’s survey articles [Kle68, Kle94] for details and to the books[And04, MNP13] for a modern account.Let X be a smooth projective variety of dimension n defined over K and let H X be a fixedample divisor on X . We also fix a Weil cohomology theory X H • ( X ) with coefficients ina field F of characteristic zero (e.g., Q , Q ℓ , or C ). Let L : H i ( X ) → H i +2 ( X ) , α Lα := α ∪ cl X ( H X ) be the Lefschetz operator .By the hard Lefschetz theorem (see [Del74, KM74]), for any ≤ i ≤ n , the ( n − i ) -th iterateof the Lefschetz operator L is an isomorphism L n − i : H i ( X ) ∼ −−→ H n − i ( X ) . However, L n − i +1 : H i ( X ) → H n − i +2 ( X ) may have a nontrivial kernel. Denote by P i ( X ) theset of elements α ∈ H i ( X ) , called primitive , satisfying L n − i +1 ( α ) = 0 , namely, P i ( X ) := Ker( L n − i +1 : H i ( X ) → H n − i +2 ( X )) ⊆ H i ( X ) . This will give us the following primitive decomposition: H i ( X ) = M j ≥ max( i − n, L j P i − j ( X ) . For any α ∈ H i ( X ) , write α = X j ≥ max( i − n, L j ( α j ) , α j ∈ P i − j ( X ) . We then define an operator
Λ : H i ( X ) → H i − ( X ) for ≤ i ≤ n by Λ α := X j ≥ max( i − n, L j − ( α j ) . (A.1)We also define an operator ∗ : H i ( X ) → H n − i ( X ) for ≤ i ≤ n by ∗ α := X j ≥ max( i − n, ( − ( i − j )( i − j +1)2 L n − i + j ( α j ) . (A.2)For any ≤ k ≤ n , let A k ( X ) denote the Q -vector space of algebraic cycles of codimension k on X modulo homological equivalence, i.e., A k ( X ) := Im(cl X : Z k ( X ) Q −→ H k ( X )) . (A.3) Let H k alg ( X ) denote the F -vector subspace of H k ( X ) generated by algebraic cycles, i.e., H k alg ( X ) := A k ( X ) ⊗ Q F ⊆ H k ( X ) . (A.4)Note that dim F H k alg ( X ) ≤ dim F H k ( X ) is finite. However, a priori, we do not know whether dim Q A k ( X ) < ∞ (if we do not assume the standard conjecture D ).Clearly, by the hard Lefschetz theorem, the restriction of L n − k to A k ( X ) is injective. Asan analog of the hard Lefschetz theorem, Grothendieck’s first standard conjecture of Lefschetztype predicts that it is surjective too. Conjecture A ( X, L ) . For any ≤ k ≤ n/ , the restriction map L n − k : A k ( X ) → A n − k ( X ) is an isomorphism. Now, we can view linear operators Λ and ∗ on H • ( X ) as elements of H n − ( X × X ) and H n ( X × X ) , respectively, by the Künneth formula and Poincaré duality, hence as homologicalcorrespondences in a broader sense. Below is another standard conjecture of Lefschetz type. Conjecture B ( X ) . The homological correspondence Λ is algebraic, i.e., Λ = cl X × X ( Z ) forsome Z ∈ Z n − ( X × X ) Q . Equivalently, the homological correspondence ∗ is also algebraic. Recall that by the Künneth formula we have the following decomposition of the diagonalclass: cl X × X (∆ X ) = n X i =0 π i , where π i ∈ H i ( X ) ⊗ H n − i ( X ) corresponds to the i -th projection operator π i : H • ( X ) → H i ( X ) via pullback for each i . Below is the standard conjecture of Künneth type. Conjecture C ( X ) . The Künneth components π i of the diagonal class cl X × X (∆ X ) are al-gebraic. Namely, there are rational algebraic cycles ∆ i ∈ Z n ( X × X ) Q such that π i =cl X × X (∆ i ) . Proposition A.1 (cf. [Kle94, §4]) . We have the following implications: (1)
Conjecture B ( X ) = ⇒ Conjecture A ( X, L ) +
Conjecture C ( X ) . (2) Conjecture B ( X ) + Conjecture B ( Y ) = ⇒ Conjecture B ( X × Y ) . (3) Conjecture C ( X ) + Conjecture C ( Y ) = ⇒ Conjecture C ( X × Y ) . (4) Conjecture A ( X × X, L ⊗ ⊗ L ) = ⇒ Conjecture B ( X ) .Remark A.2 . (1) Conjecture B ( X ) holds for all X ⇐⇒ Conjecture A ( X, L ) holds for all X .(2) Conjecture B ( X ) holds if X is a curve, a surface (see [Kle68, Corollary 2A10]), or anAbelian variety by Lieberman (see, e.g., [Kle68, Theorem 2A11]). Over C , in general,Conjecture B ( X ) follows from the famous Hodge conjecture.(3) If X is defined over a finite field F q , Conjecture C ( X ) holds as a consequence of Weil’sRiemann hypothesis proved by Deligne [Del74] (see [KM74, Theorem 2]). TANDARD CONJECTURES AND DYNAMICAL DEGREES 65
Conjecture D k ( X ) . Homological equivalence of algebraic cycles of codimension k on X co-incides with numerical equivalence. Precisely, for any rational algebraic cycle Z ∈ Z k ( X ) Q ,if Z ≡ in N k ( X ) Q , then cl X ( Z ) = 0 in H k ( X ) . Proposition A.3 (cf. [Kle68, Proposition 3.6]) . For any ≤ k ≤ n , the following assertionson X are equivalent. (1) Conjecture D k ( X ) holds. (2) The canonical pairing A k ( X ) ⊗ Q A n − k ( X ) → Q is non-degenerate. (3) The cycle class map cl X induces a homomorphism, necessarily injective, N k ( X ) ⊗ Z F ֒ −→ H k ( X ) . Conjecture D ( X ) . Conjecture D k ( X ) hold for all ≤ k ≤ n . By Poincaré duality, one can define the "transcendental" part of H k ( X ) by H k tr ( X ) := H n − k alg ( X ) ⊥ := { α ∈ H k ( X ) : α ∪ β = 0 , for all β ∈ H n − k alg ( X ) } . (A.5)In other words, H k tr ( X ) is the kernel of the following natural linear map: H k ( X ) −→ Hom F ( H n − k alg ( X ) , F ) , α ( β α ∪ β ) . (A.6) Lemma A.4.
Suppose that Conjecture D k ( X ) holds. Then we have a direct sum decompositionof H k ( X ) as follows: H k ( X ) = H k alg ( X ) M H k tr ( X ) . Proof.
Note that H n − k alg ( X ) = A n − k ( X ) ⊗ Q F . Hence there is a Q -basis of H n − k alg ( X ) consisting of algebraic k -cycle classes β j . By Proposition A.3, there exists a dual basis of H k alg ( X ) , denoted by α i so that α i ∪ β j = δ ij . Using this one can easily verify that the right-hand side is a direct sum. Again by Proposition A.3, we know that the homomorphism eq. (A.6)is surjective since H k ( X ) ⊇ H k alg ( X ) ≃ Hom F ( H n − k alg ( X ) , F ) . It follows that dim F H k ( X ) = dim F H k tr ( X ) + dim F Hom F ( H n − k alg ( X ) , F )= dim F H k tr ( X ) + dim F H n − k alg ( X )= dim F H k tr ( X ) + dim F H k alg ( X ) . This shows that H k ( X ) = H k alg ( X ) ⊕ H k tr ( X ) . (cid:3) Remark
A.5 . (1) Conjecture D ( X ) holds for divisors on X (see Matsusaka [Mat57]).(2) In characteristic zero, Conjecture D ( X ) holds for Abelian varieties by Conjecture Hdg( X ) and Lieberman’s result on Conjecture B ( X ) ; see Remark A.7(1) and Remark A.2(2).(3) For Abelian varieties over finite fields of prime characteristic p , Clozel [Clo99] proved thatthere are infinitely many primes ℓ = p (in fact, a set of primes of positive density), suchthat Conjecture D ( X ) holds for ℓ -adic étale cohomology. (4) Over C , Lieberman proved that Conjecture D ( X ) holds, not only for divisors, but also forcycles of codimension (see [Lie68, Corollary 1]).For any ≤ k ≤ n/ , denote by A k prim ( X ) := A k ( X ) ∩ P k ( X ) = { α ∈ A k ( X ) : L n − k +1 ( α ) = 0 } the set of primitive classes in H k ( X ) generated by algebraic cycles of codimension k . Wedefine a symmetric bilinear form on A k prim ( X ) as follows: A k prim ( X ) × A k prim ( X ) −→ Q ( α, β ) ( − k L n − k ( α ) ∪ β. (A.7) Conjecture
Hdg( X ) . The above bilinear form is positive definite whenever k ≤ n/ . Proposition A.6 (cf. [Kle94, §5] and [And04, §5.4]) . We have the following implications: (1)
Conjecture D ( X ) = ⇒ Conjecture A ( X, L ) . (2) Conjecture B ( X ) + Conjecture
Hdg( X ) = ⇒ Conjecture D ( X ) . (3) Conjecture D ( X × X ) = ⇒ Conjecture B ( X ) .Remark A.7 . (1) In characteristic zero, the standard conjecture of Hodge type follows from theLefschetz principle, Artin’s comparison theorem, and the classical Hodge theory (precisely,the Hodge–Riemann bilinear relations). Hence we have the following implications:Conjecture D ( X × X ) ⇐⇒ Conjecture B ( X ) = ⇒ Conjecture D ( X ) . In particular, over C , Conjecture D ( X ) follows from the Hodge conjecture as does B ( X ) .(2) In arbitrary characteristic, Conjecture Hdg( X ) holds for surfaces by the classical Hodgeindex theorem of Segre and Grothendieck (see [Gro58]); in particular, it holds in higherdimensions when k = 1 by the weak Lefschetz theorem.(3) Very recently, Ancona [Anc21] proves a version of Conjecture Hdg( X ) for Abelian four-folds in positive characteristic. There is a subtlety worth mentioning: Conjecture Hdg( X ) here, following [Kle68, And04, MNP13], is stated under homological equivalence, whilethe version proven in [Anc21] is under numerical equivalence (see [Anc21, Remark 6.9]).The following theorem could be regarded as the initial motivation of the standard conjectures.It provides the existence of the so-called Weil form on H i ( X ) (that indeed exists over C by theclassical Hodge theory), while the latter directly implies Weil’s Riemann hypothesis. Theorem A.8 (cf. [Kle68, Theorem 3.11]) . Let X be a smooth projective variety of dimension n over K . Suppose that Conjecture B ( X ) and Conjecture Hdg ( X × X ) hold. Then for all i ,the non-degenerate bilinear form H i ( X ) × H i ( X ) −→ F ( α, β )
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