Smooth subvarieties of Jacobians

We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.


Introduction
Let c, d be nonnegative integers and let X be a smooth projective complex variety of dimension n := c + d. An important geometric invariant of X is the subgroup H 2c (X, Z) alg ⊂ H 2c (X, Z) of algebraic cohomology classes, which is generated by the cycle classes of codimension c algebraic subvarieties Y ⊂ X. Since the subvarieties Y may be singular, the following question going back to Borel and Haefliger [BH61, Section 5.17] naturally arises. In this article, we build on the counterexample given in [Deb95]. There, Debarre considers the Jacobian X of a smooth projective complex curve C of genus n, polarized by its theta divisor class θ ∈ H 2 (X, Z). The minimal cohomology class θ c c! ∈ H 2c (X, Z) is the cycle class of the image W n−c (C) ⊂ X of the symmetric power C (n−c) by the Abel-Jacobi map, hence is algebraic. However, when n ≥ 2c + 2, the variety W n−c (C) is singular and there is no general reason why its class θ c c! should be a Z-linear combination of classes of smooth subvarieties of X. Debarre shows that this is indeed not the case if n ≥ 7, c = 2, and X is very general.
Our main theorem extends this result to many other values of (c, n). Recall that α(m) is the number of ones in the binary expansion of m. Theorem 1.2 (Theorem 3.7). Let (X, θ) be a very general complex Jacobian of dimension n. Let c be a nonnegative integer such that α(c + α(c)) > α(c) and n ≥ 4c − 2. Then the integral classes λ θ c c! with λ odd are algebraic but are not Z-linear combinations of cycle classes of smooth subvarieties of X.
The weird condition α(c + α(c)) > α(c) springs naturally from our proof. It holds for integers c in the set {2, 4, 5, 8, 9, 12, 16, 17, . . . }. We have nothing to say when c = 3: we do not know if there exist Jacobians (X, θ) such that θ 3 3! is not a Z-linear combination of cycle classes of smooth subvarieties of X. Applied with c = 2 and n = 6, Theorem 1.2 implies the following result. Corollary 1.3. There exists a smooth projective complex variety X of dimension 6 such that H 4 (X, Z) alg is not generated by classes of smooth subvarieties of X.
As noted above, the groups of algebraic cohomology classes of smooth projective varieties of dimension at most 5 are generated by classes of smooth subvarieties. The dimension of the variety X in Corollary 1.3 is thus the lowest possible. We do not know, however, if Question 1.1 has a positive answer if c = d = 3.
The proof of Theorem 1.2 when c = 2 and n ≥ 7 given in [Deb95] relies on a Barth-Lefschetz-type theorem for abelian varieties, on the Hirzebruch-Riemann-Roch theorem, and on the Serre construction. We can still rely on the same Barth-Lefschetz-type theorem under our more general hypotheses. However, computations based on the Hirzebruch-Riemann-Roch theorem become untractable in high codimension (and do not yield a proof when c = 2 and n = 6), and we cannot use the Serre construction in general because it is specific to codimension 2 subvarieties.
Both difficulties are overcome by resorting to complex cobordism, whose use in the theory of algebraic cycles was pioneered by Totaro [Tot97]. We replace the integrality results derived from the Hirzebruch-Riemann-Roch theorem with divisibility properties of Chern numbers (Proposition 2.5). Those are obtained in Section 2 as a consequence of a detailed understanding of the structure of the complex cobordism ring. In [Deb95], the Serre construction was used to infer restrictions on the cohomology class of a smooth subvariety Y ⊂ X. Instead, we consider the class of Y in the complex cobordism of X and use the description of the complex cobordism of an abelian variety (in Sections 3.1-3.2). These tools are combined in Section 3.3 to prove Theorem 1.2.
In low codimension, a different method based on complex topological K-theory and on the Grothendieck-Riemann-Roch theorem, closer to the one used in [Deb95], gives small improvements on Theorem 1.2. We use this method in Section 4 to prove the following.

The Hurewicz morphism of MU
In this paragraph, we recall the structure of the Hurewicz morphism of the spectrum MU representing complex cobordism. The computation of the cohomology of complex Grassmannians and the Thom isomorphism combine to show that H * (MU, Z) is a polynomial ring with integral coefficients with one generator in degree 2i for each i ≥ 1 (see [Ada74, Section I.3]). A deep theorem of Milnor shows that π * (MU) is also a polynomial ring with integral coefficients with one generator in degree 2i for each i ≥ 1 and that H is injective (see [Ada74, Theorem II.8.1 and Corollary II.8.11]). Quillen's theorem identifying π * (MU) with the coefficient ring of the universal formal group law [Ada74, Theorem II.8.2] and Hazewinkel's explicit construction of a universal formal group law [Haz78] allow us to be more precise.
for some t ≥ 1 and some prime number p, and λ i = 1 otherwise.
Proof. Let u i ∈ π 2i (MU) be the polynomial generators of π * (MU) specified in [Haz78,Section 34.4.1]. Induction on i using the formula [Haz78,(34.4.3)] shows that there exist v i ∈ H 2i (MU, Z) such that H(u i ) = λ i v i . It then follows from [Ada74, Lemmas II.7.9(iii) and II.8.10] that v i generates H 2i (MU, Z) modulo its decomposable elements. Consequently, For each e ≥ 1, define the ideal I e ⊂ π * (MU) to be the kernel of the composition of H and the reduction modulo 2 e . Working in the monomial bases associated with the u i and the v i given by Proposition 2.1 shows at once the following result.

Chern numbers
Let MU * be the complex cobordism ring, whose degree d elements are complex cobordism classes of d-dimensional compact stably almost complex manifolds. The Thom-Pontrjagin construction gives an identification We define the Segre classes s i ∈ Z[c j ] j≥1 to be the unique elements such that s i has degree i and The Chern numbers associated with s i have the following properties.

A congruence result for the top Segre class
The next proposition is the main goal of this section. It simultaneously generalizes the theorem of Rees and Thomas recalled in Lemma 2.4(b) (when e = 0) and [Ben20, Theorem 3.6] (when e = 1).
Proposition 2.5. Fix e ≥ 0, h ≥ 1, and i ≥ 1. The following assertions are equivalent: such that the Chern number Proof. Assertion (i) implies that s i : MU 2i → Z is divisible by 2 h . So does assertion (ii) by Lemma 2.4(b). We may thus assume that the function Identify MU 2i and π 2i (MU) using (2.2). Consider the following statements: (a) The function s i 2 h : MU 2i → Z coincides modulo 2 e with a Chern number. Proof. The first assertion follows from the Künneth formula in ordinary homology and the second assertion from the Künneth formula in complex cobordism (apply [Swi75,Theorem 13.75 Proof. Let M be a 2i-dimensional compact stably almost complex manifold representing x ∈ MU 2i . Let g E : M ×T E → X be the composition of the second projection and f E : T E → X. We will also let h E : M ×T E → M denote the first projection. Since the stably almost complex structure on the tangent bundle on the torus T E is stably trivial, one has c j (T E ) = 0 for j > 0. It follows from the Whitney sum formula that

Complex cobordism of abelian varieties
Now let X be a complex abelian variety of dimension n with a principal polarization θ ∈ H 2 (X, Z). We identify X and (S 1 ) N for N = 2n by means of a Lie group isomorphism X ≃ (S 1 ) N . By Lemma 3.1(a), there exists for each k ≥ 0 a unique Z-linear combination is Poincaré-dual to the integral class θ n−k (n−k)! or, in other words, such that Proposition 3.3. Let (X, θ) be a principally polarized complex abelian variety of dimension n. Assume that the group Hdg 2k (X, Z) of Hodge classes is generated by θ k k! for each k ≥ 0, and let τ k ∈ MU 2k (X) be as in (3.1). Let f : Y → X be a morphism of smooth projective varieties with Y of pure dimension d. Then there exists, for each i ∈ {0, . . . , d}, an element x i ∈ MU 2i such that Proof. Let R i be the rank of the free Z-module MU 2i , and let (y i,r ) 1≤r≤R i be a Z-basis of it. Since the MU * -module MU * (X) is free with basis ([f E ]) E⊂{1,...,N } by Lemma 3.1(b), there exist unique integers ν i,r,E such that is an inclusion of free Z-modules of the same rank R i by Proposition 2.1, it follows from Lemma 2.3 that there exists a degree i element P ∈ Z[c j ] j≥1 such that P (y i,s ) is nonzero if and only if s = r. In view of Lemma 3.2 and the projection formula, the characteristic number of (3.4) associated with P and ω ∈ H 2d−2i (X, Z) reads As ω is arbitrary, Poincaré duality on X implies The left side of (3.5) is algebraic, hence is a Hodge class. As a consequence, so is the right side. Since P (y i,r ) 0, we deduce from our hypothesis that the class |E|=2d−2i ν i,r,E f E, * (1) is an integral multiple of Proof. Let us compute the characteristic number of f associated with P and θ d−l (d−l)! . Combining (3.3), (3.1), and Lemma 3.2 shows that it is Since deg X (θ n ) = n!, the proposition is proven. □

Smooth subvarieties of abelian varieties
The next proposition is an application of a Barth-Lefschetz-type theorem proved by Sommese [Som82].
Proposition 3.5. Let (X, θ) be a principally polarized complex abelian variety of dimension n such that Hdg 2k (X, Z) is generated by θ k k! for all k ≥ 0. Let f : Y → X be the inclusion of a smooth projective subvariety of pure codimension c. Assume that n ≥ 4c − 2l for some l ≥ 1 . Then there exist a 0 , . . . , a c Proof. Since the subvariety Y ⊂ X is algebraic, the homology class f * [Y ] is Poincaré-dual to a Hodge class, which is necessarily of the form a c θ c c! by hypothesis. This proves (b). By the self-intersection formula [Ful98,Corollary 6.3], the top Chern class of the normal bundle N Y /X is a c f * ( θ c c! ). Since the tangent bundle T X is trivial, one has c(N Y /X ) = c(Y ) −1 = s(Y ). This shows (a) for i = c.
If the abelian variety X were not simple, pulling back an ample divisor from a nontrivial quotient would produce a nonample divisor on X, contradicting the fact that Hdg 2 (X, Z) is generated by θ. We deduce that X is simple. One may thus apply [Som82, Corollary 3.5 and (3.6.1)] with B = Y to obtain π j (X, Y , y) = 0 for j ≤ n − 2c + 1 and all y ∈ Y . It follows from the version [Hat02,Theorem 4.37] of the Hurewicz theorem, from the universal coefficient theorem, and from the long exact sequence of relative cohomology of the pair (X, Y ) that the restriction map H j (X, Z) → H j (Y , Z) is an isomorphism for j ≤ n − 2c. For 0 ≤ i ≤ c − l, one may apply this fact with j = 2i because n ≥ 4c − 2l. This shows that the class s i (Y ) ∈ H 2i (Y , Z), which is Hodge because it is algebraic, is the restriction to Y of a class in Hdg 2i (X, Z). The latter is necessarily of the form a i θ i i! by hypothesis. The proof is now complete. □ Proposition 3.6. Keep the hypotheses and notation of Proposition 3.5, assume that l = 1, and suppose in addition that α(c + α(c)) > α(c). Then a c is even.
Proof. Let Q ∈ Z[c j ] j≥1 be the degree c homogeneous polynomial obtained by applying Proposition 2.5 with i = c, e = α(c), and h = 1. Applying Proposition 3.4 with l = c and P = s c + 2Q yields the identity Using that Chern classes may be expressed as polynomials with integral coefficients in Segre classes by (2.3), it follows from Proposition 3.5(a) that Applying Proposition 3.5(a) again, we get P (c j (Y )) = (a c + 2b)f * ( θ c c! ). Rewriting the left side of (3.6) using the projection formula and Proposition 3.5(b), we obtain Using deg X (θ n ) = n!, we finally get (3.7) P (x c ) = 2c c a c (a c + 2b).
Our choice of Q implies that the left side of (3.7) is divisible by 2 α(c)+1 . The formula for the 2-adic valuation of the factorial given in [Rob00, Lemma, Section 5.3.1, p. 241] implies that the 2-adic valuation of 2c c is equal to α(c). We deduce that a c (a c + 2b) is even, hence that a c is even. □ Theorem 3.7. Let (X, θ) be a very general complex Jacobian of dimension n. Let c ≥ 0 be such that α(c + α(c)) > α(c) and n ≥ 4c − 2. Then the classes λ θ c c! with λ odd are algebraic but are not Z-linear combinations of cycle classes of smooth subvarieties of X.
Proof. The integral class θ c c! is algebraic by [BL04, Poincaré's formula 11.2.1]. The hypothesis that Hdg 2k (X, Z) is generated by θ k k! for all k ≥ 0 is satisfied by [BL04,Theorem 17.5.1] and because the integral class θ k k! is primitive. One may thus combine Propositions 3.5(b) and 3.6 to show that the cycle classes of smooth codimension c subvarieties Y ⊂ X are even multiples of θ c c! . This concludes the proof. □ Remarks 3.8.
(i) In Theorem 3.7, the hypothesis that (X, θ) is very general is only used to ensure that Hdg 2k (X, Z) is generated by θ k k! for k ≥ 0. As there exist Jacobians over Q whose Mumford-Tate group is the full symplectic group (see [And96, Théorème 5.2 3) and Remarque (vii) below it] which apply as all Hodge classes on abelian varieties are absolute Hodge by [Del82, Main Theorem 2.11]), one may find such an (X, θ) that is defined over Q.
(ii) The proof of Theorem 3.7 actually shows that the class of any smooth subvariety of codimension c of X is divisible by 2 in the group H 2c (X, Z) alg .

Codimension 4 cycles
Theorem 3.7 is not optimal in several respects. When c is fixed, it is sometimes possible to give stronger restrictions on the cycle classes of smooth subvarieties of codimension c of X or results for lower values of the dimension n of X.
To obtain such improvements, one may work with complex topological K-theory instead of complex cobordism and replace the divisibility result for Chern numbers given in Proposition 2.5 with an application of the Grothendieck-Riemann-Roch theorem and an integrality property of the Chern character (cf. Lemma 4.1). This works well when c is low but becomes intractable for high values of c. We illustrate the method when c = 4.
We start with a lemma. Let X be a topological space, and let K * (X) = K 0 (X) ⊕ K 1 (X) be its Z/2graded complex topological K-theory defined in [AH61, Section 1.9]. We consider the Chern character ch : K * (X) → H * (X, Q) as in [AH61, Section 1.10]. Proof. First, suppose that N = 1. The morphism ch : K 0 (S 1 ) → H 0 (S 1 , Q) = Q has image Z because it associates with each vector bundle its rank, and it is injective because K 0 (S 1 ) ≃ Z. That ch : K 1 (S 1 ) → H 1 (S 1 , Q) is an isomorphism onto H 1 (S 1 , Z) follows from the definition of this morphism using suspension and Bott periodicity (see [AH61, Section 1.10]) and from the fact the Chern character sends the Bott element which is a generator of K 0 (S 2 ) to a generator of H 2 (S 2 , Z) (see [ Proof. Let f : Y → X be the inclusion map. Proposition 3.5 shows the existence of integers a i such that s i (Y ) = a i f * ( θ i i! ) for i ∈ {0, 1, 2, 4} (if n ≥ 12), or i ∈ {0, 1, 2, 3, 4} (if n ≥ 14), and such that, in addition, the cohomology class of Y in X is equal to a 4 θ 4 4! . As the class f * s 3 (Y ) is Hodge, we may also write f * s 3 (Y ) = b θ 7 7! for some b ∈ Z.
By Lemma 4.1, which applies because X is diffeomorphic to (S 1 ) 2n , the left side ch(χ) of (4.2) is an integral cohomology class, hence so is its right side. Assume by way of contradiction that a 4 is odd. The integrality of the class ch 5 (χ) = a 1 a 4 θ 5 /48 implies that a 1 is even, and the integrality of the class ch 6 (χ) = (4a 2 1 − a 2 )a 4 θ 6 /576 shows that a 2 is divisible by 4. All terms appearing in the expression for ch 8 (χ) given in (4.2) are multiples of θ 8 8! with coefficients a rational number of nonnegative 2-adic valuation, with the exception of a 2 4 θ 8 /414 720 = 7a 2 4 72 θ 8 8! since a 4 is assumed to be odd. It follows that ch 8 (χ) is not an integral cohomology class, which gives a contradiction. This proves (a). Now suppose n ≥ 14. Then bθ 7 /7! = f * s 3 (Y ) = f * f * (a 3 θ 3 /3!) = a 3 a 4 θ 7 /144, so that b = 35a 3 a 4 . Assume by way of contradiction that a 4 is not divisible by 4. By the integrality of ch 5 (χ) = a 1 a 4 θ 5 /48, we see that a 1 a 4 is even. The integrality of ch 6 (χ) = (4a 2 1 − a 2 )a 4 θ 6 /576 shows that a 2 is even. These pieces of information imply that in the right side of the equality