aa r X i v : . [ m a t h . AG ] A ug POSITIVITY OF THE DIAGONAL
BRIAN LEHMANN AND JOHN CHRISTIAN OTTEM
Abstract.
We study how the geometry of a projective variety X is reflected in the posi-tivity properties of the diagonal ∆ X considered as a cycle on X × X . We analyze when thediagonal is big, when it is nef, and when it is rigid. In each case, we give several implicationsfor the geometric properties of X . For example, when the cohomology class of ∆ X is big,we prove that the Hodge groups H k, ( X ) vanish for k >
0. We also classify varieties of lowdimension where the diagonal is nef and big. Introduction
The geometry of a projective variety X is determined by the positivity of the tangentbundle T X . Motivated by the fact that T X is the normal bundle of the diagonal ∆ X in theself-product X × X , we will in this paper study how the geometry of X is reflected in thepositivity properties of ∆ X itself, considered as a cycle on X × X . The prototypical exampleof a variety with positive diagonal is projective space; the central theme of the paper isthat positivity of the diagonal forces X to be similar to projective space. In dimension 1,this perspective is already quite vivid: P is the only curve where the diagonal is an ampledivisor; elliptic curves have nef, but not big diagonals; and for higher genus, the diagonal iscontractible, hence ‘negative’ in a very strong sense.In general, when X has dimension n , the diagonal determines a class in the space N n ( X × X ) of n -dimensional cycles modulo numerical equivalence, and we are interested in how thisclass sits with respect to the various cones of positive cycles of X × X . Note that in theabsence of the Hodge conjecture, we often do not even know the dimension of the space N n ( X × X ). Thus we develop techniques to prove positivity or rigidity without an explicitcalculation of the positive cones.The subsections below recall several different types of positivity and give a number oftheorems illustrating each. At the end of the introduction we will collect several examplesof particular interest. Big diagonal.
A cycle class α is said to be big if it lies in the interior of the closed cone gen-erated by classes of effective cycles. Bigness is perhaps the most natural notion of positivityfor cycles. We will also call a cycle homologically big if it is homologically equivalent to thesum of an effective Q -cycle and a complete intersection of ample Q -divisors. Homologicalbigness implies bigness, and equivalence of the two notions would follow from the standardconjectures.The primary example of a variety with (homologically) big diagonal is projective space.In this case, the diagonal has a K¨unneth decomposition of the form∆ X = X p + q = n π ∗ h p · π ∗ h q where h is the hyperplane divisor, and this class is evidently big. Of course, the sameargument applies also for the fake projective spaces , that is, smooth varieties = P n withthe same betti numbers as P n . In dimension 2, there are exactly 100 such surfaces [PY07],[CS10], and they are all of general type. Thus unlike the case of curves, we now allow BRIAN LEHMANN AND JOHN CHRISTIAN OTTEM examples with positive Kodaira dimension, but which are still ‘similar’ to projective spacein the sense that they have the same Hodge diamond.More generally, homological bigness of the diagonal implies the vanishing of the ‘outer’Hodge groups of X . Following ideas of [Fu12], we show: Theorem 1.1.
Let X be a smooth projective variety. If ∆ X is homologically big, then H i, ( X ) = 0 for i > . An interesting feature of this result is that the proof makes use of non-algebraic cohomologyclasses to control effective cycles. When X is a surface with a big diagonal, Theorem 1.1implies the existence of a cohomological decomposition of the diagonal; we discuss thisrelationship in more depth in Section 9. Example 1.2.
Let X denote the blow-up of P along a planar elliptic curve which doesnot admit complex multiplication. In Example 3.8 we verify that ∆ X is big even though h , ( X ) = 0. Thus the vanishing results for Hodge groups as in Theorem 1.1 are optimal forthreefolds.We emphasize that even amongst varieties satisfying the hypotheses of Theorem 1.1 thereare very few with big diagonal. We will prove several additional strong constraints onthe geometry of a variety with big diagonal. For example, such varieties can not admit amorphism to variety with smaller dimension. Nevertheless, the complete classification ofvarieties with big diagonal seems subtle (see Section 10).1.1. Big and nef diagonal.
A cycle class is said to be nef if it has non-negative intersectionagainst every subvariety of the complementary dimension. Diagonals which are both bigand nef are positive in the strongest possible sense, and we classify such varieties in lowdimensions.
Theorem 1.3.
Let X be a smooth projective variety. • If dim X = 2 and ∆ is nef and big then X has the same rational cohomology as P :it is either P or a fake projective plane. • If dim X = 3 and ∆ is nef and homologically big then X has the same rationalcohomology as P : it is either P , a quadric, a del Pezzo quintic threefold V , or theFano threefold V . It is interesting to compare this result to Mori’s theorem that the only smooth varietywith ample tangent bundle is P n . By switching to the perspective of numerical positivity of∆ X , we also include varieties with the same cohomological properties as projective space.In higher dimensions we make partial progress toward a classification. In particular, weshow that N k ( X ) ∼ = R for every 0 ≤ k ≤ dim X , provided that the diagonal is big and universally pseudoeffective (this is a stronger condition than nefness, in the sense that π ∗ ∆ X is required to be pseudoeffective for every morphism π : Y → X × X ). Dual positivity.
We also study nefness or universal pseudoeffectiveness in the absence ofbigness.
Theorem 1.4.
Let X be a smooth projective variety. If ∆ X is nef (resp. universally pseu-doeffective) then every pseudoeffective class on X is nef (resp. universally pseudoeffective). For example, a surface with nef diagonal must be a minimal surface.
Example 1.5. If X has a nef tangent bundle, then ∆ X is nef. Campana and Peternellpredict that any Fano manifold with nef tangent bundle is in fact rational homogeneous. OSITIVITY OF THE DIAGONAL 3
Note that Theorem 1.4 is compatible with this conjecture: on a homogeneous variety everypseudoeffective class must be nef.While varieties with nef tangent bundle will have nef diagonal, the converse is not true;for example, fake projective planes have anti-ample tangent bundle but their diagonals areuniversally pseudoeffective.It is interesting to look for other sources of feedback between nefness of the diagonal andnefness of the tangent bundle. For example:
Theorem 1.6.
Let S be a smooth surface of Kodaira dimension ≤ whose diagonal is nef.Then T S is a nef vector bundle, except possibly when S is a minimal properly elliptic surfacewith no section. The exception is necessary: Example 6.11 constructs a hyperelliptic surface with no sectionwhich has nef diagonal. Also, the natural extension to general type surfaces is false: a fakeprojective plane has nef diagonal.
Examples.Example 1.7 (Toric varieties) . Let X be a smooth toric variety. Theorem 3.9 shows that∆ X is big if and only if every nef cycle on X is big. One might expect that the only toricvarieties with big diagonal are the projective spaces, but this turns out not to be the case.For example, [FS09] gives an example of a toric threefold of Picard rank 5 with big diagonal.By combining our work with results of [FS09] we can classify toric varieties with nefdiagonal: Proposition 1.8.
Let X be a smooth projective toric variety. Then ∆ X is nef if and onlyif X is a product of projective spaces. Example 1.9 (Hypersurfaces) . Let X be a smooth hypersurface of degree ≥ ≥
2. It is easy to see that the diagonal of X is not nef. For bigness, we show: Theorem 1.10.
For a smooth Fano hypersurface of degree ≥ and dimension ≤ , thediagonal is not big. For a quadric hypersurface, ∆ X is big if and only if the dimension is odd, in which caseit is a fake projective space (see Section 7.1). Example 1.11 (K3 surfaces) . By Theorem 1.1 the diagonal of a K3 surface is not big. Weprove the diagonal of a K3 surface is never nef by using the birational geometry of Hilb ( X )as described by [BM14]. For general K3 surfaces we can say more: using a deformationargument we show Theorem 1.12.
For a very general K3 surface, the diagonal is the unique effective R -cyclein its numerical class and it lies on an extremal ray of the pseudoeffective cone. We expect the statement holds for every K3 surface, and we prove it for some specificclasses (for example, for K3 surfaces of degree divisible by 4 with Picard rank 1).1.2.
Acknowledgements.
We want to thank M. Fulger for his input and for numerouscorrections and improvements. We thank F. Catanese for a discussion about fake quadrics,E. Macr`ı for a discussion about Hilb of K3 surfaces, and X. Zhao for alerting us to the workof Lie Fu [Fu12]. BL was supported by an NSA Young Investigator Grant and by NSF grant1600875, and JCO was supported by RCN grant 250104. BRIAN LEHMANN AND JOHN CHRISTIAN OTTEM Background
Throughout we work over C . For a projective variety X , we will let ∆ X denote thediagonal in the self-product X × X . The two projections of X × X will be denoted by π and π respectively.2.1. Cones of positive cycles.
Let X be a projective variety. We let N k ( X ) Z denote thegroup of k -cycles modulo numerical equivalence. The numerical class of a cycle Z is written[ Z ] and we use ≡ to denote numerical equivalence. The abelian group N k ( X ) Z forms alattice inside the numerical group N k ( X ) := N k ( X ) Z ⊗ Z R , which is a finite dimensional realvector space. We define N k ( X ) to be the vector space dual to N k ( X ). When X is smooth ofdimension n , capping against [ X ] defines an isomorphism N n − k ( X ) → N k ( X ), and we willswitch between subscripts and superscripts (of complementary dimension) freely. Definition 2.1.
We say that a numerical class is effective if it is the class of an effective R -cycle. The pseudoeffective cone Eff k ( X ) in N k ( X ) is the closure of the cone generated byeffective classes. A class is big when it lies in the interior of Eff k ( X ).The nef cone Nef k ( X ) in N k ( X ) is the dual of the pseudoeffective cone, and a cycle iscalled nef if its class belongs to this cone. That is, a cycle is nef if it has non-negativeintersection numbers with all k -dimensional subvarieties.The basic properties of these cones are verified in [FL17]: they are full-dimensional, con-vex, and contain no lines. Pseudo-effectiveness is preserved by pushforward, and nefness ispreserved by pullback. It is useful to have more a restrictive form of dual positivity: Definition 2.2 ([FL17]) . Let X be a projective variety. A cycle class α ∈ N k ( X ) is said tobe universally pseudoeffective if π ∗ α is pseudoeffective for every morphism π : Y → X .The primary examples of such cycles are complete intersections of ample divisors, or moregenerally, Chern classes of globally generated vector bundles. As suggested by the superscriptdemarcation, the universally pseudoeffective cone is naturally contravariant for morphismsand should be thought of as a “dual” positive cone by analogy with the nef cone.2.2. Positive homology classes.
Let H k ( X ) alg ⊆ H k ( X ) denote the subspace of alge-braic homology classes, i.e., the image of the cycle class map cl : CH k ( X ) ⊗ R → H k ( X ).Let E k ( X ) ⊂ H k ( X ) alg denote the cohomological effective cone . Definition 2.3.
We say a k -cycle Γ is homologically big if its cohomology class [Γ] lies inthe interior of E k ( X ).In general, for smooth complex projective varieties, cohomological implies numerical equiv-alence, so any homologically big cycle is big in the usual sense. If Grothendieck’s standardconjecture D holds on X , namely that numerical and cohomological equivalence coincide,then N k ( X ) = H k ( X ) alg and the two notions of ‘big’ coincide. In the special case of a self-product, it is known that D holds on X × X if and only if the Lefschetz standard conjectureholds on X (i.e. the inverse of the hard Lefschetz isomorphism is induced by a correspon-dence). This is known to hold for surfaces [Lie68], and for threefolds not of general type byresults of Tankeev [Tan11]. We will in this paper be mostly interested in surfaces, and usethe fact that the two notions coincide in this case without further mention.We will also require the following result of [Ott15] which follows from the theory of relativeHilbert schemes: Proposition 2.4.
Let f : X → T be a smooth family of projective varieties over a smoothvariety T and suppose that α ∈ H k,k ( X , Z ) has that the restriction to a very general fiber isrepresented by an effective cycle. Then α | X t is an effective class for any fiber X t . OSITIVITY OF THE DIAGONAL 5
We can use this result when X is a family of varieties for which homological and numericalequivalence coicide (e.g., fourfolds). In this case, the theorem also implies that a class whichrestricts to be big on a very general fiber has big restriction on every fiber.3. Varieties with big diagonal
In this section we consider the geometric implications of big diagonals.
Lemma 3.1.
Let X be a projective variety. If X carries a universally pseudoeffective class α ∈ N k ( X ) that is not big, then ∆ X is not big. In particular, if X carries a nef divisor that is not big, then ∆ X is not big. Proof.
Let n denote the dimension of X . Since α is not big, there is some non-zero nefclass β ∈ N n − k ( X ) that has vanishing intersection with α . Then consider γ := π ∗ α · π ∗ β on X × X . Clearly γ is a nef class: if E is an effective cycle of dimension n , then π ∗ α · E isstill pseudoeffective, so that it has non-negative intersection against the nef class π ∗ β . Since γ · ∆ X = 0, we see that ∆ X can not be big. (cid:3) Corollary 3.2.
Let X be a projective variety of dimension n . If X admits a surjectivemorphism f : X → Y to a variety of dimension < n , then ∆ X is not big. It is sometimes helpful to consider non-algebraic classes as well. In this setting, we recallthat a (1 , α is defined to be nef if it is the limit of K¨ahler classes. Theorem 3.3.
Let X be an n -dimensional smooth projective variety admitting a non-zeronef cohomology class α ∈ H , ( X, R ) such that α n = 0 . Then ∆ X is not homologically big.Proof. Let ω be a K¨ahler form on X × X . Let α be a nef (1 , X and let 0 < k < n be an integer so that α k = 0, but α k +1 = 0. The two pullbacks π ∗ α k and π ∗ α ∪ ω n − k − areweakly positive forms on X × X , and hence their product β = π ∗ α k ∪ π ∗ α ∪ ω n − k − is a weakly positive ( n, n )-class on X × X [Dem07, Ch. III]. Now the main point is that β isnef, in the sense that R Z β ≥ Z ⊂ X × X . This is because β restricts toa non-negative multiple of the volume form on Z for every smooth point on it (cf. [Dem07,Ch. III (1.6)]). Note however that it is not in general the case that the product of two nefclasses remains nef, as shown in [DELV11].If ∆ X is homologically big, then we can write [∆ X ] = ǫh n + Z where ǫ > h is an ampleline bundle and Z is an effective cycle. Moreover, since h is ample and α is nef, the followingtwo inequalities hold:(1) Z Z β ≥ Z X h n ∪ β > β · ∆ X = 0, which holds by our assumptions on α and k . (cid:3) Bigness of the diagonal is compatible with pushforward:
Lemma 3.4.
Let f : X → Y be a surjective morphism of projective varieties. If ∆ X is big,then so is ∆ Y . Note that by Lemma 3.1 the hypothesis is never satisfied if 0 < dim Y < dim X , so themain interest is in the generically finite case. BRIAN LEHMANN AND JOHN CHRISTIAN OTTEM
Proof.
Let n denote the dimension of X and d denote the dimension of Y . Fix an ampledivisor H on X . Then ∆ X · H n − d is a big class on X .Consider the induced map f × f : X × X → Y × Y . The set-theoretic image of ∆ X is∆ Y ; in particular, ( f × f ) ∗ : [∆ X ] · H n − d is proportional to ∆ Y . Since the pushforward ofa big class under a surjective map is still big, we see that ∆ Y is also big. (cid:3) Cohomological criteria.
The main result of this section is the following theorem.
Theorem 3.5.
Let X be a smooth projective variety with homologically big diagonal. Then H k, ( X ) = 0 for all k > . In particular, no varieties with trivial canonical bundle can have homologically big diag-onal.
Proof.
Following [Voi10] and [Fu12], we will utilize the Hodge–Riemann relations to findfaces of the effective cones of cycles. To set this up, let ω be a K¨ahler form on a smoothprojective variety W . Note that a cohomology class in H k, ( W ) is automatically primitive.Thus by the Hodge–Riemann bilinear relations, the bilinear form on H k, ( W ) given by q ( a, b ) = ε Z W a ∪ ¯ b ∪ ω n − k is positive definite. Here ε = 1 if k is even, and ε = √− k is odd.Now fix a K¨ahler form ω on X × X and let σ be a non-zero closed ( k, X .Consider the product β = ε ( π ∗ σ − π ∗ σ ) ∪ ( π ∗ ¯ σ − π ∗ ¯ σ ) ∪ ω n − k . This is a non-zero ( n, n )-form on X × X , which by construction vanishes on the diagonal.Now, if Z ⊂ X × X is an n -dimensional subvariety, the Hodge–Riemann relations (appliedon a resolution of Z ) imply that β · Z ≥
0. Similarly, β · h n > h on X × X . Finally, since ∆ X · β = 0, it follows that ∆ cannot be homologically big. (cid:3) Remark 3.6.
The above theorem can also be deduced from [Fu12, Lemma 3.3], which isproved using a similar argument.
Example 3.7.
Even when the diagonal is only (numerically) big, we can still show that H , ( X ) vanishes. First suppose that A is an abelian variety of dimension n . Then thediagonal is the fiber over 0 of the subtraction map f : A × A → A . In particular, ∆ A hasvanishing intersection against the nef class f ∗ L · H n − where L is an ample divisor on A and H is an ample class on A × A . Under suitable choices, the class β in the proof of Theorem3.5 constructed from H , ( A ) will be exactly this n -cycle.More generally, when X is a smooth projective variety with non-trivial Albanese, wehave a subtraction map X × X → A . The diagonal will have vanishing intersection againstthe pullback of an ample divisor from A under the subtraction map intersected with anappropriate power of an ample divisor on X × X . Again, this is essentially the same as theclass β constructed in the proof above. Example 3.8.
We give an example of a smooth Fano threefold X with homologically bigdiagonal which satisfies h , ( X ) = 0. Thus Theorem 3.5 is optimal in the sense that theother Hodge groups need not vanish.Let X be the blow-up of P along a planar elliptic curve C which does not have complexmultiplication. Let H denote the pullback of the hyperplane class to X and E denote the OSITIVITY OF THE DIAGONAL 7 exceptional divisor. It is easy to verify that:Eff ( X ) = h H − E, E i Nef ( X ) = h H − E, H i Eff ( X ) = h HE, H − HE i Nef ( X ) = h H , H − HE i On X × X let H i , E i denote the pullbacks of H and E under the i th projection. Since C does not have complex multiplication, N ( X × X ) has dimension 11: it is spanned by ∆ X and the non-zero products of H , E , H , E .Recall that C × C has three-dimensional Neron-Severi space spanned by the fibers F , F of the projections and the diagonal ∆ C . Let Z a,b,c denote the class in N ( X × X ) obtainedby pulling the divisor aF + bF + c ∆ C back from C × C to E × E and then pushing forwardto X × X . An intersection calculation shows that Z a,b,c = a H E E + b H E E + c ( H + H H + H H + H − ∆ X ) . Applying this to the effective divisor 2 F + 2 F − ∆, we obtain∆ X = Z , , − + 16 H E E + 16 H E E + 56 H E ( H − E ) + 56 H E ( H − E )+ 56 H H ( H − E ) + 56 H H ( H − E ) + 16 H ( H − E ) + 16 H ( H − E )+ 16 H E + 16 H E + H + H and since the terms are all effective and together span N ( X × X ) we see ∆ X is big (andhence homologically big, since X is a rational threefold). We also note in passing that ∆ X is not nef, since it has negative intersection against the effective cycle H E E .3.2. Criteria for bigness.
There is one situation where it is easy to test for bigness of thediagonal, namely when the effective cones of X × X are as simple as possible. Theorem 3.9.
Let X be a smooth projective variety of dimension n . Suppose that for every k Eff k ( X × X ) = X i + j = k π ∗ Eff i ( X ) · π ∗ Eff j ( X ) . Then ∆ X is big if and only if every nef class on X is big.Proof. We first claim that the nef cone has the expressionNef k ( X × X ) = X i + j = k π ∗ Nef i ( X ) · π ∗ Nef j ( X ) . The containment ⊇ is clear from the description of the pseudoeffective cone. Conversely, itsuffices to show that every class generating an extremal ray of Eff k ( X × X ) has vanishingintersection against some element of the right hand side. By hypothesis such classes havethe form π ∗ α i · π ∗ α k − i where α ∈ Eff i ( X ) and α k − i ∈ Eff k − i ( X ) both lie on extremalrays. Choose nef classes β i ∈ Nef i ( X ) and β k − i ∈ Nef k − i ( X ) satisfying α i · β i = 0 and α k − i · β k − i = 0. Then ( π ∗ α i · π ∗ α k − i ) · ( π ∗ β i · π ∗ β n − i ) = 0Now suppose that ∆ X is not big. Then it must have vanishing intersection against some α ∈ Nef n ( X × X ) which lies on an extremal ray. By the expression above, such a class hasthe form α = π ∗ β j · π ∗ β n − j where for some constant j we have β j ∈ Nef j ( X ) and β n − j ∈ Nef n − j ( X ). But then β j · β n − j =0 as classes on X . Since β j has vanishing intersection against a nef class, it can not be big. BRIAN LEHMANN AND JOHN CHRISTIAN OTTEM
Conversely, suppose that there is a nef class in N k ( X ) which is not big. Since there arealso big nef classes in N k ( X ), by convexity of the nef cone we can find a nef class β ∈ N k ( X )on the boundary of the pseudoeffective cone. Thus there is another nef class β ′ such that β · β ′ = 0. Arguing as above, we see that π ∗ β · π ∗ β ′ is a nef class with vanishing intersectionagainst ∆ X . (cid:3) Two typical situations where one can apply Theorem 3.9 are when: • X is a toric variety. • N k ( X × X ) = ⊕ i + j = k π ∗ N i ( X ) · π ∗ N j ( X ), every pseudoeffective cone on X is simpli-cial, and every nef class on X is universally pseudoeffective.The first fact is well-known. To see the second, note that the hypothesis on universalpseudo-effectivity shows that any external product of nef cycles is nef. The simplicial hy-pothesis then implies that the external product of the pseudoeffective cones is dual to theexternal product of the nef cones. Thus the external product of the pseudoeffective cones isin fact the entire pseudoeffective cone of X × X .We will apply Theorem 3.9 to examples where one can prove directly that all nef classesare universally pseudoeffective (e.g., fake projective spaces, Grassmannians,. . . ). However,it seems relatively rare in general for the condition on pseudoeffective cones in Theorem 3.9to hold. Here is a basic example: Example 3.10.
Let S be the blow-up of P in r general points for some r ≥
5. There is astrict containment R ≥ [ F ] ⊕ π ∗ Eff ( S ) · π ∗ Eff ( S ) ⊕ R ≥ [ F ] ( Eff ( S × S ) . In fact, a lengthy but straightforward computation shows that the diagonal does not lie inthe cone on the left. 4.
Dual positivity
We next turn to the “dual” forms of positivity: nefness and universal pseudoeffectiveness.The main examples are varieties with nef tangent bundle. For such varieties the class of ∆ X is nef, but not all varieties with nef diagonal have nef tangent bundle; for example, a fakeprojective plane has nef diagonal even though the tangent bundle is antiample.We emphasize that only “dual-positivity” of the tangent bundle should be inherited bythe diagonal. The bigness of the tangent bundle T X is quite different from the bigness ofthe class ∆ X . For example, a product of at least two projective spaces has big and neftangent bundle, but by Lemma 3.1 the diagonal class is not big. More generally, a smoothtoric variety has big tangent bundle by [Hsi15], but it is rare for a toric variety to have bigdiagonal. Proposition 4.1.
Let X be a smooth variety. If ∆ X is nef (resp. universally pseudoeffective)then every pseudoeffective class on X is nef (resp. universally pseudoeffective). In fact, the proposition is true for any property preserved by pullback and flat pushforward.This proposition strengthens [CP91, Proposition 2.12], which shows the analogous statementfor divisors on a variety with T X nef. Proof.
We focus on nefness; the proof for universal pseudoeffectiveness is identical, using theproperties of positive dual classes proved in [FL17].It suffices to show nefness for the class of an irreducible cycle Z on X . Since π is flat, Z ′ = π − ( Z ) represents π ∗ [ Z ]. The restriction of ∆ X to Z ′ is nef; since nefness is preservedby flat pushforward onto a smooth base, ( π | Z ′ ) ∗ [∆ X ] | Z ′ = [ Z ] is also nef on X . (cid:3) OSITIVITY OF THE DIAGONAL 9
Corollary 4.2.
Let X be a smooth projective variety.(1) If ∆ X is big and nef, then N ( X ) ∼ = R .(2) If ∆ X is big and universally pseudoeffective, then N k ( X ) ∼ = R for every k .Proof. Combine Proposition 4.1, Lemma 3.1, and the fact that nef divisors are universallypseudoeffective. (cid:3)
Lemma 4.3.
If a smooth variety X admits a surjective map to a curve C of genus ≥ ,then ∆ X is not nef.Proof. Denote the morphism by π : X → C . Let H be an ample divisor on X × X . Letting n denote the dimension of X , we have ∆ X · H n − · ( π × π ) ∗ ∆ C < C has negative self-intersection. (cid:3) We can also give a necessary condition for nefness based on the gonality of X . Proposition 4.4.
Let X be a smooth projective variety of dimension n admitting a surjectivegenerically finite map f : X → Y of degree d to a smooth projective variety Y . Suppose that c n ( X ) > dc n ( Y ) . Then ∆ X is not nef.Proof. If f contracts a curve, then X carries a curve that is not nef, and hence ∆ X is notnef. Thus it suffices to consider the case when f is finite.Consider the map F = ( f × f ) : X × X → Y × Y . This is finite surjective, hence flat.Note that F | ∆ X = f , so F ∗ ∆ X = d ∆ Y . Moreover, by flatness, F ∗ ∆ Y is an effective cyclecontaining ∆ X in its support. The intersection of F ∗ ∆ Y − ∆ X with ∆ X is F ∗ ∆ X · ∆ Y − ∆ X = dc n ( Y ) − c n ( X )which is negative by assumption, so that ∆ X is not nef. (cid:3) Corollary 4.5.
Let X be a smooth projective variety of dimension n admitting a surjectivegenerically finite map f : X → P n of degree d . Suppose that c n ( X ) > ( n + 1) d . Then ∆ X isnot nef. Rigidity
The results of the previous sections indicate that it is quite rare for a variety to have bigdiagonal. In this section we will study varieties where ∆ X is as far away from big as possible,and in particular, when [∆ X ] spans an extremal ray in the pseudoeffective cone. Definition 5.1.
Let Z be an effective R -cycle on a projective variety X of dimension k . Wesay that Z is:(1) strongly numerically rigid , if Z is irreducible and for every infinite sequence of effec-tive R -cycles Z i such that lim i →∞ [ Z i ] = [ Z ], the coefficient a i of Z in Z i limits to1.(2) exceptional for a morphism π : X → Y , if reldim( π | Z ) > reldim( π ).Exceptional classes are studied in [FL16] and are closely related to the notion of anexceptional divisor. Among other nice properties, an exceptional numerical class can not berepresented by a cycle whose deformations cover X .If Z is strongly numerically rigid then it spans an extremal ray of the pseudoeffectivecone and is the unique effective cycle in its numerical class. A typical example of a stronglynumerically rigid class is an irreducible divisor of numerical dimension 0. A related conceptis discussed briefly in [Nak04, Page 93 Remark]. Blowing up.Lemma 5.2.
Let X be a smooth projective variety and let Z be an k -dimensional subvariety.Suppose that there is an open neighborhood U ⊂ N k ( X ) of [ Z ] such that Z appears withpositive coefficient in any effective R -cycle with class in U . Then Z is strongly numericallyrigid. The point is that there is no assumed lower bound for the coefficient with which Z appearsin the cycles. Proof.
We first show that, perhaps after shrinking U , there is a constant ǫ > ǫZ ≤ T for any effective R -cycle T with numerical class in U . Suppose otherwise for acontradiction. Choose β in the interior of the movable cone (that is, the closure of the coneof classes subvarieties which deform to cover X ). For some sufficiently small τ we havethat β + τ [ Z ] is still in the interior of the movable cone. Thus, if α ∈ U has an effectiverepresentative where Z appears with coefficient c , the class α + cτ β is represented by aneffective R -cycle in which Z has coefficient 0. If there is an open neighborhood U ′ of [ Z ]with U ′ ⊂ U and admitting representatives with arbitrarily small coefficients of Z , we obtaina contradiction.We can now argue as in [Nak04, Page 93 Remark]: we define a function σ Z : Eff ◦ k ( X ) → R that records the infimum of the coefficients of Z appearing in any effective R -cycle ofclass α . This function is continuous on the big cone; by taking limits we extend it toa lower semicontinuous function on the entire pseudoeffective cone. Furthermore, for any α ∈ Eff k ( X ) and β in the interior of the movable cone, the restriction of σ Z to the ray α + tβ isstrictly decreasing in t . We deduce that σ Z ([ Z ]) >
0. An easy rescaling argument shows that σ Z ([ Z ]) = 1, and we conclude the strong numerical rigidity of [ Z ] by lower semi-continuityof σ Z . (cid:3) We can then test for the strong numerical rigidity of ∆ by blowing up ∆.
Proposition 5.3.
Let X be a smooth projective variety of dimension n . Let φ : W → X × X denote the blow-up of the diagonal and let i : E → W denote the inclusion of the exceptionaldivisor. Suppose that α ∈ N n ( X × X ) is a non-zero class such that φ ∗ α = M + i ∗ N where M ∈ N n ( W ) is a nef class and N ∈ N n ( E ) is a nef class.(1) If α · ∆ X = 0 then ∆ X is not big.(2) If α · ∆ X < then ∆ X is strongly numerically rigid.Proof. Let T be an effective n -cycle on W . If T is not supported on E then T · φ ∗ α is non-negative. Pushing forward, we see that the only effective n -cycle on X which can possiblyhave negative intersection with α is ∆ X itself. Thus:(1) Suppose α · ∆ X = 0. Then α is nef and thus ∆ X can not be big.(2) Suppose α · ∆ X <
0. Then also α · β < β sufficiently closeto [∆ X ]. This means that any effective representative of such a β must contain ∆ X in itssupport with positive coefficient. We conclude that ∆ X is strongly numerically rigid byLemma 5.2. (cid:3) For surfaces, we have the following criterion:
Proposition 5.4.
Let S be a smooth surface. Let φ : Y → S × S denote the blow-up of ∆ S .If φ ∗ [∆ S ] is not pseudoeffective, then ∆ S is strongly numerically rigid. OSITIVITY OF THE DIAGONAL 11
Proof.
We let E denote the exceptional divisor of φ and let g : E → ∆ S denote the projectivebundle map and ξ the class of the relative O (1) on E . We denote by i : E → Y the inclusion.Suppose that φ ∗ [∆ S ] is not pseudoeffective, and let η be a nef class in N ( Y ) such that η · φ ∗ [∆ S ] <
0. Choose a sufficiently small open subset U ⊂ N ( S × S ) of [∆ S ] such that η · φ ∗ β < β ∈ U . Let Z be any effective R -cycle on S × S such that [ Z ] ∈ U . Let T be any effective R -cycle on Y that pushes forward to Z ; after removing vertical components,we may suppose that T does not have any components contracted by φ . Let α denote theclass of T . We can write α = φ ∗ φ ∗ α + i ∗ g ∗ L for some (not necessarily effective) R -divisorclass L on S . Then since η · α ≥
0, and η · φ ∗ φ ∗ α <
0, we have η · i ∗ g ∗ L > g ∗ ( η | E ) · L > . Now, since η | E is nef and g is flat, g ∗ ( η | E ) is the class of a nef curve e η on S . Then, if π denotes the projection to the first factor, we find E · φ ∗ π ∗ e η · α = E · φ ∗ π ∗ e η · i ∗ g ∗ L = ( − ξ ) · g ∗ ( e η · L ) < . By the nefness of e η (and hence φ ∗ π ∗ e η ), we see that some component of T must be containedin E , and furthermore (since we removed all π -contracted components) this component mustdominate ∆ S under π . Pushing forward, we see that ∆ S must be contained in Z with positivecoefficient. We conclude by Lemma 5.2. (cid:3) Rigidity via the Hilbert scheme.
Using the rational map S × S Hilb ( S ), onecan study the positivity of ∆ S via the geometry of the Hilbert scheme. This approach issurprisingly successful, allowing us to use results arising from Bridgeland stability. Theorem 5.5.
Let S be a surface and let B ′ denote the divisor on Hilb ( S ) such that B ′ parametrizes non-reduced subschemes. For nef divisors H and A on X , consider D := H [2] − b B ′ and D := A [2] − b B ′ on Hilb ( S ) . If c ( S ) > and • D and D are movable and b b > A · Hc ( S ) then ∆ S is not nef. • D is nef, D is movable, and b b > A · Hc ( S ) then ∆ S is strongly numerically rigid.Proof. Let φ : Y → S × S be the blow-up along the diagonal. The exceptional divisor E isisomorphic to P (Ω S ) with projection g : E → S . Letting ξ denote the class of the relative O (1) and i : E → Y the injection, we have that φ ∗ ∆ S = i ∗ ( ξ − g ∗ K S ).Let ψ : Y → Hilb ( S ) denote the 2 : 1-map. Then we compute intersections by restrictingto E : ψ ∗ D · ψ ∗ D · φ ∗ ∆ S = (2 g ∗ H + b ξ ) · (2 g ∗ A + b ξ ) · ( ξ − g ∗ K S )= − b b c ( S ) + 4 A · H First suppose that D and D are movable and the inequality holds. Since ψ is finite, ψ ∗ D and ψ ∗ D are also movable, and hence their intersection is pseudoeffective. The assumedinequality shows that ψ ∗ D · ψ ∗ D · φ ∗ ∆ S <
0, so that ∆ is not nef.
Next suppose that D is nef and D is movable. Then ψ ∗ D · ψ ∗ D is nef and by the samecalculation as before we deduce that φ ∗ ∆ S is not pseudoeffective. By Proposition 5.4 ∆ S isstrongly numerically rigid. (cid:3) It would be interesting if Theorem 5.5 could be improved by a more in-depth study of thegeometry of the Hilbert scheme Hilb ( S ).5.3. Albanese map.
Let X be a smooth projective variety and let alb : X → A be theAlbanese map (for a chosen basepoint). By the subtraction map for X , we mean the com-position of alb × : X × X → A × A with the subtraction map for A . Note that this mapdoes not depend on the choice of basepoint. Proposition 5.6.
Let X be a smooth projective variety of dimension n . Suppose that theAlbanese map alb : X → A is generically finite onto its image but is not surjective. Then ∆ X is exceptional for the subtraction map.Proof. Note that the diagonal is contracted to a point by the subtraction map. Thus, itsuffices to prove that a general fiber of the subtraction map f : X × X → A has dimension < n . Let X ′ denote the image of the albanese map. Since alb is generically finite onto itsimage, it suffices to prove that the general fiber of the subtraction map f : X ′ × X ′ → A hasdimension < n .Suppose otherwise for a contradiction. For every closed point p ∈ f ( X ′ × X ′ ) the fiber F p denotes pairs of points ( x , x ) ∈ X ′ × X ′ such that x = p + x . If this has dimension n , then it must dominate X ′ under both projections. In other words, X ′ is taken to itselfunder translation by every point of f ( X ′ × X ′ ). Recall that X ′ contains the identity of A ,so that in particular X ′ ⊆ f ( X ′ × X ′ ). Thus, the subgroup of A fixing X ′ is all of A . Thisis a contradiction when X ′ = A . (cid:3) There are many other results of a similar flavor. For example, if the diagonal is theonly subvariety of dimension ≥ n contracted by the Albanese map then ∆ X is stronglynumerically rigid using arguments similar to those of [FL16, Theorem 4.15]. This situationholds for every curve of genus ≥ Surfaces
We now discuss positivity of the diagonal for smooth surfaces. First, by combining Theo-rem 3.5 with Corollary 4.2 (and using the equality of homological and numerical equivalencefor surface classes) we obtain:
Theorem 6.1.
The only smooth projective surfaces with big and nef diagonal are the pro-jective plane and fake projective planes.
In this section we discuss each Kodaira dimension in turn. We can summarize the discus-sion as follows: • The only possible surfaces with big diagonal are P or a surface of general typesatisfying p g = q = 0. In the latter case, the only example with big diagonal that weknow of is a fake projective plane. • If the Kodaira dimension of X is at most 1, then ∆ X is nef if and only if X has neftangent bundle, with the exception of some properly elliptic surfaces which admit nosection. Surfaces with nef tangent bundle are classified by [CP91].Note that any surface with nef diagonal must be minimal by Proposition 4.1. OSITIVITY OF THE DIAGONAL 13
Kodaira dimension −∞ .Proposition 6.2. Let X be a smooth surface of Kodaira dimension −∞ . Then(1) ∆ X is big if and only if X = P .(2) ∆ X is nef if and only if X has nef tangent bundle, or equivalently, if X is either P , P × P , or a projective bundle P ( E ) over an elliptic curve where E is either an unsplitvector bundle or (a twist of ) a direct sum of two degree line bundles.Proof. (1) Let S be a smooth uniruled surface and let g : S → T be a map to a minimalmodel. If T is not P , then T (and hence also S ) admits a surjective morphism to a curve.If T = P and g is not an isomorphism, then g factors through the blow up of P at a point,which also admits a surjective morphism to a curve. In either case Corollary 3.2 shows thatthe diagonal of S is not big.(2) We only need to consider minimal surfaces. Using the classification, we see that anyminimal ruled surface besides the ones listed carries a curve with negative self-intersectionor maps to a curve of genus ≥
2. By Proposition 4.1 and Lemma 4.3 such surfaces can nothave nef diagonal. (cid:3)
Kodaira dimension .Proposition 6.3. The diagonal of a surface of Kodaira dimension is not big.Proof. By Lemma 3.4 it suffices to prove this for minimal surfaces. Using the classificationand Theorem 3.5, the only surface which could have a big diagonal would be an Enriquessurface. However, such surfaces always admit a map to P and thus can not have big diagonalby Lemma 3.1. (cid:3) We next turn to nefness of the diagonal. Recall that any surface with nef diagonal must beminimal, and we argue case by case using classification. Abelian surfaces and hyperellipticsurfaces both have nef tangent bundles, and thus nef diagonal. For K3 surfaces, Theorem6.6 below verifies that the diagonal is never nef.Finally, any Enriques surface admits an ample divisor D with D = 2 which defines adouble ramified cover. Hence there is an involution i : S → S exchanging the two sheets.Then if Γ i is the graph, we have ∆ S · Γ i = − C < . , and so ∆ S is not nef.6.2.1. K3 surfaces.
K3 surfaces are perhaps the most interesting example, and in this sub-section we discuss them at some length. We first discuss nefness, and we start with a couplelow degree examples.
Example 6.4.
Let S → P be a degree 2 K3 surface. As for the Enriques surface, thereis an involution i : S → S , and intersecting ∆ S with the graph of the involution gives anegative number, so ∆ S is not nef. Example 6.5.
Let S be a surface in P , and let W = ^ S × S be the blow-up along thediagonal. Consider the divisor H + H − E , where H i is the pullback of the hyperplanesection via the i -th projection. This divisor is base-point free, and defines a morphism φ : W → Gr (2 , . Geometrically, this is the morphism obtained by sending a pair of points on S to the linethey span; it is finite when S contains no lines.Now suppose that S is a quartic K3 surface. Then( H + H − E ) π ∗ ∆ S = ( H + H + O (1)) O (1) = (2 H ) −
24 = − . In particular, ∆ S has negative intersection with the images of the fibers of φ . The previous example shows how knowledge of the nef cone of the blow-up ^ S × S , orequivalently, Hilb ( S ), can be used to produce interesting subvarieties of S × S havingnegative intersection with ∆ S . By the work of Bayer–Macr`ı, we can use similar argumentsalso for higher degrees. Theorem 6.6.
Let S be a K3 surface. Then ∆ S is not nef. We first prove a special case:
Lemma 6.7.
Let S be a K3 surface of Picard rank polarized by an ample divisor H ofdegree d ≥ . Then ∆ S is not nef.Proof. We start by recalling the results of [BM14] on the geometry of Hilb ( S ). Supposethat d/ x − ( d/ y = 1 must have x ≥ p d/
2, so that the fundamental solution yields a ratio yx = r d r − x ≥ r d r d − d . Set b d = q d − ≤ d · yx . Applying [BM14, Proposition 13.1], we see that (whether or not d/ H ′ − b d B is movable on Hilb ( S ), where H ′ is induced bythe symmetric power of H and 2 B is the exceptional divisor for the Hilbert-Chow morphism.We then apply Theorem 5.5. The only verification necessary is: b d = d − > d d in our range. (cid:3) In fact, the previous proof gives a little more: over the family of degree d K3 surfaces, wehave a class on the total space which restricts to be effective on a very general K3 surface andwhich has constant negative intersection against ∆ S for such surfaces. Applying Proposition2.4, we can take limits to deduce that for every K3 surface in the family, ∆ S has negativeintersection against a pseudoeffective class. This concludes the proof of Theorem 6.6 indegree ≥
4, and we have already done the degree 2 case in Example 6.4.6.2.2.
Rigidity for K3 surfaces.
By again appealing to the results of [BM14], we can showrigidity under certain situations.
Proposition 6.8.
Let S be a K3 surface of Picard number and degree d . Suppose that thePell’s equation x − dy = 5 has no solutions. Then the diagonal is strongly numerically rigid. For example, the theorem applies when the degree is divisible by 4, or when the degree isless than 50 except for degrees 2 , , , Proof.
By combining [BM14, Lemma 13.3] with the calculation in the proof of Lemma 6.7,we obtain the result from Theorem 5.5. (cid:3)
Finally, we will prove that the diagonal of a very general K3 of degree d is numericallyrigid, using a deformation argument.Standard results on K3 surfaces give the existence of a degree d K3 surface S which isalso a quartic surface. It follows by the computation in Example 6.5 that π ∗ (∆ S ) is notpseudoeffective. Now take a family S → T of polarized degree d surfaces in a neighbourhoodof S . Let π : ^ S × T S → S × T S be the blow-up of the diagonal. The induced family OSITIVITY OF THE DIAGONAL 15 ^ S × T S → T is a smooth morphism. Consider the cycle class ( π t ) ∗ (∆ S t ) = ( π ∗ ∆ S /T ) t . Sincethis is not pseudoeffective on the special fiber, π ∗ (∆ S t ) is not pseudoeffective for t verygeneral, by Proposition 2.4. So applying again Proposition 5.4, we see that ∆ is stronglynumerically rigid on the very general K3 surface of degree d . Theorem 6.9.
Let S be a very general polarized K3 surface. Then the diagonal is stronglynumerically rigid. A posterori, this result is intuitive in light of the Torelli theorem, at least for subvarietiesof S × S which are graphs of self-maps f : S → S : if Γ is such a graph and [Γ] = [∆], then f induces the identity on H ( S, Z ), and hence has to be the identity, and so Γ = ∆ S .6.3. Kodaira dimension . Let S be a surface of Kodaira dimension 1 and let π : S → C be the canonical map. By Corollary 3.2, we have: Corollary 6.10.
A surface of Kodaira dimension does not have big diagonal. We next show that the diagonal is not nef when π admits a section. As usual we mayassume S is minimal so that K S is proportional to some multiple of a general fiber of π .Using Lemma 4.3, we see that if the diagonal is nef then the base C of the canonical mapmust have genus either 0 or 1. If T is a section of π , then by adjunction we see that T < S is not nef (since as before ∆ S · ( π ∗ T · π ∗ T ) < Example 6.11.
When S → C does not admit a section, it is possible for the diagonalto be nef. Indeed, let E be an elliptic curve without complex multiplication and let C be a hyperelliptic curve of genus g which is very general in moduli. The product E × C admits an involution i which acts on E as translation by a 2-torsion point and on C by thehyperelliptic involution. The quotient surface S = ( E × C ) /i is a properly elliptic surfaceof Kodaira dimension 1. The elliptic fibration S → C/i = P has a non-reduced fiber, andtherefore can not admit a section. We claim that the diagonal of S is nef.Let S ′ = E × C and let Γ denote the graph of the involution i . By the projection formulait is enough to check that ∆ S ′ + Γ is nef on S ′ × S ′ . Indeed, if π : S ′ → S is the quotientmap, the map π × π is flat, and ( π × π ) ∗ ∆ S = ∆ S ′ + Γ is nef if and only if ∆ S is.Let f : S ′ × S ′ → C × C denote the projection map π × π . Claim:
If an irreducible surface T ⊂ E × C × E × C is not nef, then it maps to a curve D in C × C with negative self-intersection. Furthermore it can only have negative intersectionwith surfaces contained in f − ( D ). Proof.
It is clear that T is nef if it maps to a point in C × C . We next prove nefness if T maps dominantly onto C × C . Fix an irreducible surface V ; we will show T · V ≥
0. Itsuffices to consider the case when V is not a fiber of the map to C × C . In this situation wecan deform T using the abelian surface action so that it meets V in a dimension 0 subset.Indeed, the set Y ⊂ C × C of points y such that the fiber T ∩ f − ( y ) is 1-dimensional isfinite. For a general translation in E × E , this curve will meet V ∩ f − ( y ) in a finite setof points. Next consider the open set U = C × C \ Y . Let W ⊂ T be the subset lying over U . We have a finite map from E × E × W → E × E × U given by ( a, b, w ) ( a, b ) · w . Inparticular, the preimage of V in E × E × W will be a surface, and thus will meet a generalfiber of E × E × W → E × E properly. Altogether we see a general translation of T willmeet V properly. Thus T is nef.Finally, suppose that T maps to a curve D and let F = f − ( D ). Suppose V is a surfacenot contained in F . Then V · T can be computed by restricting V to F . This restriction iseffective, and hence nef (by the group action on F ), showing that T · V ≥
0. If V is alsocontained in F , then V · T in X is the same as f ∗ ( v · t ) · D on C × C , where V = i ∗ v , T = i ∗ t . Note that f ∗ ( v · t ) will be a non-negative multiple of D in C × C . So if D ≥
0, we againfind that T · V ≥
0. This completes the proof of the claim above. (cid:3)
Note that if the diagonal ∆ S is not nef, there is a surface T ⊂ S ′ × S ′ with T · (Γ+∆ S ′ ) < · T < S ′ · T <
0; replacing T by i ( T ), we may assume that thelatter is the case.Arguing as above, we see ∆ S ′ and T are both contained in the preimage L of ∆ C . Theintersection Γ · L is transversal; it consists of the points of Γ over the 2-torsion points of C .In particular, the the restriction of Γ to L is numerically equivalent to(∆ C · Γ C )Γ E = (2 g + 2)Γ E where Γ E is the pushforward of the graph of the involution from a fiber E × E . In contrast, i ∗ i ∗ ∆ S will be i ∗ i ∗ ∆ S = (2 − g )∆ E where ∆ E is the pushforward of the diagonal from a fiber E × E . Since Γ E and ∆ E arenumerically proportional, we see that ∆ S ′ + Γ is nef when restricted to this threefold. Henceits intersection with T is non-negative, and so ∆ S ′ + Γ is nef overall.6.4. Surfaces of general type.
Surfaces with vanishing genus.
By Castelnuovo’s formula, a surface of general typesatisfying p g = 0 also must satisfy q = 0. The minimal surfaces satisfying these conditionsare categorized according to K S , which is an integer satisfying 1 ≤ K S ≤
9, and have Picardrank 10 − K S . It is interesting to look for examples where bigness holds or fails.For such surfaces Theorem 3.9 shows: Corollary 6.12.
Let S be a smooth surface satisfying p g ( S ) = q ( S ) = 0 . If Eff ( S ) issimplicial, then ∆ S is big if and only if every nef divisor is big. However, determining bigness can still be subtle. K = : The surfaces here are exactly the fake projective planes, and we saw in theintroduction that ∆ S is both big and nef.Suppose we blow-up a very general point to obtain a surface Y . The results of [Ste98] onSeshadri constants show that Y carries a divisor which is nef and has self-intersection 0, sothe diagonal for Y is neither nef or big. K = : Since these surfaces have Picard rank 2, the pseudoeffective cone is automaticallysimplicial. Thus we have an interesting trichotomy of behaviors: • Eff ( S ) = Nef ( S ). Let D , D be generators of the two rays of the pseudoeffectivecone, and set a = D · D . Then∆ S = F + F + 1 a π ∗ D · π ∗ D + 1 a π ∗ D · π ∗ D is nef. However, ∆ S is not big since S carries a non-zero nef class with self-intersection0. • If exactly one extremal ray of Eff ( S ) is nef, then S carries both a curve of negativeself-intersection and a nef class with vanishing self-intersection. Thus ∆ S is neitherbig nor nef. • If no extremal rays of Eff ( S ) are nef, then ∆ S is big by Lemma 3.9 but is not nef.There are a few known geometric constructions of such surfaces. First, there are the surfacesconstructed explicitly via ball quotients which are classified in [Dˇza14] and [LSV15]. Second,there are the surfaces admitting a finite ´etale cover which is a product of two curves. Suchsurfaces are classified in [BCG08] and are further subdivided into two types. Write S = OSITIVITY OF THE DIAGONAL 17 ( C × C ) /G for some finite group G acting on the product. If no element of G swaps thetwo factors, then G acts on each factor separately. This is known as the “unmixed” case.There is also the “mixed” case, when C ∼ = C and some elements of G swap the two factors.In the unmixed case, we have C /G ∼ = C /G ∼ = P , and we obtain two maps S → P suchthat the pullbacks of O (1) generate the pseudoeffective cone. In particular ∆ S is nef but notbig. However, we do not know what happens in the other two situations. Burniat surfaces : These are certain surfaces constructed as Galois covers of weak delPezzos (see for example [Ale16]). By pulling back from the del Pezzo we obtain nef divisorswith self-intersection 0, so that ∆ S is not big. The Godeaux surface : This surface is the quotient of the Fermat quintic by a Z / P (see for example the second-to-last paragraphon page 3 of [GP02]), so ∆ S is not big.6.4.2. Surfaces with non-vanishing genus.
We next discuss several classes of surfaces of gen-eral type where we can apply our results. These examples have p g >
0, and so ∆ S cannotbe big.We note that by the computations of [BC13], if H is a very ample divisor on a surface S then (maintaining the notation of Section 5.2) H [2] − B ′ is nef. Example 6.13 (Surfaces in P ) . Suppose that S is a smooth degree d hypersurface in P . Then c ( S ) = d − d + 6 d . Thus Theorem 5.5 shows that the diagonal is stronglynumerically rigid and is not nef as soon as d ≥ Example 6.14 (Double covers) . Suppose that S is a double cover of P ramified over asmooth curve of even degree d . Then S is of general type once d ≥
8. These surfaces have c ( S ) = d − d + 6 and carry a very ample divisor of degree d . Thus Theorem 5.5 showsthat the diagonal is strongly numerically rigid and is not nef as soon as d ≥ Example 6.15 (Horikawa surfaces) . Minimal surfaces of general type satisfying q ( S ) = 0and K S = 2 p g ( S ) − K S is not very ample, it is big andbasepoint free, which is enough to determine that K [2] S − B ′ is movable on Hilb ( S ). Usingthe equality c ( S ) = 12 + 12 p g ( S ) − K S , we see that the diagonal for such surfaces is nevernef by Theorem 5.5. 7. Higher dimensional examples
Quadric hypersurfaces.
An odd dimensional quadric is a fake projective space andthus will have big and nef diagonal as discussed in the introduction. An even dimensionalquadric will have diagonal that is nef but not big. Indeed, if X is a quadric of dimension 2 k ,then X carries two disjoint linear spaces of dimension k . These linear spaces are nef (since X is homogeneous), but not big, and hence ∆ X is not big.7.2. Nefness for hypersurfaces.
Example 6.13 shows that the diagonal of a smooth hy-persurface in P of degree at least 3 is not nef. The same is true in arbitrary dimension: Proposition 7.1.
Let X ⊂ P n +1 be a smooth degree d hypersurface. If d ≥ then thediagonal for X is not nef.Proof. The Euler characteristic of X is c n ( X ) = (1 − d ) n +2 − d + n + 2 . If n is odd, then ∆ X = c n ( X ) < X is not nef. If X has even dimension, thenCorollary 4.5 applied to the projection map from a general point outside X shows that ∆ X is not nef. (cid:3) Bigness and rigidity for hypersurfaces.
Suppose that X ⊂ P n +1 is a smooth hy-persurface of dimension n . We will apply Lemma 5.3 to show that the diagonal is not big forhypersurfaces of small dimension. Note that homological non-bigness follows from Theorem3.5 once the degree is larger than n + 1, so we will focus only on the Fano hypersurfaces.Recall that Lemma 5.3 requires us to find a class α ∈ N n ( X × X ) whose pullback underthe blow-up of the diagonal φ : W → X × X has a special form. We record the relevantinformation in the table below. The first column records the kind of hypersurface. Thesecond column records the class α – we will let H and H denote the pullback of thehyperplane class under the two projection maps.The third column verifies that Lemma 5.3 applies to α . This is done by rewriting φ ∗ α interms of Schubert classes pulled back under the natural morphism g : W → G (2 , n + 2). Wewill let i : E → W denote the inclusion of the exceptional divisor. Identifying E ∼ = P (Ω X )this divisor carries the class ξ of the relative O (1) and the pullback H of the hyperplaneclass from the base of the projective bundle. For convenience we let h : E → Gr (2 , n + 2)denote the restriction of g to E . type α φ ∗ α cubic threefold H H + H H + ∆ X g ∗ σ , + i ∗ ( h ∗ σ + 4 H )quartic threefold 2 H H + 2 H H + ∆ X g ∗ σ , + i ∗ ( h ∗ σ + 9 H )cubic fourfold H + H H + 3 H H + H H + H − ∆ X g ∗ σ + 2 g ∗ σ , + i ∗ (2 h ∗ σ , + 8 H )quartic fourfold H + H H + 7 H H + H H + H − ∆ X g ∗ σ + 6 g ∗ σ , + i ∗ (3 h ∗ σ , + 24 H )quintic fourfold H + H H + 13 H H + H H + H − ∆ X g ∗ σ + 12 g ∗ σ , + i ∗ (4 h ∗ σ , + 56 H )cubic fivefold H H + H H + H H + H H + ∆ X g ∗ σ , + i ∗ ( h ∗ σ + 4 h ∗ σ , + 12 H )quartic fivefold 2 H H + 11 H H +11 H H + 2 H H + ∆ X g ∗ σ , + 9 H H g ∗ σ , + i ∗ ( h ∗ σ + 9 h ∗ σ , + 62 H )quintic fivefold 3 H H + 35 H H +3 H H + 3 H H + ∆ X g ∗ σ , + 32 H g ∗ σ , + i ∗ ( h ∗ σ + 16 h ∗ σ , + 208 H )sextic fivefold 4 H H + 79 H H +4 H H + 4 H H + ∆ X g ∗ σ , + 75 H g ∗ σ , + i ∗ ( h ∗ σ + 25 h ∗ σ , + 500 H )cubic sixfold H + H H + 3 H H + 3 H H +3 H H + H H + H − ∆ X g ∗ σ + 2 g ∗ σ , + i ∗ (2 h ∗ σ , + 8 Hh ∗ σ , + 24 H ) In all cases the class α satisfies α · ∆ ≤
0, and in all but the first the intersection isnegative. By Lemma 5.3, a smooth cubic threefold has non-big diagonal and in all othercases the diagonal is strongly numerically rigid.It seems very likely that the same approach will work for all hypersurfaces of degree ≥ ≥
3. Indeed, as the degree increases the coefficients are becoming morefavorable. We have verified strong numerical rigidity for several more cubic hypersurfacesbut unfortunately the combinatorics become somewhat complicated.7.4.
Grassmannians.
For a Grassmannian X = Gr ( k, n ), the tangent bundle is globallygenerated, and hence the diagonal is nef. However, when X is not a projective space, thediagonal is not big because the Schubert classes on X are universally pseudoeffective butnot big. OSITIVITY OF THE DIAGONAL 19
Toric varieties. [FS09, Proposition 5.3] shows that the only smooth toric varieties forwhich the pseudoeffective and nef cones coincide for all k are products of projective spaces.Thus products of projective spaces are the only toric varieties with ∆ X nef.Theorem 3.9 shows that for a toric variety ∆ X is big if and only if every nef class on X isbig. Although it seems reasonable to expect that the only toric varieties with big diagonalare projective spaces, this turns out to be false. Example 7.2. [FS09] constructs an example of a toric threefold which does not admit amorphism to a variety of dimension 0 < d <
3. In particular, this implies that every nefdivisor is big. Dually, every nef curve is big. Thus X has big diagonal.Note that, aside from projective space, the diagonal can not be big for smooth projectivetoric varieties of Picard rank ≤ ≤ Varieties with nef tangent bundle.
Varieties with nef tangent bundle are expectedto have a rich geometric structure. A conjecture of Campana–Peternell [CP91] predicts thatsuch varieties are (up to ´etale cover) flat bundles with rational homogeneous fibers over theiralbanese variety. To prove this result, it suffices by a result of [DPS94] to show that a Fanovariety with nef tangent bundle is rational homogeneous. This conjecture has been verifiedup to dimension 5 and in several related circumstances (see [Mok88], [Hwa01], [Mok02],[Wat14], [Kan15], [Wat15]).A variety with nef tangent bundle has nef diagonal. However, the diagonal can only bebig if the albanese map is trivial.
Example 7.3.
A variety with nef tangent bundle need not carry an action of an algebraicgroup – this can only be expected after taking an ´etale cover. An explicit example is discussedin [DPS94, Example 3.3].However, the diagonal can still be universally pseudoeffective in this situation. Indeed,suppose we have an ´etale cover f : Y → X where Y admits a transitive action of an algebraicgroup. The induced flat finite map f × : Y × Y → X × X has that f × ∗ ∆ Y = d ∆ X . Ofcourse the diagonal of Y is universally pseudoeffective and since universal pseudoeffectivenessis preserved by flat pushforward for smooth varieties by [FL17, Theorem 1.8] we deduce that∆ X is also universally pseudoeffective.8. Threefolds
In this section we discuss a couple classification results for threefolds.8.1.
Fano threefolds.
The Fano threefolds which are fake projective spaces are are P , thequadric, the Del Pezzo quintic threefolds V , and the Fano threefolds V ; see [LS86]. Asdiscussed above these will have big and nef diagonal.Any Fano threefold with nef diagonal must be primitive and the contractions correspond-ing to the extremal rays of Eff ( X ) can not be birational. Another obstruction to nefness isthe fact that ∆ X is the topological Euler characteristic of X . In addition to the fake projec-tive spaces discussed earlier, the classification of [IPP +
99] and [Ott05] leaves 7 possibilities: • Picard rank 1: V , the intersection of two quadrics in P . • Picard rank 2: a double cover of P × P with branch locus (2 , P × P of type (1 ,
2) or (1 , P × P . • Picard rank 3: P × P × P .The threefolds on this list with nef tangent bundle are classified by [CP91] and will automat-ically have nef diagonal: these are the products of projective spaces and the (1 ,
1) divisor in P × P . The divisor of type (1 ,
2) on P × P will also have nef diagonal, which we cansee as follows. Let { D , D } be the basis of N ( X ) consisting of extremal nef divisors and { C , C } denote the basis of N ( X ) consisting of extremal nef curves. Since h , ( X ) = 0 wecan write the diagonal very explicitly using products of these basis elements and it is easyto see that it is a non-negative sum of nef classes. We are unsure of what happens in theremaining 3 cases.Any Fano threefold with big diagonal must have that the contractions corresponding tothe extremal rays of Eff ( X ) are birational. We focus on the primitive Fano threefolds; goingthrough the classification of [Ott05], we see that any such threefold must have Picard rank1. Unfortunately it seems subtle to determine which of these threefolds actually have bigdiagonal.8.2. Threefolds with nef and big diagonal.Proposition 8.1.
Let X be a smooth minimal threefold of Kodaira dimension ≥ . Then X does not have homologically big diagonal.Proof. If ∆ X is homologically big, then H k, ( X ) = 0 for all k >
0. In particular, χ ( O X ) = 1.By Riemann–Roch, χ ( O X ) = c c , so we find that c c = 24. In particular, c ( X ) = 0. If0 < κ ( X ) <
3, then X admits a map to a lower-dimensional variety, contradicting bignessof ∆ X . If X has general type, we have by the Miayoka–Yau inequality,0 > − K = c ≥ c c = 64This is a contradiction. (cid:3) Corollary 8.2.
Let X be a smooth threefold such that ∆ X is nef and homologically big.Then X is a fake projective space: P , the quadric, the Del Pezzo quintic threefolds V , theFano threefolds V .Proof. Suppose first that κ ( X ) ≥
0. Since ∆ X is nef, X must be a minimal threefold. Weconclude by the previous proposition that the diagonal can not be big.Thus we know that X is uniruled. Furthermore N ( X ) = R , and so X is a Fano threefoldof Picard number 1. Note also that χ ( X ) = ∆ X ≥
1, so in particular h , is at most 1. Goingthrough the classification of such Fano threefolds [IPP +
99] reveals that the only possibilitiesare the fake projective 3-spaces. (cid:3)
Recall that when X is a threefold not of general type then numerical and homologicalequivalence coincide on X × X . Thus the corollary also classifies the threefolds not ofgeneral type which have nef and big diagonal.9. Cohomological decomposition of the diagonal and positivity
This section discusses the relationship of Theorem 1.1 with the decomposition of thediagonal in cohomology on a surface. The following result is surely well-known to experts,but we include a proof for the convenience of the reader.
Proposition 9.1.
Let S be a smooth projective surface. Then:(1) ∆ S is homologous to a sum of cycles contracted by the projection maps if and onlyif p g ( S ) = 0 .(2) ∆ S is homologous to a sum of products of pullbacks of classes from the two projectionsif and only if p g ( S ) = q ( S ) = 0 . OSITIVITY OF THE DIAGONAL 21
Proof.
We first prove (2). Note that ∆ is homologous to a sum Z + Z ′ where Z is a cyclesupported in a divisor D × S and Z ′ is supported on a fiber S × s of the projection map. By aBloch–Srinivas type argument (as in [Voi07, Theorem 10.17]), we find that p g ( S ) = q ( S ) = 0.For the reverse implication, the K¨unneth formula implies that all of H , ( S × S ) is algebraicand generated by products of pullbacks of divisors from the two projections.Now we prove (1). The forward implication follows again from the argument of [Voi07,Theorem 10.17]. Conversely, the arguments of [BKL76] using the classification theory ofsurfaces show that for surfaces with κ ( S ) < p g ( S ) = 0 there is a curve C ⊂ S suchthat CH ( C ) → CH ( S ) is surjective. Using the decomposition of the diagonal as in [BS83],we see that the reverse implication holds except possibly for surfaces of general type. But asurface of general type satisfying p g ( S ) = 0 also satisfies q ( S ) = 0 by Castelnuovo’s theorem.Thus we are reduced to (2). (cid:3) It is natural to ask whether one can obtain a tighter link between positivity and decom-positions of ∆ X than Theorem 3.5. Following an idea of [DJV13], we will prove such astatement for surfaces by perturbing the diagonal by an external product of ample divisors. Proposition 9.2.
Let S be a smooth projective surface. Then p g ( S ) = 0 if and only if thereis an ample divisor H such that ∆ S + π ∗ H · π ∗ H is big.Proof. We first prove the forward implication. By Proposition 9.1, we have an equality ofnumerical classes ∆ S = a F + b F + r X i =1 a i E i + r X j =1 b j E ′ j where a i , b j ∈ Q , each E i is an irreducible surface contracted to a curve by π , and each E ′ j is an irreducible surface contracted to a curve by π . Note that a + r X i =1 a i E i · F = 1 b + r X j =1 b j E ′ j · F = 1For each i , let C i denote the normalization of the image π ( E i ) and let D i denote C i × S .For notational convenience, we will omit the normalization and write C i and D i as if theywere subvarieties of S and S × S .Fix a small ǫ > ǫ < /r and ǫ < /r . Set c i = − a i E i · F so that a i E i + ( c i + ǫ ) F is π -relatively ample as a divisor on D i . Thus, for some sufficiently large ample H i on S ,we have that a i E i + ( c i + ǫ ) F + π | ∗ D i H i is an effective class on D i . Pushing forward to S × S and adding up as i varies, we see that( r ǫ + a − F + r X i =1 ( a i E i + π ∗ C i · π ∗ H i )is an effective class on S × S . Arguing symmetrically, with analogous notation,( r ǫ + b − F + r X j =1 (cid:0) b j E ′ j + π ∗ H ′ j · π ∗ C ′ j (cid:1) is an effective class. Of course, we can replace the C i , H i , C ′ j , H ′ j by larger ample divisorswithout affecting the effectiveness of this class. All told, there is an effective surface class Q and a positive sum of external products of ample divisors N such that∆ S + N = (1 − r ǫ ) F + (1 − r ǫ ) F + Q. Adding on a further external product of amples to both sides we can ensure that the righthand side is big: if A is ample on X , a class of the form c F + c F + c π ∗ A · π ∗ A withpositive coefficients will dominate a small multiple of the big class ( π ∗ A + π ∗ A ) . Finally,any positive sum of external products of amples is dominated by a single external productof amples, finishing the proof of the implication.Conversely, suppose h , ( X ) > α ∈ H , ( X ) be non-zero. Let β be the nef classconstructed as in Theorem 3.5 satisfying β · ∆ S = 0. This β also satisfies β · π ∗ H · π ∗ H = 0.Since β is nef, there can be no big class of the desired form (again appealing to the equalityof homological and numerical equivalence). (cid:3) Questions
We finish with a list of questions raised by our work.
Question 10.1.
Are P and fake projective planes the only smooth surfaces with big diag-onal? Question 10.2.
Suppose that S is a smooth surface S of general type with q ( S ) = 0 and p g ( S ) >
0. Is the diagonal for S numerically rigid?It is natural to ask whether some of the results for surfaces generalize for higher dimensions: Question 10.3.
Does a smooth projective variety with big and nef diagonal have the samerational cohomology as projective space?
Question 10.4.
Are the only smooth projective varieties with big diagonal either uniruledor of general type?
Question 10.5.
Is there a threefold of general type with nef diagonal?
Question 10.6.
Are there any topological restrictions on smooth varieties with nef diagonalaside from c n ( X ) ≥
0? For example, does a threefold with nef diagonal satisfy χ ( O X ) ≥ References [Ale16] Valery Alexeev. Divisors of Burniat surfaces. In
Development of moduli theory—Kyoto 2013 , vol-ume 69 of
Adv. Stud. Pure Math. , pages 287–302. Math. Soc. Japan, [Tokyo], 2016.[BC13] Aaron Bertram and Izzet Coskun. The birational geometry of the Hilbert scheme of points onsurfaces. In
Birational geometry, rational curves, and arithmetic , pages 15–55. Springer, NewYork, 2013.[BCG08] Ingrid Bauer, Fabrizio Catanese, and Fritz Grunewald. The classification of surfaces with p g = q = 0 isogenous to a product of curves. Pure Appl. Math. Q. , 4(2, Special Issue: In honor of FedorBogomolov. Part 1):547–586, 2008.[BKL76] Spencer Bloch, Arnold. Kas, and David Lieberman. Zero cycles on surfaces with p g = 0. CompositioMath. , 33(2):135–145, 1976.[BM14] Arend Bayer and Emanuele Macr`ı. MMP for moduli of sheaves on K3s via wall-crossing: nef andmovable cones, Lagrangian fibrations.
Invent. Math. , 198(3):505–590, 2014.[BS83] Spencer Bloch and V. Srinivas. Remarks on correspondences and algebraic cycles.
Amer. J. Math. ,105(5):1235–1253, 1983.[CP91] Fr´ed´eric Campana and Thomas Peternell. Projective manifolds whose tangent bundles are numer-ically effective.
Math. Ann. , 289(1):169–187, 1991.[CS10] Donald I. Cartwright and Tim Steger. Enumeration of the 50 fake projective planes.
C. R. Math.Acad. Sci. Paris , 348(1-2):11–13, 2010.[DELV11] Olivier Debarre, Lawrence Ein, Robert Lazarsfeld, and Claire Voisin. Pseudoeffective and nefclasses on abelian varieties.
Compos. Math. , 147(6):1793–1818, 2011.[Dem07] Jean-Pierre Demailly. Complex analytic and differential geometry, 2007.[DJV13] Olivier Debarre, Zhi Jiang, and Claire Voisin. Pseudo-effective classes and pushforwards.
PureAppl. Math. Q. , 9(4):643–664, 2013.
OSITIVITY OF THE DIAGONAL 23 [DPS94] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider. Compact complex manifolds withnumerically effective tangent bundles.
J. Algebraic Geom. , 3(2):295–345, 1994.[Dˇza14] Amir Dˇzambi´c. Fake quadrics from irreducible lattices acting on the product of upper half planes.
Math. Ann. , 360(1-2):23–51, 2014.[FL16] Mihai Fulger and Brian Lehmann. Morphisms and faces of pseudo-effective cones.
Proc. Lond.Math. Soc. (3) , 112(4):651–676, 2016.[FL17] Mihai Fulger and Brian Lehmann. Positive cones of dual cycle classes.
Algebr. Geom. , 4(1):1–28,2017.[FS09] Osamu Fujino and Hiroshi Sato. Smooth projective toric varieties whose nontrivial nef line bundlesare big.
Proc. Japan Acad. Ser. A Math. Sci. , 85(7):89–94, 2009.[Fu12] Lie Fu. On the coniveau of certain sub-Hodge structures.
Math. Res. Lett. , 19(5):1097–1116, 2012.[GP02] Vladimir Guletski˘ı and Claudio Pedrini. The Chow motive of the Godeaux surface. In
Algebraicgeometry , pages 179–195. de Gruyter, Berlin, 2002.[Hor76] Eiji Horikawa. Algebraic surfaces of general type with small C . I. Ann. of Math. (2) , 104(2):357–387, 1976.[Hsi15] Jen-Chieh Hsiao. A remark on bigness of the tangent bundle of a smooth projective variety and D -simplicity of its section rings. J. Algebra Appl. , 14(7):1550098, 10, 2015.[Hwa01] Jun-Muk Hwang. Geometry of minimal rational curves on Fano manifolds. In
School on VanishingTheorems and Effective Results in Algebraic Geometry (Trieste, 2000) , volume 6 of
ICTP Lect.Notes , pages 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.[IPP +
99] Vasilii Iskovskikh, Yuri Prohorov, Alexei N Parshin, Igor Rostilavovich Shafarevich, and S Tregub.
Algebraic Geometry: Fano Varieties. V . Springer-Verlag, 1999.[Kan15] Akihiro Kanemitsu. Fano 5-folds with nef tangent bundles, 2015. arXiv:1503.04579 [math.AG].[Lie68] David I. Lieberman. Numerical and homological equivalence of algebraic cycles on Hodge mani-folds.
Amer. J. Math. , 90:366–374, 1968.[LS86] Antonio Lanteri and Daniele Struppa. Projective manifolds with the same homology as P k . Monatsh. Math. , 101(1):53–58, 1986.[LSV15] Benjamin Linowitz, Matthew Stover, and John Voight. Fake quadrics, 2015. arXiv:1504.04642[math.AG].[Mok88] Ngaiming Mok. The uniformization theorem for compact K¨ahler manifolds of nonnegative holo-morphic bisectional curvature.
J. Differential Geom. , 27(2):179–214, 1988.[Mok02] Ngaiming Mok. On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties ofminimal rational tangents.
Trans. Amer. Math. Soc. , 354(7):2639–2658 (electronic), 2002.[Nak04] Noboru Nakayama.
Zariski-decomposition and abundance , volume 14 of
MSJ Memoirs . Mathemat-ical Society of Japan, Tokyo, 2004.[Ott05] Andreas Ott. On Fano threefolds with b ≥
2, 2005. Diploma thesis.[Ott15] John Christian Ottem. Nef cycles on some hyperkahler fourfolds, 2015. arXiv:1505.01477[math.AG].[PY07] Gopal Prasad and Sai-Kee Yeung. Fake projective planes.
Invent. Math. , 168(2):321–370, 2007.[RT15] Michele Rossi and Lea Terracini. Fibration and classification of a smooth projective toric varietyof low Picard number, 2015. arXiv:1507.00493 [math.AG].[Ste98] Andreas Steffens. Remarks on Seshadri constants.
Math. Z. , 227(3):505–510, 1998.[Tan11] Sergei G Tankeev. On the standard conjecture of lefschetz type for complex projective threefolds.ii.
Izvestiya: Mathematics , 75(5):1047, 2011.[Voi07] Claire Voisin.
Hodge theory and complex algebraic geometry. II , volume 77 of
Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge, english edition, 2007. Translatedfrom the French by Leila Schneps.[Voi10] Claire Voisin. Coniveau 2 complete intersections and effective cones.
Geom. Funct. Anal. ,19(5):1494–1513, 2010.[Wat14] Kiwamu Watanabe. Fano 5-folds with nef tangent bundles and Picard numbers greater than one.
Math. Z. , 276(1-2):39–49, 2014.[Wat15] Kiwamu Watanabe. Fano manifolds with nef tangent bundle and large Picard number.
Proc. JapanAcad. Ser. A Math. Sci. , 91(6):89–94, 2015.
Department of Mathematics, Boston College, Chestnut Hill, MA 02467
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