Computation of categorical entropy via spherical functors
aa r X i v : . [ m a t h . AG ] F e b COMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICALFUNCTORS
JONGMYEONG KIM
Abstract.
We study the relationship between the categorical entropy of the twist andcotwist functors along spherical functors. In particular, we prove the categorical entropyof the twist functor coincides with that of the cotwist functor in certain circumstances.We also see our results generalize the computations of the categorical entropy of sphericaltwists and P -twists by Ouchi and Fan. As an application, we apply our results to theGromov–Yomdin type conjecture by Kikuta–Takahashi. Introduction
Motivation. A (topological) dynamical system ( X, f ) consists of a topological space X and a continuous map f : X → X . One way to measure the complexity of a dynamicalsystem ( X, f ) is to investigate the asymptotic behavior of the iterations of the map f . Itgives rise to the notion of the topological entropy h top ( f ) ∈ [0 , ∞ ).For a “nice enough” dynamical system, its topological entropy coincides with a linearalgebraic quantity. The following is known as the Gromov–Yomdin theorem . Theorem 1.1 ([6],[7],[20]) . Let X be a smooth projective variety over C and f : X → X be a surjective endomorphism. Then h top ( f ) = log ρ ( f ∗ ) where f ∗ : H ∗ ( X, C ) → H ∗ ( X, C ) is the induced automorphism on the cohomology and ρ is the spectral radius, i.e., the largest absolute value of the eigenvalues. Similarly, let us consider a categorical dynamical system ( D , Φ) which means a pair ofa triangulated category D and an exact endofunctor Φ : D → D . The categorical entropy h t (Φ) (carrying a parameter t ∈ R ) of a categorical dynamical system ( D , Φ) was introducedby Dimitrov–Haiden–Katzarkov–Kontsevich [3] as a categorical analogue of the topologicalentropy.Thus, in view of the Gromov–Yomdin theorem, it is natural to expect an analogousformula to hold for the categorical entropy. . Primary 18G80; Secondary 14F08
Key Words and Phrases . Categorical entropy, Spherical functors, Spherical twists
Conjecture 1.2 ([14], Conjecture 5.3) . Let X be a smooth projective variety over C and Φ : D b ( X ) → D b ( X ) be an exact autoequivalence. Then h (Φ) = log ρ ([Φ]) where [Φ] : N ( X ) ⊗ Z C → N ( X ) ⊗ Z C is the induced automorphism on the numericalGrothendieck group (tensored with C ). Kikuta–Takahashi [14] showed that the derived pullback Φ = L f ∗ where f : X → X is asurjective endomorphism satisfies Conjecture 1.2. It, in particular, implies that h ( L f ∗ ) = h top ( f ) by Theorem 1.1. Besides that, Conjecture 1.2 has been verified for a varietyof cases such as smooth projective curves [12], smooth projective varieties with ample(anti)canonical bundles [14], abelian surfaces [21], spherical twists [16] and P -twists [5]while there are also counterexamples [4],[16],[15]. Therefore, it is important to characterizeexact autoequivalences which do or do not satisfy Conjecture 1.2.In this paper, we study the relationship between the categorical entropy of the twist andcotwist functors along spherical functors whose notion was introduced by Anno–Logvinenko[2] as a generalization of that of spherical objects [18]. In particular, we will generalize thefollowing results by Ouchi [16] and Fan [5]. Theorem 1.3 ([16], Theorem 3.1) . Let D be the perfect derived category of a smooth properdg algebra and E be a d -spherical object of D . Denote by T S E the spherical twist along E .Then (1 − d ) t ≤ h t ( T S E ) ≤ ( if t ≥ , (1 − d ) t ( if t ≤ . If moreover E ⊥ = { F ∈ Ob( D ) | Hom ∗D ( E, F ) = 0 } 6 = 0 then h t ( T S E ) = 0 for all t ≥ .Remark . Before Ouchi’s work, Ikeda proved the same formula for the spherical twistsalong the simple modules of the Ginzburg dg algebras associated with acyclic quivers [11,Proposition 4.5].
Theorem 1.5 ([5], Theorem 3.1) . Let D be the perfect derived category of a smooth properdg algebra and E be a P d -object of D . Denote by T P E the P -twist along E . Then − dt ≤ h t ( T P E ) ≤ ( if t ≥ , − dt ( if t ≤ . If moreover E ⊥ = { F ∈ Ob( D ) | Hom ∗D ( E, F ) = 0 } 6 = 0 then h t ( T P E ) = 0 for all t ≥ . Results.
The main observation is that, when we consider a d -spherical object (resp. P d -object) E as a spherical functor S in a standard way, its twist functor T S is isomorphicto the spherical (resp. P -)twist along E and its cotwist functor C S is isomorphic to theshift functor [ − − d ] (resp. [ − − d ]). Since h t ( C S [2]) = (1 − d ) t (resp. − dt ), it is OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 3 natural to expect that Theorems 1.3 and 1.5 still hold for the spherical twists along generalspherical functors after replacing (1 − d ) t (resp. − dt ) in the theorems by h t ( C S [2]).It turns out that this is the case under some mild assumptions. Theorem 1.6 (Upper bound) . Let S : C → D be a spherical functor. Then h t ( T S ) ≤ ( if h t ( C S [2]) ≤ ,h t ( C S [2]) ( if h t ( C S [2]) ≥ . Theorem 1.7 (Lower bound) . Let S : C → D be a spherical functor with right and leftadjoint functors
R, L . (1) Assume that there exist split-generators
G, G ′ of C such that δ t ( RSG, G ′ ) < ∞ . Then h t ( T S ) ≥ h t ( C S [2]) . (2) Assume that
Ker SR = 0 . Then h t ( T S ) ≥ . For instance, if there exists a split-generator G of C such that RSG is again a split-generator of C then the assumption in Theorem 1.7 (1) is satisfied. In Section 5, we willsee some examples which satisfy this condition.Combining Theorems 1.6 and 1.7, we immediately obtain the following corollary. Corollary 1.8.
Let S : C → D be a spherical functor with right and left adjoint functors
R, L . Assume that there exist split-generators
G, G ′ of C such that δ t ( RSG, G ′ ) < ∞ . Then h t ( T S ) = h t ( C S [2]) for all t ∈ R such that h t ( C S [2]) ≥ . In particular, we have h ( T S ) = h ( C S [2]) = h ( C S ) . The above results give a way to compute (or estimate) the categorical entropy of thetwist functor T S using that of the cotwist functor C S and vice versa. These are particularlyuseful when the categorical entropy of one of the twist and cotwist functors is easy tocompute while the other is not. Since every exact autoequivalences can be written as thetwist functor along a spherical functor (in various ways) [17], these results may provide anavailable tool to compute the categorical entropy.As a simple application of the above results, we give a sufficient condition for twistfunctors to satisfy Conjecture 1.2. This is equivalent to the condition that there exist split-generators
G, G ′ of C such that δ t ( LSG, G ′ ) < ∞ by Definition 2.6 (4). This is equivalent to the condition that Ker SL = 0 by Definition 2.6 (3). OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 4
Proposition 1.9 (Inverse Gromov–Yomdin type inequality) . Let C , D be triangulated cat-egories of finite type whose numerical Grothendieck groups are of finite rank and S : C → D be a spherical functor. Assume that h ( C S ) ≤ log ρ ([ C S ]) and that there exists an element v ∈ N ( C ) ⊗ Z C such that [ S ] v = 0 and [ C S ] v = λv where | λ | = ρ ([ C S ]) . Then h ( T S ) ≤ log ρ ([ T S ]) . The opposite inequalities, so-called the
Gromov–Yomdin type inequalities , are known tohold for the perfect derived categories of smooth proper dg algebras [13, Theorem 2.13].Combining this fact and Proposition 1.9, we obtain the following corollary.
Corollary 1.10 (Gromov–Yomdin type formula) . Let C be a triangulated category of finitetype whose numerical Grothendieck group is of finite rank, D be the perfect derived categoryof a smooth proper dg algebra and S : C → D be a spherical functor. Assume that h ( C S ) ≤ log ρ ([ C S ]) and that there exists an element v ∈ N ( C ) ⊗ Z C such that [ S ] v = 0 and [ C S ] v = λv where | λ | = ρ ([ C S ]) . Then h ( T S ) = log ρ ([ T S ]) . Remark . Slightly modifying the proof of Proposition 1.9, we can instead use the usualGrothendieck group for the triangulated category C in Proposition 1.9 and Corollary 1.10by merely assuming that the usual Grothendieck group of C is of finite rank. In that case,we do not need to require C to be of finite type. Similarly, we can also use the usualGrothendieck groups for both the triangulated categories C and D in Proposition 1.9 andCorollary 1.10 provided that the usual Grothendieck groups are of finite rank. In that casetoo, we do not need to require C , D to be of finite type.1.3. Structure of the paper.
In Section 2, we briefly review the basic definitions andproperties of categorical entropy and spherical functors. Then, in Section 3, we prove The-orems 1.6 and 1.7 by generalizing the proofs of [16, Theorem 3.1] (see Theorem 1.3) and[5, Theorem 3.1] (see Theorem 1.5). Bacause we consider general triangulated categoriesrather than restricting to the perfect derived categories of smooth proper dg algebras as in[16] and [5], some technical but elementary lemmas have to be established. In Section 4,we give a proof of Proposition 1.9 using Theorems 1.6 and 1.7. Finally, in Section 5, we seesome examples coming from various twist functor constructions: Seidel–Thomas’ sphericaltwists [18], Huybrechts–Thomas’ P -twists [10] and Anno–Logvinenko’s orthogonally spher-ical twists [1]. In particular, we see Theorems 1.3 and 1.5 can be deduced from Theorems1.6 and 1.7.1.4. Conventions.
Throughout the paper, all triangulated categories are linear over a field K . Moreover, whenever we deal with spherical functors, every triangulated category D isassumed to admit a dg enhancement in the sense that there exists a (pretriangulated) dgcategory A whose homotopy category H ( A ) is equivalent to D as triangulated categories. OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 5
Similarly, every exact functor Φ : H ( A ) → H ( B ) between such triangulated categories isassumed to admit a dg lift , i.e., there exists a dg functor ˜Φ : A → B which descends to Φ.
Acknowledgements.
This work was supported by the Institute for Basic Science (IBS-R003-D1). 2.
Preliminaries
Categorical entropy.
Let us begin with reviewing the notion of categorical entropyintroduced by Dimitrov–Haiden–Katzarkov–Kontsevich [3].
Definition 2.1 ([3], Definition 2.1) . Let
E, F be objects of a triangulated category D .The categorical complexity of F with respect to E is the function δ t ( E, F ) : R → [0 , ∞ ] in t defined by δ t ( E, F ) = inf k X i =1 e n i t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∗ ∗ · · · ∗ F ⊕ F ′ E [ n ] E [ n ] E [ n k ] +1 +1 +1 if F = 0, and δ t ( E, F ) = 0 if F ∼ = 0.The categorical complexity satisfies the following properties. Lemma 2.2 ([3], Proposition 2.3) . Let
E, E ′ , E ′′ be objects of a triangulated category D . (1) δ t ( E, E ′′ ) ≤ δ t ( E, E ′ ) δ t ( E ′ , E ′′ ) . (2) δ t ( E, E ′ ⊕ E ′′ ) ≤ δ t ( E, E ′ ) + δ t ( E, E ′′ ) . (3) δ t (Φ E, Φ E ′ ) ≤ δ t ( E, E ′ ) for any exact functor Φ :
D → D ′ . An object G of a triangulated category D is called a split-generator if the smallest fulltriangulated subcategory containing G and closed under taking direct summand coincideswith D itself. The categorical entropy is then defined as follows. Definition 2.3 ([3], Definition 2.5) . Let G be a split-generator of a triangulated category D . The categorical entropy of an exact endofunctor Φ : D → D is the function h t (Φ) : R → [ −∞ , ∞ ) in t defined by h t (Φ) = lim n →∞ n log δ t ( G, Φ n G ) . Remark . The categorical entropy is well-defined (i.e., the limit exists in [ −∞ , ∞ )) andindependent of the choice of a split-generator used to define it [3, Lemma 2.6].We will freely use the following fact in what follows. Lemma 2.5 ([12], Lemma 2.6) . Let
G, G ′ be split-generators of a triangulated category D and Φ :
D → D ′ be an exact functor. Then h t (Φ) = lim n →∞ n log δ t ( G ′ , Φ n G ) . OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 6
Spherical functors.
Now we review the notion of spherical functors and the (co)twistfunctors along them introduced by Anno–Logvinenko [2]. We also show some of theirproperties which will be used to prove the main theorems.
Definition 2.6 ([2], Definition 5.2) . Let C , D be triangulated categories. An exact functor S : C → D with right and left adjoint functors
R, L is called a spherical functor if it satisfiesthe following conditions:(1) The twist functor T S = Cone( SR ε → Id D ) is an exact autoequivalence of D where ε : SR → Id D is the counit of the adjoint pair S ⊣ R .(2) The cotwist functor C S = Cone(Id C η → RS )[ −
1] is an exact autoequivalence of C where η : Id C → RS is the unit of the adjoint pair S ⊣ R .(3) R ∼ = LT S [ − R ∼ = C S L [1]. Remark . In order to have functorial cones, we should take dg enhancements of tri-angulated categories C , D and dg lifts of exact functors S, R, L . However, for the sake ofsimplicity, we will refrain from using the dg category language in this paper. For a detailedargument, see [2, Section 4].
Theorem 2.8 ([2], Theorem 5.1) . Any two out of the four conditions in Definition 2.6imply the other two.
Lemma 2.9.
Let S : C → D be a spherical functor with right and left adjoint functors
R, L . Then we have the following isomorphisms: (1) T nS S ∼ = S ( C S [2]) n . (2) RT nS ∼ = ( C S [2]) n R . (3) LT nS ∼ = ( C S [2]) n L . (4) SRT nS ∼ = T nS SR . (5) SLT nS ∼ = T nS SL . (6) RS ( C S [2]) n ∼ = ( C S [2]) n RS . (7) LS ( C S [2]) n ∼ = ( C S [2]) n LS .Proof. (1) It is enough to show the isomorphism for n = 1. Let ε : SR → Id D be thecounit and η : Id C → RS be the unit of the adjoint pair S ⊣ R . By definition, they satisfy OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 7 εS ◦ Sη = Id S . Then, by the octahedral axiom, we get S SRS SC S [1] S [1] S S S [1] T S S T S SSRS [1] SC S [2] Sη εS Id S and hence T S S ∼ = SC S [2].(2), (3) These isomorphisms follow from Definition 2.6 (3), (4).(4), (5), (6), (7) These isomorphisms follow from (1), (2), (3). (cid:3) Entropy of twist and cotwist functors
Lower bound.
Let S : C → D be a spherical functor with right and left adjointfunctors
R, L . Without loss of generality, we can assume that the essential image of R contains a split-generator of C . Indeed, it can be achieved by replacing C , if necessary,by the smallest full triangulated subcategory of C containing the essential image of R andclosed under taking direct summand. We will assume C have this property throughoutthis section. Note that, under this assumption, the essential image of L also contains asplit-generator of C by Definition 2.6 (3) or (4). Proof of Theorem 1.7. (1) Let
G, G ′′′ be split-generators of D and G ′ , G ′′ be split-generatorsof C such that LG ′′′ is a split-generator of C and δ t ( LSG ′ , G ′′ ) < ∞ . Then δ t ( LG ′′′ , ( C S [2]) n G ′′ ) ≤ δ t ( LG ′′′ , LS ( C S [2]) n G ′ ) δ t ( LS ( C S [2]) n G ′ , ( C S [2]) n G ′′ ) ≤ δ t ( LG ′′′ , LS ( C S [2]) n G ′ ) δ t (( C S [2]) n LSG ′ , ( C S [2]) n G ′′ ) ≤ δ t ( G ′′′ , S ( C S [2]) n G ′ ) δ t ( LSG ′ , G ′′ ) ≤ δ t ( G ′′′ , T nS SG ′ ) δ t ( LSG ′ , G ′′ ) ≤ δ t ( G ′′′ , T nS ( G ⊕ SG ′ )) δ t ( LSG ′ , G ′′ )where the first inequality follows from Lemma 2.2 (1), the second inequality follows fromLemma 2.9 (7), the third inequality follows from Lemma 2.2 (3) and the fourth inequality OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 8 follows from Lemma 2.9 (1). Therefore, we have h t ( T S ) = lim n →∞ n log δ t ( G ′′′ , T nS ( G ⊕ SG ′ ))= lim n →∞ n log( δ t ( G ′′′ , T nS ( G ⊕ SG ′ )) δ t ( LSG ′ , G ′′ )) ≥ lim n →∞ n log δ t ( LG ′′′ , ( C S [2]) n G ′′ )= h t ( C S [2]) . (2) Take an object 0 = E ∈ Ker SR . Then, from the exact triangle SRE → E → T S E → SRE [1], it follows that T S E ∼ = E . Thus, for a split-generator G of D , we have h t ( T S ) = lim n →∞ n log δ t ( G, T nS ( G ⊕ E )) ≥ lim n →∞ n log δ t ( G, T nS E )= lim n →∞ n log δ t ( G, E )= 0 . This completes the proof. (cid:3)
Upper bound.
We say that an object F of a triangulated category D admits a conedecomposition with components ( E , E , . . . , E k ) if there is a sequence of exact triangles in D of the form 0 ∗ ∗ · · · ∗ FE E E k . +1 +1 +1 The following two lemmas follow from the well-known fact that a cone decomposition canbe “inserted” into another cone decomposition : a cone decomposition of F with components( E , E , . . . , E k ) and a cone decomposition of E i with components ( E ′ , E ′ , . . . , E ′ l ) give riseto a cone decomposition of F with components ( E , . . . , E i − , E ′ , . . . , E ′ l , E i +1 , . . . , E k ). Lemma 3.1.
Let E be an object and F ′ → F → F ′′ → F ′ [1] be an exact triangle of atriangulated category D . Then δ t ( E, F ) ≤ δ t ( E, F ′ ) + δ t ( E, F ′′ ) . Proof.
Take ε > F ′ ⊕ ˜ F ′ with components( E [ n ] , E [ n ] , . . . , E [ n k ])and a cone decomposition of F ′′ ⊕ ˜ F ′′ with components( E [ m ] , E [ m ] , . . . , E [ m l ]) OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 9 so that k X i =1 e n i t < δ t ( E, F ′ ) + ε and l X j =1 e m j t < δ t ( E, F ′′ ) + ε. Then, from the exact triangle F ′ ⊕ ˜ F ′ → F ⊕ ˜ F ′ ⊕ ˜ F ′′ → F ′′ ⊕ ˜ F ′′ → F ′ ⊕ ˜ F ′ [1], we seethat F ⊕ ˜ F ′ ⊕ ˜ F ′′ admits a cone decomposition with components( E [ n ] , . . . , E [ n k ] , E [ m ] , . . . , E [ m l ]) . It shows that δ t ( E, F ) ≤ k X i =1 e n i t + l X j =1 e m j t < δ t ( E, F ′ ) + δ t ( E, F ′′ ) + 2 ε. Since ε > (cid:3)
Lemma 3.2.
Let
F, G be objects and E ′ → E → E ′′ → E ′ [1] be an exact triangle of atriangulated category D . Then δ t ( G, F ) ≤ ( δ t ( G, E ′ ) + δ t ( G, E ′′ )) δ t ( E, F ) . Proof.
Take ε > F ⊕ ˜ F with components( E [ n ] , E [ n ] , . . . , E [ n k ]) , a cone decomposition of E ′ ⊕ ˜ E ′ with components( G [ n ′ ] , G [ n ′ ] , . . . , G [ n ′ l ])and a cone decomposition of E ′′ ⊕ ˜ E ′′ with components( G [ n ′′ ] , G [ n ′′ ] , . . . , G [ n ′′ m ])so that k X i =1 e n i t < δ t ( E, F ) + ε, l X j =1 e n ′ j t < δ t ( G, E ′ ) + ε and m X j =1 e n ′′ j t < δ t ( G, E ′′ ) + ε. Using the exact triangle E ′ → E → E ′′ → E ′ [1], we can further decompose the previouslychosen cone decomposition of F ⊕ ˜ F . Then we obtain a new cone decomposition of F ⊕ ˜ F with components ( E ′ [ n ] , E ′′ [ n ] , . . . , E ′ [ n k ] , E ′′ [ n k ])This implies that F ⊕ ˜ F ′ , where ˜ F ′ = ˜ F ⊕ ˜ E ′ [ n ] ⊕ · · · ⊕ ˜ E ′ [ n k ] ⊕ ˜ E ′′ [ n ] ⊕ · · · ⊕ ˜ E ′′ [ n k ],admits a cone decomposition with components(( E ′ ⊕ ˜ E ′ )[ n ] , ( E ′′ ⊕ ˜ E ′′ )[ n ] , . . . , ( E ′ ⊕ ˜ E ′ )[ n k ] , ( E ′′ ⊕ ˜ E ′′ )[ n k ]) OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 10 and thus one with components( G [ n + n ′ ] , . . . , G [ n + n ′ l ] , G [ n + n ′′ ] , . . . , G [ n + n ′′ m ] , . . . ,G [ n k + n ′ ] , . . . , G [ n k + n ′ l ] , G [ n k + n ′′ ] , . . . , G [ n k + n ′′ m ]) . Therefore, it follows that δ t ( G, F ) ≤ k X i =1 l X j =1 e ( n i + n ′ j ) t + k X i =1 m X j =1 e ( n i + n ′′ j ) t = l X j =1 e n ′ j t + m X j =1 e n ′′ j t k X i =1 e n i t < ( δ t ( G, E ′ ) + δ t ( G, E ′′ ) + 2 ε )( δ t ( E, F ) + ε ) . Since ε > (cid:3)
Corollary 3.3.
Let E (resp. F ) be an object of a triangulated category D (resp. C ) and Φ :
C → D be an exact functor with a right adjoint functor R . Then δ t ( E, Φ F ) ≤ (1 + δ t ( E, T Φ E [ − δ t ( RE, F ) where T Φ = Cone(Φ R → Id D ) .Proof. By definition, there is an exact triangle T Φ E [ − → Φ RE → E → T Φ E . Hence ,wesee that(1 + δ t ( E, T Φ E [ − δ t ( RE, F ) ≥ (1 + δ t ( E, T Φ E [ − δ t (Φ RE, Φ F ) ≥ ( δ t ( E, E ) + δ t ( E, T Φ E [ − δ t (Φ RE, Φ F ) ≥ δ t ( E, Φ F )where the first inequality follows from Lemma 2.2 (3) and the last inequality follows fromLemma 3.2. (cid:3) Recall that a sequence { a n } ∞ n =1 is called submultiplicative if a n + m ≤ a n a m for all n, m .If a n > n , then the sequence { b n = log a n } ∞ n =1 is subadditive , i.e., b n + m ≤ b n + b m for all n, m . Therefore the sequence { n log a n } ∞ n =1 converges andlim n →∞ n log a n = inf n ≥ n log a n by Fekete’s subadditive lemma . Lemma 3.4.
Let { a n } ∞ n =1 be a submultiplicative sequence of positive real numbers. Thenthe sequence { n log(1 + P ni =1 a i ) } ∞ n =1 converges and lim n →∞ n log n X i =1 a i ! ≤ if lim n log a n ≤ , lim n →∞ n log a n ( if lim n log a n ≥ . OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 11
Proof.
The sequence { P ni =1 a i } ∞ n =1 is submultiplicative. Indeed,1 + n + m X i =1 a i ≤ n X i =1 a i + n + m X i = n +1 a i ≤ n X i =1 a i + a n m X i =1 a i ≤ n X i =1 a i ! m X i =1 a i ! . Thus the sequence { n log(1 + P ni =1 a i ) } ∞ n =1 converges.Now let L = lim n →∞ n log a n = inf n ≥ n log a n . (Case 1) Suppose L <
0. By taking a subsequence if necessary, we can assume that a n < n . Thenlim n →∞ n log n X i =1 a i ! ≤ lim n →∞ n log(1 + n ) = 0 . (Case 2) Suppose L >
0. By definition, we have a n ≥ e Ln and hence a n → ∞ as n → ∞ .Therefore we can take an increasing subsequence which we denote by the same notation { a n } ∞ n =1 . Thenlim n →∞ n log n X i =1 a i ! ≤ lim n →∞ n log(1 + na n ) = lim n →∞ n log a n . (Case 3) Suppose L = 0. By definition, we have a n ≥ n . If the sequence { a n } is bounded, then we can argue as in (Case 1). If the sequence { a n } is unbounded, thenwe can take an increasing subsequence. A similar argument as in (Case 2) then shows theassertion. (cid:3) Proof of Theorem 1.6.
Let
G, G ′ be split-generators of D such that RG, RG ′ are split-generators of C . Applying T n − S to the exact triangle defining T S , we obtain T n − S G → T nS G → T n − S SRG [1] → T n − S G [1]and therefore δ t ( G ′ , T nS G ) ≤ δ t ( G ′ , T n − S G ) + δ t ( G ′ , T n − S SRG [1])= δ t ( G ′ , T n − S G ) + δ t ( G ′ , S ( C S [2]) n − RG [1]) ≤ δ t ( G ′ , T n − S G ) + (1 + δ t ( G ′ , T S G ′ [ − δ t ( RG ′ , ( C S [2]) n − RG [1]) OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 12 where the first inequality follows from Lemma 3.1, the second equality follows from Lemma2.9 (1) and the last inequality follows from Corollary 3.3. Continuing this process, we get δ t ( G ′ , T nS G ) ≤ δ t ( G ′ , G ) + (1 + δ t ( G ′ , T S G ′ [ − n − X i =0 δ t ( RG ′ , ( C S [2]) i RG [1]) ≤ M t n − X i =0 δ t ( RG ′ , ( C S [2]) i RG [1]) ! where M t = max { δ t ( G ′ , G ) , δ t ( G ′ , T S G ′ [ − } which is independent of n .Consequently, we have h t ( T S ) = lim n →∞ n log δ t ( G ′ , T nS G ) ≤ lim n →∞ n log M t n − X i =0 δ t ( RG ′ , ( C S [2]) i RG [1]) !! = lim n →∞ n log n − X i =0 δ t ( RG ′ , ( C S [2]) i RG [1]) ! ≤ h t ( C S [2]) ≤ , lim n →∞ n log δ t ( RG ′ , ( C S [2]) n RG [1]) = h t ( C S [2]) (if h t ( C S [2]) ≥ . Here Lemma 3.4 is used in the last inequality. (cid:3) Gromov–Yomdin type formulas
Let D be a triangulated category. The Grothendieck group K ( D ) of D is defined as K ( D ) = Z h Ob( D ) i / h E − F + G | E → F → G → E [1] i . Now suppose D is of finite type , i.e.,dim Hom ∗D ( E, F ) = X i ∈ Z dim Hom D ( E, F [ i ]) < ∞ for any E, F ∈ Ob( D ). Then the Euler form χ : K ( D ) × K ( D ) → Z given by χ ([ E ] , [ F ]) = X i ∈ Z ( − i dim Hom D ( E, F [ i ])is well-defined. We define the numerical Grothendieck group N ( D ) as N ( D ) = K ( D ) / h [ E ] ∈ K ( D ) | χ ([ E ] , − ) = 0 i . In this section, we only consider triangulated categories whose numerical Grothendieckgroups are of finite rank.
OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 13
Now let Φ :
C → D be an exact functor between triangulated categories admitting aright adjoint functor. Then the induced homomorphism [Φ] : K ( C ) → K ( D ) descends tothe numerical Grothendieck groups. Tensoring with C , we obtain the homomorphism[Φ] : N ( C ) ⊗ Z C → N ( D ) ⊗ Z C which we denote by the same notation. Proof of Proposition 1.9.
By Theorem 1.6 and the assumption, h ( T S ) ≤ h ( C S ) ≤ log ρ ([ C S ]) . Now take an element v ∈ N ( C ) ⊗ Z C such that [ S ] v = 0 and [ C S ] v = λv where | λ | = ρ ([ C S ])which exists by our assumption. Then, by Lemma 2.9 (1),[ T S ][ S ] v = [ T S S ] v = [ SC S ] v = [ S ][ C S ] v = λ [ S ] v. This shows that ρ ([ C S ]) ≤ ρ ([ T S ]) and therefore h ( T S ) ≤ log ρ ([ T S ])as desired. (cid:3) Recall that a dg algebra A is called smooth if A is perfect as a dg bimodule over itselfand proper if A has the finite dimensional total cohomology. Theorem 4.1 ([13], Theorem 2.13) . Let D be the perfect derived category of a smoothproper dg algebra and Φ :
D → D be an exact endofunctor admitting a right adjointfunctor. Then h (Φ) ≥ log ρ ([Φ]) . Proof of Corollary 1.10.
It follows directly from Proposition 1.9 and Theorem 4.1. (cid:3) Examples
Spherical objects.
Spherical objects and twists were introduced by Seidel–Thomas[18] as the mirror counterparts of Lagrangian spheres and the Dehn twists along them.
Definition 5.1 ([18], Definition 2.9) . An object E of a triangulated category D of finitetype is called a d -spherical object ( d >
0) if it satisfies the following conditions:(1) Hom ∗D ( E, E ) ∼ = K [ h ] / ( h ) where deg h = d .(2) There is a functorial isomorphism Hom ∗D ( E, − ) ∼ = Hom ∗D ( − , E [ d ]) ∨ . Theorem 5.2 ([18], Proposition 2.10) . Let E be a d -spherical object of a triangulatedcategory D of finite type. Then there is an exact autoequivalence T S E of D , called the spherical twist along E , which acts on objects by T S E F = Cone(Hom ∗D ( E, F ) ⊗ E ev −→ F ) . OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 14
Let E be a d -spherical object of a triangulated category D of finite type. Let us regard K as a dg algebra concentrated in degree 0 (so with zero differential). Denote by Perf( K )the derived category of perfect dg K -modules (which is equivalent to the derived categoryof finite dimensional vector spaces over K ). Then the functor S = − ⊗ K E : Perf( K ) → D is a spherical functor and the twist and cotwist functors along it can be written as T S ∼ = T S E and C S ∼ = [ − − d ] . Since h t ( C S [2]) = h t ([1 − d ]) = (1 − d ) t, we recover Ouchi’s result [16, Theorem 3.1] (see Theorem 1.3) from Theorems 1.6 and 1.7. Proposition 5.3.
Let E be a d -spherical object of a triangulated category D of finite typeadmitting a split-generator. Then (1 − d ) t ≤ h t ( T S E ) ≤ ( if t ≥ , (1 − d ) t ( if t ≤ . If moreover E ⊥ = 0 then h t ( T S E ) = 0 for all t ≥ . Note that we do not need to require D to be the perfect derived category of a smoothproper dg algebra.As the cotwist functor C S ∼ = [ − − d ] satisfies the Gromov–Yomdin type formula and[ S ][ K ] = [ E ], we can apply Corollary 1.10 provided that [ E ] = 0 in N ( D ). Proposition 5.4.
Let D be the perfect derived category of a smooth proper dg algebra and E be a d -spherical object of D . Assume that [ E ] = 0 in N ( D ) . Then h ( T S E ) = log ρ ([ T S E ]) = 0 . P -objects. Similarly to spherical objects and twists, P -objects and twists, introducedby Huybrechts–Thomas [10], can be considered as the mirror counterparts of Lagrangianprojective spaces and the Dehn twists along them. Definition 5.5 ([10], Definition 1.1) . An object E of a triangulated category D of finitetype is called a P d -object ( d >
0) if it satisfies the following conditions:(1) Hom ∗D ( E, E ) ∼ = K [ h ] / ( h d +1 ) where deg h = 2.(2) There is a functorial isomorphism Hom ∗D ( E, − ) ∼ = Hom ∗D ( − , E [2 d ]) ∨ . For instance, this condition holds if d is even by Definition 5.1 (1). OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 15
Theorem 5.6 ([10], Proposition 2.6) . Let E be a P d -object of a triangulated category D of finite type. Then there is an exact autoequivalence T P E of D , called the P -twist along E ,which acts on objects by T P E F = Cone(Cone(Hom ∗D ( E, F ) ⊗ E [ − h ∗ ⊗ id − id ⊗ h −−−−−−−−→ Hom ∗D ( E, F ) ⊗ E ) ev −→ F ) where h ∗ : Hom ∗D ( E, F ) → Hom ∗D ( E, F )[2] is the homomorphism given by precomposingwith h ∈ Hom D ( E, E [2]) .Remark . An object E is a P -object if and only if it is a 2-spherical object. In thatcase, there is an isomorphism T P E ∼ = ( T S E ) [10, Proposition 2.9].Let E be a P d -object of a triangulated category D of finite type. Consider the dg algebra K [ h ] with deg h = 2 (so with zero differential). Denote by D fd ( K [ h ]) the derived categoryof dg K [ h ]-modules with finite dimensional cohomology. Then the functor S = − ⊗ K [ h ] E : D fd ( K [ h ]) → D is a spherical functor and the twist and cotwist functors along it can be written as T S ∼ = T P E and C S ∼ = [ − − d ][17, Proposition 4.2],[8, Corollary 2.9]. Since h t ( C S [2]) = h t ([ − d ]) = − dt, we recover Fan’s result [5, Theorem 3.1] (see Theorem 1.5) from Theorems 1.6 and 1.7. Proposition 5.8.
Let E be a P d -object of a triangulated category D of finite type admittinga split-generator. Then − dt ≤ h t ( T P E ) ≤ ( if t ≥ , − dt ( if t ≤ . If moreover E ⊥ = 0 then h t ( T P E ) = 0 for all t ≥ . Note that we do not need to require D to be the perfect derived category of a smoothproper dg algebra.Now let us view the spherical functor S = − ⊗ K [ h ] E as a functor from Perf( K [ h ]) to D . As the cotwist functor C S ∼ = [ − − d ] satisfies the Gromov–Yomdin type formulaand [ S ][ K [ h ]] = [ E ] = 0 in N ( D ) by Definition 5.5 (1), we can apply (a modified versionof) Corollary 1.10 (see Remark 1.11). Here [ S ] is regarded as a homomorphism from K (Perf( K [ h ])) to N ( D ). Proposition 5.9.
Let D be the perfect derived category of a smooth proper dg algebra and E be a P d -object of D . Then h ( T P E ) = log ρ ([ T P E ]) = 0 . OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 16
Orthogonally spherical objects.
Spherical functors which can be expressed asFourier–Mukai transforms have been studied by many authors: Horja [9], Toda [19], Anno–Logvinenko [1] and many others. In this section, we focus on orthogonally spherical objectsand the twists along them introduced by Anno–Logvinenko [1]. In what follows, we onlyconsider equidimensional schemes for simplicity and all functors appearing in this sectionare always assumed to be derived although we will not explicitly indicate that by thenotation.Let
X, Z be separated Gorenstein schemes of finite type over C . For a closed point p ∈ Z , denote the corresponding inclusion by ι p × X : X → Z × X . For a perfect object E ∈ D b ( Z × X ), define its fiber at a closed point p ∈ Z by E p = ι ∗ p × X E . Let us alsowrite the projections by π X : Z × X → X and π Z : Z × X → Z . We define an object L E ∈ Ob( D b ( Z )) to be the cone of the composition of the morphisms O Z → π Z ∗ O Z × X → π Z ∗ H om Z × X ( E , E ) → π Z ∗ H om Z × X ( π ∗ X π X ∗ E , E )where the first morphism is induced by the adjunction unit Id D b ( Z ) → π Z ∗ π ∗ Z , the secondmorphism is induced by the adjunction unit Id D b ( Z × X ) → H om Z × X ( E , − ⊗ E ) and thethird morphism is induced by the adjunction counit π ∗ X π X ∗ → Id D b ( X ) . Definition 5.10 ([1], Definition 3.4) . Let
X, Z be separated Gorenstein schemes of finitetype over C with dim X > dim Z . A perfect object E ∈ D b ( Z × X ) is called an orthogonallyspherical object if it satisfies the following conditions:(1) Hom ∗ D b ( X ) ( π X ∗ E , E p ) ∼ = C [ h ] / ( h ) where deg h = dim X − dim Z for every closedpoint p ∈ Z .(2) E ∨ ⊗ π ! X O X ∼ = E ∨ ⊗ π ! Z L E .(3) Hom ∗ D b ( X ) ( E p , E q ) = 0 for every pair of distinct closed points p, q ∈ Z .(4) E p = 0 for every closed point p ∈ Z . Remark . The above definition is stronger than the original definition [1, Definition3.4] where
E ∈ D b ( Z × X ) is called spherical if the Fourier–Mukai transform Φ Z → X E : D b ( Z ) → D b ( X ) is a spherical functor and orthogonal if it satisfies Definition 5.10 (3).The dimension restriction (i.e., dim X > dim Z ) and Definition 5.10 (4) are included justfor the sake of simplicity. Theorem 5.12 ([1], Theorem 3.2 and Proposition 3.7) . Let
X, Z be separated Gorensteinschemes of finite type over C and E ∈ D b ( Z × X ) be an orthogonally spherical object. Thenthe Fourier–Mukai transform Φ Z → X E : D b ( Z ) → D b ( X ) with kernel E is a spherical functor.Moreover, L E [dim X − dim Z ] is isomorphic to an invertible sheaf on Z and the cotwistfunctor along the spherical functor Φ Z → X E is isomorphic to − ⊗ L E [ −
1] : D b ( Z ) → D b ( Z ) . For an orthogonally spherical object
E ∈ D b ( Z × X ), we denote the twist functor alongthe spherical functor Φ Z → X E by T O E : D b ( X ) → D b ( X ). OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 17
We can apply Theorems 1.6 and 1.7 to the following situation.
Proposition 5.13.
Let
X, Z be smooth projective varieties over C with dim X ≥ Z and E ∈ D b ( Z × X ) be an orthogonally spherical object. Then (1 − d ) t ≤ h t ( T O E ) ≤ ( if t ≥ , (1 − d ) t ( if t ≤ where d = dim X − dim Z > . If moreover T p ∈ Z E ⊥ p = 0 then h t ( T O E ) = 0 for all t ≥ .Proof. Let S = Φ Z → X E : D b ( Z ) → D b ( X ) be the Fourier–Mukai transform with kernel E and R be its right adjoint functor. In order to prove that there exist split-generators G , G ′ of D b ( Z ) such that δ t ( RS G , G ′ ) < ∞ , it is sufficient to show that there exists a split-generator G of D b ( Z ) such that RS G is again a split-generator of D b ( Z ).Let G ∈
Ob(Coh( Z )) be a split-generator of D b ( Z ). By Theorem 5.12, there is aninvertible sheaf L ′E on Z which is isomorhpic to L E [ d ]. Then we haveHom D b ( Z ) ( C S G , G ) ∼ = Hom D b ( Z ) ( G ⊗ L E [ − , G ) ∼ = Hom D b ( Z ) ( G ⊗ L ′E [ − − d ] , G ) ∼ = Hom D b ( Z ) ( G ⊗ L ′E , G [1 + d ]) ∼ = Ext dZ ( G ⊗ L ′E , G )= 0where the last equality follows from our assumption that 1 + d > dim Z . This implies thatthe exact triangle G → RS G → C S G [1] → G [1]splits and hence RS G ∼ = G ⊕ C S G [1] is a split-generator of C . The former assertion is thenfollows from Theorems 1.6 and 1.7 (1) as h t ( C S [2]) = h t ( − ⊗ L E [1]) = h t ( − ⊗ L ′E [1 − d ]) = (1 − d ) t by [3, Lemma 2.14].Let us next prove the latter assertion. More precisely, we shall show that \ p ∈ Z E ⊥ p ⊂ Ker SR OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 18 then the assertion follows from Theorem 1.7 (2). Consider the following set of projections ZX × Z Z × XX X × Z × X XX × X. π X π Z π X π Z π π π π π The functor SR : D b ( X ) → D b ( X ) is isomorphic to the Fourier–Mukai transform withkernel π ∗ ( π ∗ ( E ∨ ⊗ π ! X O X ) ⊗ π ∗ E ) ∈ Ob( D b ( X × X )). Therefore, for an object F ∈
Ob( D b ( X )), we have SR F ∼ = π ∗ ( π ∗ ( π ∗ ( E ∨ ⊗ π ! X O X ) ⊗ π ∗ E ) ⊗ π ∗ F ) ∼ = π ∗ π ∗ ( π ∗ ( E ∨ ⊗ π ! X O X ) ⊗ π ∗ E ⊗ π ∗ π ∗ F ) ∼ = π X ∗ π ∗ ( π ∗ ( E ∨ ⊗ π ! X O X ) ⊗ π ∗ E ⊗ π ∗ π ∗ X F ) ∼ = π X ∗ π ∗ ( π ∗ E ⊗ π ∗ ( E ∨ ⊗ π ! X O X ⊗ π ∗ X F )) ∼ = π X ∗ ( E ⊗ π ∗ π ∗ ( E ∨ ⊗ π ! X O X ⊗ π ∗ X F )) ∼ = π X ∗ ( E ⊗ π ∗ π ∗ ( E ∨ ⊗ π ! X F )) ∼ = π X ∗ ( E ⊗ π ∗ Z π Z ∗ ( E ∨ ⊗ π ! X F )) ∼ = π X ∗ ( E ⊗ π ∗ Z π Z ∗ H om Z × X ( E , π ! X F ))where the second and fifth isomorphisms follow from the projection formula and the seventhisomorphism follows from the base change theorem. Now suppose F ∈ E ⊥ p for every closedpoint p ∈ Z . Let us show that π Z ∗ H om Z × X ( E , π ! X F ) ∼ = 0 . Consider the following set of morphisms
X Z × X Spec C Z ι p × X π C π Z ι p where ι p × X and ι p are the inclusions corresponding to a closed point p ∈ Z . Denote by S X = − ⊗ ω X [dim X ] : D b ( X ) → D b ( X ) and S Z × X = − ⊗ π ∗ X ω X ⊗ π ∗ Z ω Z [dim X + dim Z ] : D b ( Z × X ) → D b ( Z × X ) the Serre functors. Since both X and Z are smooth projective OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 19 varieties (and so is Z × X ), we see that π ! X F ∼ = S Z × X π ∗ X S − X F = S Z × X π ∗ X ( F ⊗ ω ∨ X [ − dim X ]) ∼ = S Z × X ( π ∗ X F ⊗ π ∗ X ω ∨ X [ − dim X ])= π ∗ X F ⊗ π ∗ X ω ∨ X ⊗ π ∗ X ω X ⊗ π ∗ Z ω Z [dim Z ] ∼ = π ∗ X F ⊗ π ∗ Z ω Z [dim Z ]and thus ι ∗ p × X π ! X F ∼ = ι ∗ p × X π ∗ X F ⊗ ι ∗ p × X π ∗ Z ω Z [dim Z ] ∼ = F [dim Z ] . Consequently, we have ι ∗ p π Z ∗ H om Z × X ( E , π ! X F ) ∼ = π C ∗ ι ∗ p × X H om Z × X ( E , π ! X F ) ∼ = π C ∗ H om X ( ι ∗ p × X E , ι ∗ p × X π ! X F ) ∼ = Hom ∗ D b ( X ) ( E p , F [dim Z ])= 0where the first isomorphism follows from the base change theorem and the last equalityfollows from the assumption that F ∈ E ⊥ p = {F ∈ Ob( D b ( X )) | Hom ∗ D b ( X ) ( E , F ) = 0 } . Asthis holds for every closed point p ∈ Z , we conclude that π Z ∗ H om Z × X ( E , π ! X F ) ∼ = 0 (see[1, Lemma 2.8]). (cid:3) We can also prove the Gromov–Yomdin type formulas for the twists along orthogonallyspherical objects using Corollary 1.10.
Proposition 5.14.
Let
X, Z be smooth projective varieties over C and E ∈ D b ( Z × X ) be an orthogonally spherical object. Assume that there is a closed point p ∈ Z such that [ E p ] = 0 in N ( D b ( X )) . Then h ( T O E ) = log ρ ([ T O E ]) = 0 . Proof.
Let S = Φ Z → X E : D b ( Z ) → D b ( X ) be the Fourier–Mukai transform with kernel E .Since h ( C S ) = log ρ ([ C S ]) = 0, the assertion follows from Corollary 1.10 if we show thatthere exists an element v ∈ N ( D b ( Z )) ⊗ Z C such that [ S ] v = 0 and [ C S ] v = λv where | λ | = ρ ([ C S ]) = 1.Take a closed point p ∈ Z so that [ E p ] = 0 in N ( D b ( X )) and denote by O p the skyscrapersheaf supported at p . Then S O p = Φ Z → X E O p = E p and so [ S ][ O p ] = [ E p ] = 0. On the otherhand, choose an invertible sheaf L ′E on Z so that L E [ d ] ∼ = L ′E where d = dim X − dim Z .Then C S O p ∼ = O p ⊗ L ′E [ − − d ] ∼ = O p [ − − d ] and therefore [ C S ][ O p ] = ± [ O p ]. This showsthat we can take v = [ O p ]. (cid:3) For instance, this condition holds if dim X − dim Z is even by Definition 5.10 (1). OMPUTATION OF CATEGORICAL ENTROPY VIA SPHERICAL FUNCTORS 20
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