Pseudoeffective and nef classes on abelian varieties Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin Abstract We study the cones of pseudoeffective and nef cycles of higher codimension on the self product of an elliptic curve with complex multiplication, and on the product of a very general abelian surface with itself. In both cases, we find for instance the existence of nef classes that are not pseudoeffective, answering in the negative a question raised by Grothendieck in correspondence with Mumford. We also discuss several problems and questions for further investigation. Introduction The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension and dimensionhavestartedtocomeintofocusonlyrecently,forinstancein[17]and[20].Thepurpose ofthispaperistoexploresomeofthephenomenathatcanoccurbyworkingoutthepicturefairly completely in a couple of simple but non-trivial cases. Specifically, we study cycles of arbitrary codimension on the self product of an elliptic curve with complex multiplication, as well as two dimensional cycles on the product of a very general abelian surface with itself. Already one finds various non-classical behavior, e.g. nef cycles whose product is negative.1 We also present a number of conjectures and problems for further investigation. Turning to more details, let X be a smooth complex projective variety of dimension n. Given 0 (cid:54) k (cid:54) n, denote by Nk(X) the finite dimensional real vector space of numerical equivalence classes of codimension k algebraic cycles on X with real coefficients. We consider the closed convex cone Psefk(X) ⊆ Nk(X) generated by effective cycles. By analogy with the case of divisors, the elements of this cone are calledpseudoeffectiveclasses.ThevectorspaceNk(X)isdualtoNn−k(X),and(againextending the codimension one terminology) we define Nefk(X) ⊆ Nk(X) to be the closed convex cone dual to Psefn−k(X). Thus a class α ∈ Nk(X) is nef if and only if (cid:0)α·β(cid:1) (cid:62) 0 for all effective cycles β of dimension k. Now suppose that B is an abelian variety: write B = V/Λ where V is a complex vector space and Λ ⊂ V is a lattice. Numerical and homological equivalence coincide on B ([13], or [3, 2010 Mathematics Subject Classification 14C25, 14K99 Keywords: pseudoeffective and nef cycle, abelian variety Research of the second author partially supported by NSF grant DMS-0700774. Research of the third author partially supported by NSF grant DMS-0652845. 1This answers a question raised in 1964 by Grothendieck [9] in correspondence with Mumford: see Remark 2.3. Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin Theorem 4.11.1]), and therefore Nk(B) ⊆ Hk,k(B) ∩ H2k(B,R). So elements of Nk(B) are represented by real (k,k)-forms on V, and this leads to several further notions of positivity. Specifically, following the discussion in [6, III.1] we say that a (k,k)-form η on V is strongly positive if it is a non-negative real linear combination of forms of the type i(cid:96) ∧(cid:96) ∧···∧i(cid:96) ∧(cid:96) 1 1 k k for (cid:96) ∈ V∨, and η is weakly positive if it restricts to a non-negative multiple of the canonical j orientation form on any k-dimensional complex subspace W ⊆ V. A (k,k)-form on V is semi- positive if it is real and if the associated Hermitian form on (cid:86)kV is semipositive. This gives rise to a chain of three closed convex cones: Strongk(V) ⊆ Semik(V) ⊆ Weakk(V) sitting inside the space of real (k,k)-forms on V. When k = 1 or k = n−1 they coincide, but when 2 (cid:54) k (cid:54) n−2 the inclusions are strict. Finally, one defines cones Strongk(B) ⊆ Semik(B) ⊆ Weakk(B) in Nk(B), consisting of those classes represented by forms of the indicated type. One has Psefk(B) ⊆ Strongk(B) , Weakk(B) ⊆ Nefk(B), and in the classical case k = 1 of divisors these various flavors of positivity are actually all the same, i.e., Psef1(B) = Strong1(B) = Semi1(B) = Weak1(B) = Nef1(B) (with a similar statement for k = n−1). Our first result computes the pseudoeffective and nef cones on the self product of an elliptic curve with complex multiplication: Theorem A. Let E be an elliptic curve having complex multiplication, and set B = E ×...×E (n times). Then, for every 0 (cid:54) k (cid:54) n, Psefk(B) = Strongk(B) = Strongk(V) , Nefk(B) = Weakk(B) = Weakk(V). Here V denotes as above the vector space of which B is a quotient. It follows that B carries nef classes of every codimension 2 (cid:54) k (cid:54) n−2 that are not pseudoeffective. This implies formally the existence of nef classes whose product is not nef. In the situation of the previous theorem, the pseudoeffective and nef cones were described just in terms of positivity of forms. Our second computation shows that in general the picture can be more complicated: Theorem B. Let A be a very general principally polarized abelian surface, and let B = A×A. Then Psef2(B) = Strong2(B) = Semi2(B) (cid:36) Weak2(B) (cid:36) Nef2(B). Furthermore, Psef2(B) = S2Psef1(B), where S2Psef1(B) ⊆ N2(B) denotes the closed convex cone generated by products of elements of Psef1(B). 2 Pseudoeffective and nef classes on abelian varieties Along the way to the theorem, we exhibit inequalities defining the pseudoeffective and nef cones. The statement of the theoremremains validfor an arbitraryprincipally polarized abelian surface A provided that one intersects each term with the subspace of N2(A × A) generated by the products of certain natural divisor classes: when A is very general this “canonical subspace” fills out all of N2(A×A). By the same token, it is enough to assume in the Theorem that A is a very general abelian surface with a given polarization, or for that matter that B is isogeneous to the product A×A appearing in the statement. By a specialization argument, the Theorem also implies the following. Corollary C. Let A be a very general polarized abelian variety of dimension g. Then Semi2(A×A) = Psef2(A×A) = S2Psef1(A×A) and Strong2g−2(A×A) = Psef2g−2(A×A) = S2g−2Psef1(A×A). Theorem A follows easily from the remark that Psef1(B) = Strong1(V). By contrast, the argument leading to the second result is more computational in nature, the main point being to show that S2Psef1(A×A) = Semi2(A×A). For this we exploit a natural GL (R)-action on the 2 cones in question, and apply a classical argument in convexity theory. It would be interesting to have a more conceptual approach. In the final section of the paper, we propose some conjectures and questions dealing with positivity conditions on cycles of higher codimension. We hope that these may stimulate further work involving this relatively uncharted circle of ideas. Concerning the organization of this paper, we start in §1 with some general remarks about positivity of cycles on an abelian variety. The second section takes up the self product of an elliptic curve with complex multiplication. In §3 we begin the study of the product of an abelian varietywithitself,andintroducetherethealgebraof“canonical”classes.Themaincomputations appear in §4, while in §5 we give some complements. Finally, we propose in §6 a considerable number of questions and open problems concerning positivity of higher codimension cycles. Acknowledgements. We have profited from many conversations with T. Peternell concerning positivity of higher codimension cycles. We have also benefited from discussions with D. Edidin, M. Fulger, W. Fulton, D. Maclagen, Y. Mustopa, and Y. Tschinkel concerning some of the questions in Section 6. Many thanks also go to J.-B. Lasserre for providing the reference [1], and to K. Ribet and B. Moonen for their explanations about Proposition 3.1. Finally, this research started during the program “Algebraic Geometry” at M.S.R.I. in 2009, and the authors thank this institution for support and for excellent working conditions. 1. Positive classes on an abelian variety This section is devoted to some generalities about positivity of cycles on an abelian variety. After reviewing some facts about different notions of positivity for forms on a complex vector space, we introduce the basic cones that arise on abelian varieties. We conclude by analyzing them in the case of curves and divisors. Positivity of forms on a complex vector space We start by recalling some facts about positivity of (k,k)-forms on a complex vector space, following Chapter III.1 of Demailly’s notes [6]. 3 Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin Let V be a complex vector space of dimension n. If z = x +iy (1 (cid:54) j (cid:54) n) are complex j j j coordinates on V, then the underlying real vector space is canonically oriented by the real (n,n)- form idz ∧dz ∧···∧idz ∧dz = 2ndx ∧dy ∧···∧dx ∧dy . 1 1 n n 1 1 n n We denote by Λ(k,k)V∨ the (real) vector space of real (k,k)-forms on V. R Definition 1.1. (i). A (k,k)-form η on V is strongly positive if it is a linear combination with non-negative real coefficients of forms of the type i(cid:96) ∧(cid:96) ∧···∧i(cid:96) ∧(cid:96) 1 1 k k for linear forms (cid:96) ∈ V∨. j (ii). A (k,k)-form η is weakly positive2 if the (n,n)-form η∧ω is a real non-negative multiple of the orientation form for all strongly positive (n−k,n−k)-forms ω. Strongly and weakly positive forms are real ([6, III.1.5]). They form closed convex cones (with non-empty interiors) in Λ(k,k)V∨, which we denote by Strongk(V) and Weakk(V) respectively. R Evidently Strongk(V) ⊆ Weakk(V), and by construction there is a duality of cones: Weakk(V) = Strongn−k(V)∨. Remark 1.2. It follows from the definition that Strongk(V) = SkStrong1(V), where SkStrong1(V) denotes the closed convex cone generated by products of positive (1,1)- forms. Remark 1.3. A (k,k)-form is weakly positive if and only if it restricts to any k-dimensional complex vector subspace of V as a non-negative volume form ([6, III.1.6]). Definition 1.4. A (k,k)-form on V is semipositive if it is real and the associated Hermitian form on (cid:86)kV is semipositive. These forms form a real convex cone in Λ(k,k)V∨ that we denote R by Semik(V). Using diagonalization, we see that the cone Semik(V) is generated by the forms ik2α∧α, for α ∈ (cid:86)kV∨.ItcontainstheconeStrongk(V)andisselfdual;,hence,wehaveachainofinclusions Strongk(V) ⊆ Semik(V) ⊆ Weakk(V) (C ) k whosedualis(C ).Ifα ∈ (cid:86)kV∨,thesemipositiveformik2α∧αisstronglypositiveifandonly n−k if α is decomposable ([6, III.1.10]). Therefore, the inclusions above are strict for 2 (cid:54) k (cid:54) n−2. Classes on an abelian variety We now turn to cohomology classes on an abelian variety. Let B = V/Λ be an abelian variety of dimension n. As in the Introduction, denote by Nk(B) the real vector subspace of H2k(cid:0)B,R(cid:1) generated by classes of algebraic cycles. Thanks to [13], this coincides with the group of numerical equivalence classes of cycles. In the natural way we identify cohomology classes on B as being given by (k,k)-forms on V, and we set Strongk(B) = Strongk(V)∩Nk(B). 2We are modifying somewhat the terminology in [6]. 4 Pseudoeffective and nef classes on abelian varieties The closed convex cones Semik(B) , Weakk(B) ⊆ Nk(B) are defined similarly. Thus Strongk(B) ⊆ Semik(B) ⊆ Weakk(B). On the other hand, one defines as in the Introduction the cones Psefk(B) , Nefk(B) ⊆ Nk(B) of pseudoeffective and nef classes. Occasionally, it will be convenient to work with cycles indexed by dimension rather than codimension. As customary, we indicate this by replacing superscripts by subscripts. Thus N (B) = Nn−k(B) , Nef (B) = Nefn−k(B), k def k def and so on. As B is homogeneous, the intersection of two pseudoeffective classes is again pseudoeffective, and in particular Psefk(B) ⊆ Nefk(B). The following lemma refines this statement: Lemma 1.5. One has the inclusions Psefk(B) ⊆ Strongk(B) ⊆ Weakk(B) ⊆ Nefk(B). Proof. LetZ ⊆ B beanirreduciblesubvarietyofdimensionc,andletω ∈ Weakc(B)beaweakly positive (c,c)-form. Then (cid:82) ω (cid:62) 0 thanks to Remark 1.3. Therefore, any weakly positive class Z is nef. On the other hand, if c = n−k and η is a (k,k)-form on V representing the cohomology Z class of Z, then (cid:90) (cid:90) η ∧β = β Z B Z for every (n−k,n−k)-form β. When β ∈ Weakn−k(V) the integral in question is non-negative, and hence η ∈ Weakn−k(V)∨ = Strongk(V), Z as required. Finally, we note that the various cones in question are preserved by isogenies: Proposition 1.6. Let φ : B(cid:48) −→ B be an isogeny. Then φ induces an isomorphism Nk(B)−∼→Nk(B(cid:48)) under which each of the five cones just considered for B maps onto the corresponding cone for B(cid:48). Proof. For the pseudoeffective and nef cones this follows from the projection formula, while for the other cones defined by positivity of (k,k)-forms it follows from the fact that tangent spaces V and V(cid:48) to B and B(cid:48) at the origin are isomorphic. 5 Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin Duality Forlaterusewemakeafewremarksconcerningduality.DenotebyB(cid:98) = Pic0(B)thedualabelian variety of the abelian variety B. For each (cid:96) ∈ {0,...,2n}, there are canonical isomorphisms PD d H2n−(cid:96)(B,Z) −∼→ H(cid:96)(B,Z) −∼→ H(cid:96)(B(cid:98),Z), (*) where the first isomorphism is Poincar´e duality on B. As for the second one, we start from the canonical isomorphism H1(B,Z)−∼→H1(B(cid:98),Z) defining B(cid:98) and use the isomorphisms (cid:96) (cid:96) (cid:94) (cid:94) H(cid:96)(B,Z)−∼→ H1(B,Z) , H(cid:96)(B(cid:98),Z)−∼→ H1(B(cid:98),Z). The isomorphisms in (*) are sometimes known as the Fourier-Mukai transform. Proposition 1.7. The isomorphisms (*) have the following properties: (a). They exchange the Pontryagin product on B and the cup product on B(cid:98). (b). They are compatible with the Hodge decompositions. When (cid:96) = 2k is even, they carry classes of algebraic cycles on B to classes of algebraic cycles on B(cid:98), and therefore define isomorphisms Nn−k(B)−∼→Nk(B(cid:98)). (c). These isomorphisms preserve the strongly positive cones. Proof. ThePontryaginproductonthehomologyofB isinducedbythesummapσ : B×B → B. Its iterations provide the canonical isomorphisms (cid:86)iH (B,Z)−∼→H (B,Z). Similarly, the cup 1 i product on the cohomology of B(cid:98) provides the isomorphisms i (cid:94) H1(B(cid:98),Z)−∼→Hi(B(cid:98),Z). Hence (a) follows from the definition of d. On H (B,Z), the map d is given by the interior product with the Hodge class 1 c1(P) ∈ H1(B,Z)⊗H1(B(cid:98),Z) ⊆ H2(B×B(cid:98),Z) of the Poincar´e line bundle P. Therefore d acts on H (B,Z) via the cup product with a multiple (cid:96) of c (P)(cid:96), which implies the assertions in statement (b). 1 For k = 1 or k = n − 1, item (c) follows from the explicit representation of d as induced by the interior product with c (P), and by choosing an explicit representative of c (P) as in 1 1 (3.4) below. For general k, we observe that Strongn−k(V) is the convex cone generated by cup products of n − k elements of Strong1(V) of rank 2. But the image under d ◦ PD of such a decomposable class is the Pontryagin product of n−k elements of Strongn−1(V(cid:98)) and hence is strongly positive. Divisors and one-cycles We conclude this section by describing the cones in question in the classical cases of divisors and curves. As above B denotes an abelian variety of dimension n. To begin with, nef and pseudoeffective divisors coincide on any homogeneous variety. There- fore Lemma 1.5 yields the (well known) equalities Psef1(B) = Strong1(B) = Semi1(B) = Weak1(B) = Nef1(B). (1.1) 6 Pseudoeffective and nef classes on abelian varieties It follows dually that Nefn−1(B) = Psefn−1(B), which implies similarly the analogue of (1.1) for the cones of curves. The next proposition asserts that any pseudoeffective curve class can be written as a positive R-linear combination of intersections of pseudoeffective divisor classes. Proposition 1.8. Let B be an abelian variety of dimension n. One has Psefn−1(B) = Sn−1Psef1(B), where Sn−1Psef1(B) ⊆ Nn−1(B) is the closed convex cone generated by cup products of pseu- doeffective divisor classes. The proof will use the following Lemma, which involves the Pontryagin self products of a curve class on B: given γ ∈ H (B,R) = H2n−2(B,R), we write γ∗(k) ∈ H2n−2k(B,R) for the 2 k-fold Pontryagin product of γ with itself. Lemma 1.9. Let B be an abelian variety of dimension n and let k be an integer, with 0 (cid:54) k (cid:54) n. (a). For any α ∈ H2(B,R), one has (cid:16)αn(cid:17)n−k−1αk (αn−1)∗(n−k) = (n−k)!(n−1)!n−k . n! k! (b). For any β ∈ H2n−2(B,R), one has (cid:16)β∗(n)(cid:17)n−k−1βk (β∗(n−1))n−k = (n−k)!(n−1)!n−k . n! k! Proof. When α is an ample class, “Poincar´e’s Formula” ([3], 16.5.6) reads αk d (cid:18) αn−1 (cid:19)∗(n−k) = k! (n−k)! d(n−1)! where d = deg(α) = αn/n!. Since the ample cone has non-empty interior in N1(B), this implies the equality in (a) for all classes α in N1(B) (which is the only case where we will use it). But this equality can also be checked by representing any class in H2(B,R) by a skew-symmetric form on V, very much as in [3, 4.10], and it is easily seen that the forms represented by either side of the equality are proportional, with a proportionality constant depending only on n and k. This constant must then be the one we just obtained. Item (b) follows from (a) and Proposition 1.7 (a). Proof of Proposition. The issue is to show that Psefn−1(B) ⊆ Sn−1Psef1(B). To this end, con- sider a cohomology class β ∈ N2n−2(B,R) which lies in the interior of Psefn−1(B). Then β can be represented as a positive R-linear combination of an effective curve and a complete inter- section of very ample divisors; in particular, β generates B and β∗(n) is non-zero. The effective divisor class β∗(n−1) is then ample, and we are done by the formula (b) in the lemma. 2. Products of CM elliptic curves Inthissection,weconsidercyclesofarbitrarycodimensionontheselfproductofanellipticcurve with complex multiplication. In this case the global cones coincide with those defined by linear algebra. 7 Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin Let E = C/Γ be an elliptic curve admitting complex multiplication, and put (cid:8) (cid:9) S = s ∈ C | s·Γ ⊆ Γ = End(E). Thus S is an order in an imaginary quadratic extension of Q. We view the elements of S inter- changeably as complex numbers or as endomorphisms of E. Note that the R-span of S, seen in the first light, is all of C. Denote by B the n-fold product E×n, which we write as usual B = V/Λ. We establish Theorem 2.1. One has Psefk(B) = Strongk(B) = Strongk(V) , Nefk(B) = Weakk(B) = Weakk(V). The proof appears at the end of the section. First we record a corollary concerning the product of nef classes. Specifically, we have seen that when 2 (cid:54) k (cid:54) n−2 there is a strict inclusion Strongk(V) (cid:36) Weakk(V). Therefore, we have the following corollary. Corollary 2.2. For any 2 (cid:54) k (cid:54) n − 2, B carries nef cycles of codimension k that are not pseudoeffective. In particular, the product of nef cycles is not in general nef. Remark 2.3 Grothendieck’s questions. The corollary answers in the negative some questions raised by Grothendieck in 1964 in correspondence with Mumford [9]. In this letter, Grothendieck starts by proposing some conjectures (subsequently settled by Mumford and Kleiman [10]) con- cerning numerical characterizations of amplitude for divisors. He goes on to write: Iwouldlikeevenalotmoretobetrue,namelytheexistenceofanumericaltheoryofampleness forcyclesofanydimension.AssumeforsimplicityX projectivenon-singularconnectedofdim. n, let Ai(X) be the vector space over Q deduced from numerical equivalence for cycles of codimension i (presumably this is of finite dimension over Q), and A (X)=An−i(X) defined i by cycles of dimension i, presumably A and Ai are dual to each other. Let A+ be the cone i i generated by positive3 cycles, and let Pi ⊂Ai be the polar cone. The elements of Pi might be called pseudo-ample, and those in the interior of Pi ample (which for i=1 would check with the notion of ample divisor, if for instance the strengthening of Mumford-Nakai’s conjecture considered above is valid4). The strongest in this direction I would like to conjecture is that the intersection of pseudo-ample (resp. ample) cycles is again pseudo-ample (ample), thus the intersection defines Pi×Pj −→Pi+j. Ifiandj arecomplementary,i+j =n,thisalsomeansthatthenaturalmapu :Ai −→A i n−i maps Pi into A+ (and one certainly expects an ample cycle to be at least equivalent to a n−i positive one!). For i and j arbitrary, the above inclusion can also be interpreted as meaning that the intersection of an ample cycle with a positive cycle is again (equivalent to) a positive cycle.Ofcourse,onewouldexpectanamplepositivecycletomovealotwithinitsequivalence class, allowing to consider proper intersections with another given positive cycles. I wonder if you have any material against, or in favor of, these conjectures? 3i.e., effective. 4Earlier in the letter, Grothendieck had asked whether it is enough to test positivity against curves in Nakai’s criterion: Mumford’s celebrated counterexample appears in his reply to Grothendieck. Recall however that it is the content of Kleiman’s work [10] that the interior of P1 is in fact the cone of ample divisor classes. 8 Pseudoeffective and nef classes on abelian varieties Needless to say, in the years since this letter was written it has become abundantly clear that intuition from divisors is often a poor guide for higher codimensions. The computations of the present paper give yet another illustration of this principle. Finally, we give the Proof of Theorem 2.1. It suffices to show Strong1(V) ⊆ Psef1(B). (*) Indeed, this implies that Strongk(V) = SkStrong1(V) ⊆ Psefk(B) thanks to the fact that the product of pseudoeffective classes on an abelian variety is pseudo- effective. The reverse inclusion being automatic, we obtain Strongk(V) = Psefk(B), and hence (dually) Nefk(B) = Weakk(V). For (*), write B = C×n/Γ×n, and denote by z the coordinate function on the j-th com- j ponent of C×n. Thus we may view (dz ,...,dz ) as a basis for the complex vector space W of 1 n holomorphic 1-forms on B. Consider the subgroup M ⊆ W given by (cid:8) (cid:9) M = s dz +···+s dz | s ∈ S . 1 1 n n j Note that any holomorphic one-form in W is a non-negative R-linear combination of elements in M. Consequently the cone Strong1(V) is generated by elements of the form i(cid:96)∧(cid:96), where (cid:96) ∈ M. So (*) will follow if we show that i(cid:96)∧(cid:96) ∈ Psef1(B) for any (cid:96) ∈ M. Suppose to this end that (cid:96) = s dz +···+s dz ∈ M, where s ∈ S. Consider the endomor- 1 1 n n i phism α : E×n −→ E×n , α(x ,...,x ) = (s x ,...,s x ). 1 n 1 1 n n Composing with the map E×n −→ E given by summation, we arrive at a morphism β : E×n −→ E with β∗(dz) = (cid:96). Now D = β−1(0) is an effective divisor in B. On the other hand, the cohomology class of [0] def is given by a positive real scalar multiple of idz∧dz, and hence the cohomology class of D is a positive multiple of i(cid:96)∧(cid:96). Thus i(cid:96)∧(cid:96) represents a pseudoeffective class, and we are done. Remark 2.4. One could streamline a little the argument just concluded by using the fact that when B is isogeneous to the self product of an elliptic curve with complex multiplication, then N1(B) is equal to the n2-dimensional real vector space of Hermitian forms on V (cf. [3, Exercise 10 in §5.6]). Therefore Strong1(V) = Strong1(B), while quite generally Strong1(B) = Psef1(B) thanks to (1.1). However we preferred to give a down-to-earth direct argument. 3. Canonical cycles on the self product of an abelian variety In this section we begin our investigation of cycles on the product of a higher-dimensional prin- cipally polarized abelian variety with itself. In order to obtain uniform statements, we will work always with the algebra generated by some natural divisor classes on this product. We introduce and study these here, and check that for a very general abelian variety they span the whole numerical equivalence ring. 9 Olivier Debarre, Lawrence Ein, Robert Lazarsfeld and Claire Voisin We start with some notation. Let (A,θ) be a principally polarized abelian variety and write p ,p : A×A −→ A for the two projections. In H2(A×A,Q), we consider the three classes 1 2 θ = p∗θ , θ = p∗θ , λ = c (P), 1 1 2 2 1 where P is the Poincar´e bundle. We denote by N• (A×A) the subalgebra of H•(A×A,Q) can Q generated by these classes, and we set N• (A×A) = N• (A×A) ⊗ R ⊆ N•(A×A) ⊆ H•(A×A,R). can can Q Q Hodge classes on the self product of a very general abelian variety When (A,θ) is very general, results of Tankeev and Ribet imply that the rational canonical classes on A×A are exactly the rational Hodge classes. Proposition 3.1 Tankeev, Ribet. Let (A,θ) be a very general principally polarized abelian variety. Then Nk (A×A) = Hk,k(A×A) ∩ H2k(A×A,Q) can Q for all integers k. In particular, Nk (A×A) = Nk(A×A). can Proof. By a result of Tankeev ([19]; see also [18]), the algebra of Hodge classes on A × A is generated by the Hodge classes of type (1,1). So we only need to show that classes of divisors are spanned by θ , θ , and λ. To this end, let D ⊂ A×A be a prime divisor that dominates A via 1 2 the first projection. The cohomology class of the general fiber of p : D → A is constant, so we 1 get a map A −→ Pic0(A) (cid:39) A mapping a ∈ A to the class of O (D −D ). Since End(A) (cid:39) Z, A a 0 this map is multiplication by an integer n. The restriction of O (D−p∗D )⊗P−n to a general A 2 0 fiber of p is then trivial; hence, this line bundle must be a pull-back via p . 1 1 The GL (R)-action 2 (cid:18) (cid:19) a c Let M = ∈ M (Z) be an integer matrix. We associate to M the endomorphism b d 2 u : A×A −→ A×A , (x,y) (cid:55)→ (ax+by,cx+dy), M i.e., u (x,y) = (x,y)M. This induces right actions of GL (R) on N• (A×A), N•(A×A), and M 2 can H•(A×A,R). Note that θ , λ, and θ are each in one piece of the Ku¨nneth decomposition 1 2 H2(A×A,R) (cid:39) (cid:0)H2(A,R)⊗H0(A,R)(cid:1) ⊕ (cid:0)H1(A,R)⊗H1(A,R)(cid:1) ⊕ (cid:0)H0(A,R)⊗H2(A,R)(cid:1), (cid:18) (cid:19) a 0 and that acts by multiplication by a2, a, and 1 on the respective pieces. 0 1 Moreover, the addition map σ : A×A → A satisfies ([14, p. 78]) σ∗θ = θ +θ +λ, 1 2 andthisimpliesinturnthattheinvolution(x,y) (cid:55)→ (y,x)swapsθ andθ andleavesλinvariant. 1 2 It follows that the representation of GL (R) on N1 (A×A) is isomorphic to S2W, where 2 can W is the tautological 2-dimensional representation. More precisely, if (e ,e ) is a basis for W, 1 2 the correspondence is θ ↔ e2 , θ ↔ e2 , λ ↔ 2e e . 1 1 2 2 1 2 10