Chern-Dold character in complex cobordisms and theta divisors

We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.


Introduction
In the complex cobordism theory, going back to the foundational works of Milnor and Novikov [39], [46], a prominent role is played by the Chern-Dold character introduced by the first author in [6].
By definition, the Chern-Dold character ch U is a natural multiplicative transformation of cohomology theories where U * (X) is the complex cobordism ring of a CW -complex X and Ω U = U * (pt), where pt is a point, is the cobordism ring of the stably complex manifolds (or, in short, U -manifolds).It is uniquely defined by the condition that when X = pt is the canonical homomorphism of Ω U to Ω U ⊗ Q.This implies that for any finite CW -complex X the transformation is an isomorphism of Ω U -modules.The fundamental Milnor-Novikov result says that the coefficient ring of the theory U * (X) is the graded polynomial ring Ω U = Z[y 1 , . . ., y n , . . .], deg y n = −2n of infinitely many generators y n , n ∈ N.
Let u ∈ U 2 (CP ∞ ) and z ∈ H 2 (CP ∞ ) be the first Chern classes of the line bundle associated to the hyperplane section over CP ∞ in the complex 1 cobordisms and cohomology theory respectively.The Chern-Dold character is uniquely defined by its action where B 2n are certain U -manifolds, characterised by their properties in [6].
The series β(z) is the exponential of the commutative formal group of the geometric complex cobordisms introduced by Novikov in [48], so that Quillen identified this group with Lazard's universal one-dimensional commutative formal group [51].The inverse of this series is the logarithm of this formal group, which can be given explicitly by the Mischenko series [48,7]: The question whether there are smooth irreducible algebraic representatives of the cobordism classes [B 2n ] in the exponential of the formal group given by the Chern-Dold character was open for a long time since 1970 (see [6]).
In this paper we give the following answer to this question, presenting an explicit form of the series (3) as where Θ n is a smooth theta divisor of a general principally polarised abelian variety A n+1 , considered as the complex manifold of real dimension 2n.The cobordism class of the theta divisor does not depend on the choice of such abelian variety provided Θ n is smooth, which is true in general case [2].
Theorem 1.1.The theta divisor Θ n of a general principally polarised abelian variety A n+1 is a smooth irreducible projective variety, which can be taken as an algebraic representative of the coefficient [B 2n ] in the Chern-Dold character.
As a corollary we have the following representation of the cobordism class of any U -manifold M 2n in terms of the theta divisors: where the sum is over all partitions λ = (i 1 , . . ., i k ) of (λ + 1)! := (i 1 + 1)! . . .(i k + 1)! and c ν λ (M 2n ) ∈ Z are the Chern numbers of M 2n corresponding to the normal bundle ν(M 2n ) (see the next section for details).
As another corollary we have the following explicit expression of the exponential generating function of any Hirzebruch genus Φ of theta divisors: where Q(z) = 1+ n∈N a n z n is the characteristic power series of Hirzebruch genus Φ (see Section 3 for details).
In particular, for the Todd genus we have Q(z) = z 1−e −z , so (n + 1)! , so that the Todd genus of the theta divisors is (cf. [6]).Thus we have the following formula for the Todd genus for any Since the Todd genus is integer, this implies the divisibility condition on the Chern numbers c ν λ (M 2n ) of U -manifolds.As it follows from the results of Stong [55] and Hattori [24], all divisibility conditions for U -manifolds one can get applying to formula (6) the Landweber-Novikov operations and taking the Todd genus (see the discussion in the last section).
The action of the Landweber-Novikov operations on the theta divisors can be described explicitly in the following way.
Let λ = (i 1 , . . ., i k ) be a partition of |λ| := i 1 + • • • + i k with (k) = (k, 0, . . ., 0) being a one-part partition.Let S λ [M ] be the result of the action of the Landweber-Novikov operation S λ on U -manifold M defined in terms of its stable normal bundle (see [34,48]).Consider a smooth complete intersection ) of Θ n with k general translates Θ n (a i ), a i ∈ A n+1 of the theta divisor Θ n , which for k < n is an irreducible algebraic variety (see Section 3 below).
Let D = c 1 (L) ∈ H 2 (A n+1 , Z) be the first Chern class of the principal polarisation bundle L, which is the cohomology class Poincaré dual to the cycle defined by Note that for a general principally polarised abelian variety the cohomology class D p generates the corresponding Hodge group ) for all p, due to Mattuck [38], who proved the Hodge (p, p)-conjecture in this case (see section 17.4 in [5]).
Theorem 1.2.Let λ be a partition with |λ| < n and S λ be the corresponding Landweber-Novikov operation, then the cobordism class S λ [B 2n ] has a smooth irreducible algebraic representative.More precisely, if λ is not a one-part partition, then S λ [B 2n ] = S λ [Θ n ] = 0, while for λ = (k), k ≤ n we have As a corollary we have the following expression for [Θ n−k k ] as a residue at zero with β(z) given by (5).Moreover, the cobordism class with positive integer coefficients, which implies that the polynomial subring Θ U ⊂ Ω U generated by the theta-divisors: is invariant under the Landweber-Novikov operations (see section 4 below).
We use the Landweber-Novikov algebra S over Q, which is a graded Hopf algebra generated as vector space by S λ , to define the quantum complex cobordism theory as the extraordinary cohomology theory with QU * (pt) = QΩ * := Ω U ⊗ S * , where S * = Hom(S, Q) is the dual Landweber-Novikov algebra, considered here as the deformation parameter space.The cohomology theory QU * admits a geometric realisation as double cobordism theory DU * , which was introduced in [9] in relation with Drinfeld's quantum double and studied in more detail in [10].
From the results of Landweber [34] and Novikov [48] (see also [8]) it follows that there is a canonical isomorphism of algebras σ : S * ∼ = Ω U ⊗ Q.We show that the image of the dual basis S λ ∈ S * can be given explicitly as where Θ λ are the products of theta divisors (7) with the cobordism classes [Θ λ ] giving the canonical basis in our ring Θ U .
Inspired by constructions from [9]), we introduce the quantisation map where the sum here is over all non-empty partitions.Define also a dequantisation map where µ : U * (X) → H * (X, Z) is the cycle realisation homomorphism, which is defined uniquely by the property µ(u) = z for the same u ∈ U 2 (CP ∞ ) and z ∈ H 2 (CP ∞ , Z) as before.This allows us to interpret Chern-Dold character in complex cobordisms as the composition Theorem 1.3.The Chern-Dold character in the complex cobordisms is the composition of quantisation and dequantisation maps: For X = pt this composition is the canonical embedding Ω U → Ω U ⊗ Q, but even in this case this leads to a non-trivial formula (6) (see Section 4).
We should mention here very interesting work by Coates and Givental [14,15], who considered an analogue of quantum cohomology with the corresponding Gromov-Witten invariants [60] taking values in complex cobordisms.Some important relations of complex cobordisms with conformal field theory and integrable systems were discussed also in [30,33,43].
We consider also the most degenerate case of abelian variety A n+1 = E n+1 , where E is an elliptic curve.In that case the theta-divisor is singular, but we show that there is a smooth real-analytic representative of the same homology class in H 2n (A n+1 , Z).More precisely, we have the following result (see more details in Section 5).

Theorem 1.4.
There is a smooth real-analytic U -manifold M 2n W ⊂ E n+1 given in terms of classical Weierstrass functions, which can be used as a representative of the cobordism class [Θ n ].
For every k > 1 the cobordism class k n+1 [Θ n ] can be realised by an irreducible algebraic subvariety of E n+1 .
The structure of the paper is following.We start with a review of the main notions and results in complex cobordism theory, including the Chern-Dold character and Riemann-Roch-Grothendieck-Hirzebruch theorem.
In the central section 3 we describe the topological properties of the smooth theta divisors and use them to express explicitly the Todd class and Chern-Dold character in complex cobordism theory.We introduce also the corresponding dual complex bordism classes and study their properties.
In section 4 we discuss the Landweber-Novikov algebra and use its dual algebra as the deformation parameter space for certain quantisation of the complex cobordism theory, giving a different interpretation of the Chern-Dold character.We describe the action of the Landweber-Novikov operations on the cobordism classes of the theta divisors in terms of the algebraic cycles in general abelian varieties.
In section 5 we present some real-analytic representatives of the cobordism classes of theta divisors written explicitly in terms of the classical Weierstrass sigma and zeta functions.
In the last section we discuss the link with the Milnor-Hirzebruch problem about description of possible characteristic numbers of the smooth irreducible algebraic varieties.

Complex bordisms and cobordisms
We present now a brief review of the complex cobordism theory, referring for the details to Stong's lecture notes [56], or, for more algebraic view, to Quillen's work [52].For the theory of the characteristic classes in cohomology theory we refer to Milnor and Stasheff [41], in K-theory -to Atiyah [3] and in cobordism theory -to Conner and Floyd [16,17], for relations with the theory of algebraic cobordisms [36] -to Panin et al [49] and survey [11].
Let M m be a smooth closed real oriented manifold.By stable complex structure (or, simply U -structure) on M m we mean an isomorphism of the real oriented vector bundles T M m ⊕ (2N − m) R ∼ = rξ, where T M m is the tangent bundle of M m , (2N −m) R is trivial naturally oriented real (2N −m)dimensional bundle over M m , ξ is a complex vector bundle over M m and rξ is its real form.A manifold M m with a chosen U -structure is called Umanifold.Note that a complex structure on ξ determines complex structure in the stable normal bundle νM m .
Two closed smooth real oriented m-dimensional U -manifolds M 1 and M 2 are called U -cobordant if there exists a real (m + 1)-dimensional U -manifold W with boundary such that the boundary ∂W is a disjoint union of M m 1 with given orientation and M m 2 with the opposite orientation, and such that the restriction of the stable complex normal bundle νW to M i coincides with the stable complex normal bundles νM i , i = 1, 2. The notion of bordisms of U -manifolds is a bit more involved, see details in [17], Ch. 1.
Define the following operations in the corresponding equivalence classes of U -manifolds.The sum of bordism classes of two closed U -manifolds M m 1 and Similarly define the product of the bordism classes of and the rings Ω U and Ω U are isomorphic.This isomorphism can be extended to Poincare duality between complex bordisms and cobordisms for any U -manifold.
Let λ = (i 1 , . . ., i k ), i 1 ≥ • • • ≥ i k be a partition of n = i 1 + • • • + i k , and p(n) be the number of such partitions.Using the standard splitting principle one can define the Chern classes c λ (T M ) ∈ H 2n (M, Z) of a U -manifold M corresponding to the monomial symmetric functions m λ (t) = t i 1 1 . . .t i k k + . . .(see [41]).We would like to emphasize the choice of monomial symmetric functions here, which is essential for us.
The Chern number c λ (M 2n ), |λ| = n of U -manifold M 2n is defined as the value of the cohomology class c λ (T M 2n ) on the fundamental cycle M 2n : We have p(n) Chern numbers c λ (M 2n ), which depend only on the bordism class of M 2n .
The following fundamental result is due to Milnor and Novikov.
The choice of suitable algebraic representatives of the bordism classes y k ∈ Ω U , k ∈ N was discussed starting from the work of Milnor and Novikov, see the references and latest results in [53].
It will be more convenient for us to use the Chern numbers c ν λ (M 2n ) defined using the stable normal bundle νM 2n : They can be expressed through the usual Chern numbers c λ (M 2n ) and contain the same information about U -manifold M 2n .We will use the following convenient class of U -manifolds from [13].
Let M 2n be a smooth real manifold of dimension 2n.A complex framing of M 2n is a choice of complex line bundle L on M 2n , such that the direct sum T M 2n ⊕ L admits a structure of trivial complex vector bundle.Thus complex framing is a U -structure of very special type.The examples of such structures is given by the following natural construction.
Let X be a complex manifold of (complex) dimension n + 1 with holomorphically trivial tangent bundle and L be a complex line bundle over X.Let S be a real-analytic section S : X → L, transversal to the zero section and consider M 2n = {x ∈ X : S(x) = 0} ⊂ X, which is a smooth real-analytic submanifold of X.Then the line bundle L = i * (L), where i : M 2n → X is the natural embedding, determines the complex framing on M.
In our main example X is a principally polarised abelian variety and L is the canonical line bundle, with holomorphic section given by the θ-function (see next section).A more explicit example of a real-analytic submanifold for X being a product of elliptic curves is discussed in section 5.
In complex cobordism theory there exists an analogue of the celebrated Riemann-Roch formula [6].To explain it recall first its Hirzebruch version.
Let X be a CW -complex and ξ → X be a complex vector bundle over X.The characteristic Todd class T d(ξ) ∈ H * (X, Q) of ξ is uniquely defined by following properties: , where η is the line bundle associated to the hyperplane section over CP n and z = c 1 (η) ∈ H 2 (CP n ; Z) for every n.
The Todd class T d : K(X) → H * (X, Q) in K-theory and the classical Chern character ch : K(X) → H * (X, Q) are related by the fundamental relation where ) is the first Chern class of the line bundle η in K-theory (see [6,16]).
The Todd genus of a U -manifold M 2n is the characteristic number where M 2n is the fundamental cycle of manifold M 2n .Todd genus defines the ring homomorphism Ω U → Z, which, due to Thom's results [57], is uniquely determined by the condition that T d(CP n ) = 1 for all n.Let M be a smooth complex algebraic variety and g i be the complex dimension of the space of holomorphic forms of degree i on M .Then Todd genus T d(M ) of M is equal to its arithmetic genus, or holomorphic Euler characteristic: T d(M ) = i 0 (−1) i g i (see [26,27]).
More generally, let ξ be a holomorphic vector bundle over M , ch(ξ) be its Chern character [41] and H i (M, O(ξ)) be the corresponding cohomology groups [27], then we have the following generalisations of the Riemann-Roch formula due to Hirzebruch and Grothendieck.
The Grothendieck's version reduces to Hirzebruch's one when Y is a point (see e.g.[20]).To describe its extension in the theory of complex cobordisms we recall that the characteristic Todd class T d U (ξ) of complex vector bundle over CW -complex X with values in H * (X, Ω U ⊗ Q) is uniquely defined by the following properties (see [6]): • For every two vector bundles ξ 1 and ξ 2 over where T M 2n is the tangent bundle of M 2n and [M 2n ] is its bordism class.Note that from the first condition the Todd class is uniquely defined by its value on the line bundle η over CP N : where the coefficients A n ∈ Ω −2n U ⊗Q are determined by the second condition.One of the results of this paper is the formula where β(z) is expressed in terms of the theta divisors by (5).
For any complex vector bundle ξ over CW -complex X we have where the sum is over all partitions λ = (i 1 , . . ., i k ), c λ (ξ) ∈ H 2|λ| (X, Z) are the corresponding Chern classes and The analogue of the fundamental relation (21) has the form where the cobordism class c U 1 (η) ∈ U 2 (CP n ) is the first Chern class of the canonical bundle η in complex cobordisms [16], which is dual to the bordism class of the canonical embedding CP n−1 ⊂ CP n .This relation is a key ingredient in the proof of the following analogue of Riemann-Roch formula in complex cobordisms [6].
For U -manifolds we have the Poincaré-Atiyah duality in complex cobordisms [56]: ) be the standard bordism homomorphism, and be the corresponding Gysin homomorphism in complex cobordisms [6].In particular, when a ∈ U 2k (M 2n ) is the cobordism class dual to the bordism class of a smooth U -submanifold M 2n−2k ⊂ M 2n , then for the mapping f : where In particular, when so in the left hand side we have simply f ♯ x.Since for the mapping f : X → pt we have f !(a) = (a, X ), the formula (28) reduces to So we see that both Chern-Dold character ch U and Todd class T d U play a prominent role in these formulas.
We will show now that they both can be explicitly described in terms of the theta divisors (see Theorems 3.1 and 3.5 below).

Theta divisors, Todd class and Chern-Dold character in complex cobordisms
Let A n+1 = C n+1 /Γ be a principally polarised abelian variety with lattice Γ generated by the columns of the (n + 1) × 2(n + 1) matrix (I, τ ) with complex symmetric (n + 1) × (n + 1) matrix τ having positive imaginary part [21].Its polarisation line bundle L has one-dimensional space of sections generated by the classical Riemann θ-function The corresponding theta divisor Θ n ⊂ A n+1 given by θ(z, τ ) = 0 is known (after Andreotti and Mayer [2]) to be smooth for a general principally polarised abelian variety A n+1 .The topology of the smooth theta divisor does not depend on the choice of such abelian variety.
In particular, for n = 1 a generic abelian surface A 2 is the Jacobi variety of a smooth genus 2 curve C with theta divisor Θ 1 ∼ = C.For n = 2 an indecomposable A 3 is Jacobi variety of a smooth genus 3 curve C; in that case Θ 2 ∼ = S 2 (C) is smooth for all non-hyperelliptic curves C, which must be then trigonal.For n ≥ 3 the general case of A n+1 is not Jacobian, and the theta divisor is smooth outside a locus in the moduli space of the abelian varieties of complex codimension 1.For more detail on the geometry of theta divisors we refer to the survey [23] by Grushevsky and Hulek.
The line bundle L is ample with L ⊗3 known (after Lefschetz [5]) to be very ample, so that the sections of L ⊗3 determine the embedding of A n+1 into corresponding projective space P N , N = 3 n+1 − 1.The corresponding quadratic and cubic equations, defining the image in P N , were described by Birkenhake and Lange [4] (see also Ch. 7 in [5]).For the elliptic curves this reduces to the Hasse cubic equation Note that the line bundle L ⊗2 is not very ample; its sections define the quotient A n+1 /Z 2 by the involution z → −z, which is known as Kummer variety.When n = 1 this is the famous Kummer quartic surface in P 3 with 16 singular points, see e.g.[5].
Let i : Θ n → A n+1 be the natural embedding and since the tangent bundle of an abelian variety is trivial.This means that the topological Euler characteristic since D n = (n + 1)! (see next section).Alternatively, we can use formula (8) with Q(z) = 1 + z corresponding to the Euler characteristic Φ = χ: The Betti numbers of the theta divisors Θ n are not difficult to compute, see e.g.[29,44].Indeed, by the Lefschetz hyperplane theorem the embedding i : Θ n → A n+1 induces the isomorphisms for k < n, while for k = n these homomorphisms are surjections [35,40].
In particular, for n ≥ 2 the theta divisor has the fundamental group except the middle one b n , which can be found using the formula (32) for the Euler characteristic: where n is the n-th Catalan number, see [54].Since the cohomology groups of Θ n have no torsion [29], this defines them uniquely, but it seems that the multiplication structure remains to be understood.Note that we can compute the signature τ (Θ n ) of the corresponding quadratic form on the middle cohomology for even n using our general formula (8) (see also [13]).Indeed, the signature corresponds to the Hirzebruch L-genus with from (8) we have that the signature of the theta divisor Θ n for even n is where B n are the classical Bernoulli numbers: = −2.Note that the integrality of the right hand side of ( 34) is not obvious.The appearance of both Bernoulli and Catalan numbers looks quite intriguing and invites further study here.
We should mention here very interesting work by Nakayashiki and Smirnov on the computation of cohomology groups of the complement of the (singular) theta divisor in hyperelliptic Jacobi variety [44,45].
For all n the smooth theta divisor Θ n is a projective variety of general type.Indeed, since the canonical class of abelian variety is zero, by the adjunction formula [21] the canonical bundle K Θ n = i * (L) = L, which is ample.In particular, L is known to have an (n + 1)-dimensional space of sections generated by the partial derivatives ∂ ξ θ(z, τ ) of the theta function.
Consider the intersection of Θ n with k general translates Θ n (a i ), a i ∈ A n+1 of the theta divisor Θ n .
Proposition 3.1.For all k < n and general a i ∈ A n+1 , i = 1, . . ., k the variety Θ n−k k is smooth and irreducible of general type.
Proof.The abelian variety A n+1 is a group variety and hence is homogeneous.Each of the subvarieties Θ n (a i ) is a translate of the smooth subvariety Θ n .Therefore for general general a i ∈ A n+1 the intersection Θ n−k k is smooth by Kleiman's theorem [20].
For k < n this variety is irreducible, since by the Lefschetz hyperplane theorem (see Th.3.1.17 in [35]) the Betti number b 0 (Θ n−k k ) = 1 and thus Θ n−k k is connected.The translate Θ n (a i ) is the zero set of section of the shifted line bundle L(a i ), so the adjunction formula [21] shows that the canonical bundle K Θ n−k k is given by In particular, Θ 0 n consists of (n + 1)! points and Θ 1 n−1 is a curve with Euler characteristic χ = −n(n + 1)!.
The cobordism class [Θ n−k k ] does not depend on a 1 , . . ., a k and the choice of abelian variety.Alternatively, as a representative of this cobordism class we can choose the variety given by the system of equations Since the partial derivatives ∂ ξ θ(z, τ ) are the sections of the line bundle L on Θ n , this bundle is base-point free by smoothness of Θ n .Therefore by Bertini theorem [21] for generic ξ 1 , . . ., ξ k ∈ C n+1 the equation ( 36) determines a smooth irreducible algebraic variety for all k < n.
It turns out that the theta divisors can be used to represent Todd class of complex vector bundles in quite general situation.
Theorem 3.2.The characteristic Todd class of complex vector bundle ξ over CW-complex X is given by the formula where the sum is over all partitions λ = (i 1 , . . ., i k ) and as in (7).In particular, for any U -manifold M we have Proof.Denote the right hand side of formula (37) as T(ξ) and check that this characteristic class satisfies both defining properties of the Todd class.
The first property T(ξ 1 + ξ 2 ) = T(ξ 1 )T(ξ 2 ) follows from the well-known formula for the Chern classes We claim that it is the identity.Lemma 3.3.The Chern numbers (20) of the theta divisor Θ n are for any partition λ of n different from the one-part partition λ = (n), and Proof.Since the tangent bundle of abelian variety is trivial, the normal bundle νΘ n is stably equivalent to the line bundle L = i * (L), where i : Θ n → A n+1 is natural embedding and L is the principal polarisation line bundle on A n+1 .This immediately implies (39).
To prove condition (40) we need only to use the well-known fact that where D ∈ H 2 (A g , Z) is the Poincare dual cohomology class of the theta divisor Θ ⊂ A g of any principally polarised abelian variety (see e.g.[5]).
Geometrically, this means that the intersection of g generic shifts of theta divisor Θ of abelian variety A g consists of g! points.One can see this easily in the degenerate case when X g = E g is the product of g elliptic curves.
Due to the results of Milnor and Novikov, since c (n) (Θ n ) = 0 the theta divisors can be chosen as multiplicative generators of the algebra Ω U ⊗ Q. Hence to prove that (T(T M ), M ) = [M ] it is enough to check this for all theta divisors, which immediately follows from the lemma.By uniqueness T(ξ) = T d U (ξ), which completes the proof.
Corollary 3.4.The cobordism class of any U -manifold M 2n can be given by formula (6).Indeed, we have formula (6) by evaluating formula (38) on the fundamental cycle of M 2n and using the second property of the Todd class.
Recall now that the Todd class is uniquely defined by its value on the line bundle η over CP ∞ by formula (23): where some coefficients A n ∈ Ω −2n U ⊗ Q.Now we can describe these coefficients in terms of theta divisors.
Corollary 3.5.The Todd class of η has the form Indeed, this follows from the relation T d U (η)T d U (−η) = 1 and formula (37) applied to ξ = −η.Now we are ready to prove Theorem 1.1.
Theorem 3.6.The Chern-Dold character is uniquely defined by the formula where z = c 1 (η) ∈ H 2 (CP N , Z) and u ∈ U 2 (CP N ) is the first Chern class of the line bundle η over CP N in the complex cobordisms.
Proof.We use the fundamental relation (26), which in these notations has the form Comparing this formula with (41) we have formula (42) and the claim.
As a corollary we have formula (8) for the exponential generating function of any Hirzebruch genus Φ of all theta divisors.
The Hirzebruch genus in complex cobordisms [27] is a homomorphism Φ : Ω U → A, where A is some algebra over Q, determined by its characteristic power series For any Hirzebruch genus Φ with characteristic power series Q(z) we have the following relations with the Chern-Dold character and Todd class (see [6]).Combining this with Theorem 3.6 we have Corollary 3.7.The exponential generating function of Consider now the special case of A = Ω ⊗ Q and Φ : Ω → A = Ω ⊗ Q being the natural embedding.Then from (45) we see that the corresponding series Q(z) = z/β(z) with β(z) given by (5).
Define the cobordism classes v n ∈ Ω U ⊗ Q as the coefficients of Q v (z) written in the form with the series In the theory of symmetric functions [37] this corresponds to the duality ω between elementary symmetric functions e n and complete symmetric functions h n (see formula (2.6) in [37]), where we substitute The determinantal formula for this duality (see e.g.page 28 in [37]) allows to express v n as a polynomial of t k := [Θ k ], k = 1, . . ., n with rational coefficients.For example, In fact, since the series in (48) are the exponential generating functions of y n = vn n+1 and x n = [Θ n ] n+1 respectively, we can apply the results about Hurwitz series [28,50] to deduce that y n ∈ Z[x 1 , . . ., x n ] is a polynomial with integer coefficients (which is clearly not the case in the formulae for v n above). Let . ., i k ) then the cobordism classes v λ considered as elements of S * form a basis dual to the Landweber-Novikov operations Sλ defined using the tangent bundles (see Novikov [48]).In particular, the (usual, tangent) characteristic numbers c λ (v n ) = 0 for any λ = (n), and Theorem 3.8.For any U -manifold M 2n we have the following analogue of formula ( 6): where instead of c ν λ (M 2n ) we use the characteristic numbers c λ (M 2n ) of the tangent bundle T M 2n .
The Hirzebruch genus Φ(v n ) with characteristic power series Q(z) can be found from the generating function The proof follows directly from Corollary 3.7 and formulas (47), (48).
In particular, for the Euler characteristic with Q(z) = 1 + z we conclude that χ(v n ) = 0 for all n > 1 with χ(v 1 ) = −2.Similarly, for the Todd genus we have where B n are the Bernoulli numbers.This implies that is zero for odd n > 1 and non-zero rational for even n.In particular, T d(v 2 ) = 1 2 , so the cobordism class v 2 cannot be represented by a Umanifold.
A natural question is what is the minimal integer k n ∈ N such that k n v n ∈ Ω U is a cobordism class of some U -manifold.The following result gives the answer to this question.
Let q n be the denominator of the fraction (n+1)B n written in the simplest form, where for odd n > 1 with B n = 0 we put q n = 1.
Theorem 3.9.The cobordism class for some U -manifold V 2n , and this is the smallest multiple of v n with such property.
Proof.Let k n v n = [M 2n ] ∈ Ω U , then the Todd genus k n (n + 1)B n must be an integer, which means that k n is divisible by q n , and thus k n = q n must be minimal.
To prove that the cobordism class q n v n = [V 2n ] for some U -manifold V 2n we use the fundamental result of Hattori [24] and Stong [55,56], saying that element a ∈ Ω U ⊗ Q belongs to Ω U if and only if the Todd genus of all the results S λ (a) of the Landweber-Novikov operations applied to a are integer.This is obviously true for the Todd genus of a = q n v n since T d(a) = q n (n + 1)B n .We have the following important result.Lemma 3.10.Every Landweber-Novikov operation S λ , different from the identity, maps the cobordism class v n to the subring In particular, We will use the following properties of the Landweber-Novikov operations: where β(z) is given by ( 5), and S λ (β(z)) = 0 for all non one-part partitions (see the next section for more details).We will prove Lemma by induction in the length of λ.For length one partition λ = (k) we apply the operation S (k) to the relation (48), which we rewrite as Q v (z)β(z) = z, to have where we have used (57).Since Q v (z)β(z) = z this implies that In particular, for k = 1 we have and use the induction in k to conclude the proof for one-part partitions.
The general case follows from the multiplicative property (60) of the Landweber-Novikov operations and the fact that S λ (β(z)) = 0 for all partitions λ of length more than one.Now the claim of the Theorem follows from Lemma and Stong and Hattori results, since the Todd genus of any U -manifold is integer.Remark 3.11.It is natural to introduce also the following realisation of the power sums in complex cobordisms p n = (−1) n−1 (n−1)!w n , where w n ∈ Ω U ⊗ Q are defined by the formula (see [37] and formulae ( 47), (49) above).Applying the Landweber-Novikov operation S (k) to both sides and using (67) we have which implies that Up to a multiple, w n coincide with the cobordism classes of the manifolds N 2n from Theorem 1.4 of the paper [6].

The Landweber-Novikov algebra and quantisation of complex cobordisms
The Landweber-Novikov algebra introduced in [34,48] is an important subalgebra of all cohomological operations in complex cobordisms with additive basis given by the Landweber-Novikov operations S λ , λ ∈ P, where P is the set of all partitions λ = (i 1 , . . ., i k ) (see [37]).
Recall that the cobordism class α ∈ U 2 (X) is called geometric if it belongs to the image of the natural homomorphism H 2 (X, Z) → U 2 (X) (see [48]).The action of the Landweber-Novikov operations on any geometric cobordism class α is defined as follows: for any one-part partition λ = (k), k ∈ N the Landweber-Novikov operation S (k) (α) = α k+1 , while for all other partitions S λ (α) = 0 (see Lemma 5.6 in [48]).Together with the rule this uniquely defines the operations S λ : U * (X) → U * (X) for any CWcomplex X.
The action of S λ on the cobordism class of U -manifold M 2n can be given as where p ♯ : U * (M 2n ) → Ω U is the Gysin homomorphism for p : M 2n → pt and c U λ (νM 2n ) ∈ U 2|λ| (M 2n ) are the Conner-Floyd characteristic classes of the normal bundle of M 2n [16].
In particular, when are the characteristic numbers (20).
In this paper we consider the Landweber-Novikov algebra as the graded algebra S over Q with a special basis S λ , λ ∈ P, where the degree of S λ equals to 2|λ| = 2(i 1 + • • • + i k ) and the identity is defined as S ∅ .It is also a Hopf algebra with the diagonal given by The diagonal is symmetric, so the dual graded Hopf algebra S * = Hom(S, Q) is commutative.
The following result, which can be extracted from [34,48], plays an important role in constructions from [8].Proposition 4.1.There is a canonical isomorphism of algebras Proof.Let µ : U * (X) → H * (X, Z) be the cycle realisation homomorphism, which is defined uniquely by the property µ(u) = z, where as before u ∈ U 2 (CP ∞ ) and z ∈ H 2 (CP ∞ , Z) be the first Chern classes of the universal line bundle η over CP ∞ in the complex cobordisms and cohomology theory respectively.In particular, when X = pt is a point we have the augmentation ring homomorphism µ : Ω U → Z, which is 1 on the identity and 0 on elements with non-zero grading.
We have the canonical graded pairing between Landweber-Novikov algebra S and Ω U ⊗ which is known to be non-degenerate [48].This implies the canonical isomorphism σ : S * ∼ = Ω U ⊗ Q, which is also the algebra isomorphism as it follows from ( 60) and (63).
A natural question is what is the basis S λ ∈ Ω U ⊗Q dual to the Landweber-Novikov operations S λ , λ ∈ P. We can now give an explicit answer to this question.
Theorem 4.2.The dual basis to Landweber-Novikov operations S λ , λ ∈ P in Ω U ⊗ Q is given by where Θ λ is the product of theta divisors (7).
We can now describe the action of the Landweber-Novikov operations on the theta divisors and to prove Theorem 1.2.
Let [Θ n−k k ] be the cobordism class of the intersection divisor (11).
Proof.Recall that the normal bundle νΘ n can be identified with L = i * (L), where i : Θ n → A n+1 and L is the principal polarisation bundle of A n+1 .Since L is of rank one from formula (61) and properties of Conner-Floyd characteristic classes it immediately follows that S λ ([Θ n ]) = 0 if λ is not a one-part partition.
For the proof of formula (67) we need the following properties of the Gysin homomorphism f ♯ : U * (M 1 ) → U * (M 2 ) for any mapping of U -manifolds f : M 1 → M 2 (see e.g.Proposition D.3.6 in [12]): and for f : 61) and the properties of Gysin homomorphisms where f and p = f • i are the mappings to a point of A n+1 and Θ n respectively.Since [1] is the identity in the ring U * (Θ n ) we have using property (68) with x = [1] . By the definition of the Gysin homomorphism the cobordism class f ♯ ([D] k+1 ) is dual to the transversal intersection of k + 1 copies of cycles dual to [D], which can be realised as the intersection of generally shifted theta-divisors (11).This means that f which proves the theorem.
Let us introduce now the quantum complex cobordism theory as the extraordinary cohomology theory QU * := U * ⊗ S * with QU * (pt) = QΩ * := Ω U ⊗ S * .The algebra S * is used here as the deformation parameter space.
Introduce also the following quantum analogue of cycle realisation homomorphism as where we used the natural isomorphism It can be viewed as a kind of "dequantisation" map.
We claim that the Chern-Dold character in complex cobordisms is the composition of the quantisation and dequantisation maps: Theorem 4.4.The quantisation map defined by ( 69) is the algebra homomorphism.The Chern-Dold character ch U : Proof.The algebra homomorphism property q ⋆ = q ⋆ (x)q ⋆ (y) follows from the multiplicative property (60) of the Landweber-Novikov operations.
To prove the rest it is enough to check (72) only for u = c U 1 (η) ∈ U 2 (CP ∞ ).In that case by definition due to Theorem 3.6.
Corollary 4.5.The Chern-Dold character in complex cobordisms can be written as where µ : U * (X) → H * (X, Z) is the cycle realisation homomorphism.For X = pt and x = [M 2n ] ∈ Ω U we have in agreement with (1) and (6).
Note that the composition of the quantisation and dequantisation maps for X = pt is non surprisingly the identical map, but leads to a non-trivial formula (74).
According to general construction of Dold [19] every extraordinary cohomology theory h * has its analogue of Chern character, which is the transformation of cohomology theories with characteristic property that for X = pt it is the canonical homomorphism Ω h → Ω h ⊗ Q.
In our case of quantum complex cobordism theory QU * the corresponding quantum Chern-Dold character ch ⋆ U : QU * (X) = U * ⊗ S * → H * (X, QΩ * ) has the following explicit form Let us deduce now our formula (13).
Theorem 4.6.The cobordism class of Θ n−k k can be given as the residue integral at zero , where β(z) is given by (5).
Proof.We use the fact that the Chern-Dold character ch U : U * (X) → H * (X, Ω U ⊗ Q) commutes with the action of the Landweber-Novikov algebra, which is trivial on the cohomology H * (X, Q) (cf.[6]).
Applying Landweber-Novikov operation S λ to both sides of the relation (42) we have S λ ch U (u) = ch U (S λ u) = β(z) k+1 if λ = (k) for some k ∈ N and 0 otherwise.On the right hand side for λ = (k) we have and zero for any λ = (k).Comparison proves the claim and formula (13).The polynomial subring Θ U ⊂ Ω U generated by the theta divisors: is invariant under the Landweber-Novikov operations.
Indeed, since β(z) is an exponential generating function, this follows from the properties of Hurwitz series [28], see also problems 174-177 in [50], Vol. 2, Ch. 8.In particular, The following important interpretation of the Landweber-Novikov algebra was found by Buchstaber and Shokurov [8].
Consider the group Dif f 1 of the formal diffeomorphisms of the line given by f Its Lie algebra diff 1 is the Lie algebra of the corresponding formal vector fields.
Theorem 4.8.(Buchstaber and Shokurov [8]) The real version S ⊗ R of the Landweber-Novikov algebra is isomorphic to the universal enveloping algebra of the Lie algebra diff 1 , which can be identified with the algebra of the leftinvariant differential operators on the group Dif f 1 .
In particular, as an algebra S ⊗R is generated by two Landweber-Novikov operations S (1) and S (2) , corresponding to two vector fields on the group Dif f 1 , which can be written in coordinates α k , k ∈ N as The algebra A U of all cohomological operations in complex cobordisms was introduced in [48] and is known as Novikov algebra.It is Z-graded algebra, which is a free left Ω U -module with generators S λ and the commutation relations (see Lemma 5.4 in Novikov [48]).
Let us introduce two new families of elements of Novikov algebra where for partition λ = (λ 1 , . . ., λ k ) we define being U -manifolds (56) from the previous section.
From our results it follows that they give additive bases in two proper subalgebras of Novikov algebra, generating this algebra over Q.Our Theorem 1.2, Corollary 4.7 and Lemma 3.9 can be used to describe the multiplication in these bases, which could be useful to study the representations of Novikov algebra.
The important elements of Novikov algebra, motivated by the Conner and Floyd results [16], describing the K-theory in terms complex cobordisms, are the Adams-Novikov operations Ψ k U (see [48]).They are defined uniquely as the multiplicative operations in complex cobordisms, which satisfy the relation For any cobordism class [M 2n ] of U -manifold M 2n we have Let Θ n (k) be a smooth zero locus of generic section of the k-th tensor power L ⊗k of the line bundle L, defining the principal polarisation of an abelian variety A n+1 .Note that for k ≥ 2 the zero locus of generic section of L ⊗k is smooth for any abelian variety by Bertini theorem.We would like to mention that the subvarieties Θ n (k) ⊂ A n+1 have appeared in [42] as the spectral varieties of certain commutative algebras of differential operators.
The cohomology and the corresponding cobordism class [Θ n (k)] do not depend on the choice of such section.As for the theta divisor, by Lefschetz hyperplane theorem the corresponding Betti numbers are except the middle one b n , which can be found using the formula for the Euler characteristic χ(Θ n (k)) = (−1) n k n+1 (n + 1)! : The signature of τ (Θ n (k) = k n+1 τ (Θ n ), so for even n we have where B n are the Bernoulli numbers.
Theorem 4.9.The cobordism class [Θ n (k)] can be expressed as Proof.The normal bundle to Θ n (k) can be identified with L ⊗k = i * L ⊗k induced from L ⊗k by the embedding i : Θ n (k) → A n+1 .According to formula (6) we have )!, which together with (80) proves the claim.

Real-analytic elliptic representatives
Consider now the most degenerate case of the abelian variety, when A n+1 is the product of n + 1 copies of an elliptic curve E = C/Γ, where Γ is the lattice with periods 2ω 1 , 2ω 2 (in the classical notations from [59]).
Let L be the canonical line bundle on it with the holomorphic section where σ is the classical Weierstrass sigma function [59].The corresponding theta divisor defined by S 0 (u) = 0 is the union of n + 1 coordinate hypersurfaces given by u i = 0.It is singular, but we can show now that one can find a smooth realanalytic representative of the same homology class, which can be determined in terms of classical elliptic functions.
Consider the classical Weierstrass zeta function ζ(z) with simple poles at the lattice points and the transformation properties where η i = ζ(ω i ), i = 1, 2, see [59].
Introduce the following non-holomorphic function (inspired by the theory of periodic vortices [25,58]) where a, b is the unique solution of the following linear system or, explicitly , where we have used the Legendre identity [59] η 1 ω 2 − η 2 ω 1 = πi 2 .
In the lemniscatic case with ω 1 = ω ∈ R, ω 2 = iω we have which has zeros precisely at the 3 half-periods.From Legendre identity we have iωη 1 − ωη 2 = 2iωη 1 = πi 2 , so in this case which gives a = 0, b = − π 4ω 2 and the relation (86).Note that the function ξ(z) satisfies the equation ∂ ∂ξ(z, z) = 0 and thus is indeed a complex-valued harmonic function.The zeros of such functions were extensively studied, see [18] and references therein.
The number of the zeros of such functions depends on the position of zero in relation with the caustic defined as the image ξ(Σ) ⊂ C of the critical set Σ := {z ∈ C : J ξ (z, z) = 0}, where J ξ is the Jacobian of the map ξ : R 2 → R 2 .The real Jacobian of the harmonic function f (z, z) = g(z) + h(z) with holomorphic g, h is

Thus in our case
is a genus 2 curve with non-commutative π 1 (C) and i * : π 1 (C) → Z 4 being the abelianisation map and all other homotopy groups being trivial.The manifolds with free abelian fundamental group were studied by Novikov[47] in relation with the famous problem of topological invariance of rational Pontryagin classes.The theta divisors give non-trivial examples of such manifolds with non-zero Pontryagin classes.Using Poincare duality we obtain now all Betti numbers of Θ n as b In particular, for n = 2 we have b 2 (Θ 2 ) = 16 and τ (Θ 2 ) = 2 4 (2 4 −1)B 4 4