Chern-Dold character in complex cobordisms and theta divisors
aa r X i v : . [ m a t h . A T ] J u l CHERN-DOLD CHARACTER IN COMPLEX COBORDISMSAND ABELIAN VARIETIES
V.M. BUCHSTABER AND A.P. VESELOV
Abstract.
We show that the theta divisors of general principally po-larised abelian varieties can be chosen as smooth irreducible algebraicrepresentatives of the coefficients of the Chern-Dold character in com-plex cobordisms and describe the action on them of the Landweber-Novikov operations. The link with Milnor-Hirzebruch problem aboutalgebraic representatives in the complex cobordisms is discussed. Introduction
In the complex cobordism theory going back to the foundational worksof Milnor and Novikov [15], [16] a crucial role is played by the Chern-Doldcharacter introduced by the first author in [4]. In particular, it appears in theformulation of an analogue of the Riemann-Roch-Grothendieck-Hirzebruchtheorem in the theory of complex cobordisms, see [6].By definition, the Chern-Dold character ch U is a natural multiplicativetransformation ch U : U ∗ ( X ) → H ∗ ( X ; Ω U ⊗ Q ) , where U ∗ ( X ) is the complex cobordism ring of a CW -complex X andΩ U = U ∗ ( pt ) is the coborodism ring of the stably complex manifolds (or, inshort, U -manifolds). The fundamental Milnor-Novikov result says that thecoefficient ring of the theory of U ∗ ( X ) is the polynomial ringΩ U = Z [ y , . . . , y n , . . . ] , deg y n = − n of infinitely many generators y n , n ∈ N . Let u ∈ U ( C P ∞ ) and z ∈ H ( C P ∞ ) be the first Chern classes of theuniversal line bundle over C P ∞ in the complex cobordisms and cohomologytheory respectively. The Chern-Dold character is uniquely defined by itsaction ch U : u → β ( z ) , β ( z ) := z + ∞ X n =1 [ B n ] z n +1 ( n + 1)! , (1)where B n are certain U -manifolds (see [4]).The series β ( z ) is the exponential of the commutative formal group F ( u, v ) = u + v + X i,j a i,j u i v j of the geometric complex cobordisms introduced by Novikov in [17], so that F ( β ( z ) , β ( w )) = β ( z + w ) . Quillen identified this group with Lazard’s universal one-dimensional com-mutative formal group [18]. he inverse of this series is known as the logarithm of this formal groupand can be explicitly given by the Mischenko series [17, 5]: β − ( u ) = u + ∞ X n =1 [ C P n ] u n +1 n + 1 . (2)The question whether there are natural algebraic representatives of thecobordism classes [ B n ] in the exponential of the formal group given bythe Chern-Dold character was open for a long time since 1970 work of thefirst author [4].In this paper we give the following answer to this question, presenting anexplicit form of the series (1) as β ( z ) = z + ∞ X n =1 [Θ n ] z n +1 ( n + 1)! , (3)where Θ n is the theta divisor of a general principally polarised abelian vari-ety A n +1 , considered as real manifold of dimension 2 n. The cobordism classof the theta divisor does not depend on the choice of such variety providedΘ n is smooth, which is true in general case [1]. Theorem 1.1.
The theta divisor Θ n of a general principally polarised abelianvariety A n +1 is a smooth irreducible projective variety, which can be taken asa natural algebraic representative of the coefficient [ B n ] in the Chern-Doldcharacter. As a corollary we have the following representation of the cobordism classof U -manifold M n in terms of theta divisors[ M n ] = X ω : | ω | = n c νω ( M n ) [Θ nω ]( ω + 1)! , (4)where the sum is over all partitions ω = ( i , . . . , i k ) of | ω | = i + · · · + i k = n, Θ nω := Θ i . . . Θ i k , ( ω + 1)! := ( i + 1)! . . . ( i k + 1)! and c νω ( M n ) ∈ Z are the Chern numbers ofthe normal bundle of M n (see next section for details).Since according to [4] the Todd genus T d ( B n ) = ( − n we have thefollowing formula for Todd genus for any U -manifold M n T d ( M n ) = X ω : | ω | = n c νω ( M n ) ( − n ( ω + 1)! . (5)This implies the divisibility condition on the Chern numbers c νω ( M n ) of U -manifolds. More divisibility conditions we can get applying to formula(4) the Landweber-Novikov operations and taking the Todd genus (see thelast section).The action of the Landweber-Novikov operations on the theta divisorscan be described explicitly.Let ω = ( i , . . . , i k ) be a partition of | ω | := i + · · · + i k and S ω [ M ] be theresult of the action of the Landweber-Novikov operation S ω on U -manifold M defined in terms of its stable normal bundle (see [14, 17]). et ( k ) = ( k, . . . ,
0) be a one-part partition and consider the smoothcomplete intersectionΘ n − kk = Θ n ∩ Θ n ( a ) ∩ . . . Θ n ( a k )of Θ n with k general translates Θ n ( a i ) , a i ∈ A n +1 of the theta divisor Θ n . Let D = c ( L ) ∈ H ( A n +1 , Z ) be Poincare dual cohomology class of Θ n , then Θ n − kk is the Poincare dual to D k +1 ∈ H k +2 ( A n +1 , Z ) . Note that, dueto a theorem of Mattuck, for a general abelian variety D m generates thecorresponding Hodge group H mHodge ( A n +1 ) for all m = 1 , . . . , n + 1 (seesection 17.4 in [3]). Theorem 1.2.
For every partition ω with | ω | < n the Landweber-Novikovcobordism class S ω [ B n ] has a smooth irreducible algebraic representative.More precisely, if ω = ( k ) for some k ∈ N then S ω [ B n ] = S ω [Θ n ] = 0 , while for ω = ( k ) we have S ( k ) [ B n ] = S ( k ) [Θ n ] = [Θ n − kk ] . (6)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, butwe show that there is a real-analytic representative of the same cohomologyclass given in terms of the classical elliptic functions.In the last section we discuss the link with Milnor-Hirzebruch problemabout characteristic numbers of smooth irreducible algebraic varieties.2. Complex bordisms and cobordisms
Let M m be a smooth closed real manifold. By stable complex structure (or, simply U - structure ) on M m we mean an isomorphism of real vectorbundles i M : T M m ⊕ (2 N − m ) R → rξ, where T M m is the tangent bundle of M m , (2 N − m ) R is trivial real (2 N − m )-dimensional bundle over M m and rξ is a complex vector bundle over M m considered over reals. A manifold M m with a chosen U -structure is called U - manifold .Note that complex structure in the stable tangent bundle T M m deter-mines complex structure in the stable normal bundle νM m . Two closed smooth real m -dimensional manifolds M and M are called bordant if there exists real ( m + 1)-dimensional U -manifold W such that theboundary ∂W is a disjoint union of M m and M m and the restriction of thestable tangent bundle T W to M i coincides with T M i , i = 1 , . Similarly, Two closed smooth real m -dimensional manifolds M and M are called cobordant if there exists real ( m + 1)-dimensional U -manifold W such that the boundary ∂W is a disjoint union of M m and M m and therestriction of the stable normal bundle νW to M i coincides with νM i , i =1 , . Define the following operations in corresponding equivalence classes of U -manifolds.Disjoint union M m ∪ M m of two closed m -dimensional U -manifolds is U -manifolds. Define the sum of the corresponding bordism classes as[ M m ] + [ M m ] = [ M m ∪ M m ] . imilarly define the product of bordism classes by[ M m ][ M m ] = [ M m × M m ] , where M m × M m is the direct product of M m and M m . This defines thecommutative graded ring Ω U ∗ = X m ≥ Ω Um , where Ω Um is the group of bordism classes of m -dimensional U -manifolds.Similarly, we have the graded ring Ω ∗ U = P m ≥ Ω − mU , where Ω − mU is thegroup of cobordism classes of m -dimensional U -manifolds.From the correspondence between stably complex structures in tangentand normal bundles it follows that the groups Ω Um and Ω − mU and the ringsΩ ∗ U and Ω U ∗ are isomorphic. This isomorphism can be extended to Poincareduality between complex bordisms and cobordisms.The following fundamental result is due to Milnor and Novikov. Theorem 2.1. (Milnor [15] , Novikov [16] ) The graded complex bordismring Ω U ∗ is isomorphic to the graded polynomial ring Z [ a , a , . . . , a n , . . . ] of infinitely many variables a n , n ∈ N , where deg a n = 2 n. In particular, Ω U n − = 0 . Let ω = ( i , . . . , i k ) , i ≥ i ≥ · · · ≥ i k be a partition of n : i + · · · + i k = n. Using the standard splitting principle one can define the Chern classes c ω ( T M ) ∈ H n ( M, Z ) of a U -manifold M corresponding to the monomialsymmetric functions m ω ( t ) = t i . . . t i k k + . . . .The Chern numbers c ω ( M n ) , | ω | = n of U -manifold M n are definedas the value of the cohomology class c ω ( T M n ) on the fundamental cycle < M n > : c ω ( M n ) := ( c ω ( T M n ) , < M n > ) . (7)We have π ( n ) Chern numbers c ω ( M n ), where π ( n ) is the number of parti-tions of n, which depend only on the bordism class of M n . Corollary 2.2.
Two closed n -dimensional U -manifolds M and M are U -bordant if and only if all the corresponding Chern numbers are the same. It will be more convenient for us to use the Chern numbers c νω ( M n )defined using the stable normal bundle νM n : c νω ( M n ) := ( c ω ( νM n ) , < M n > ) . (8)They can be expressed through the usual Chern numbers c ω ( M n ) and con-tain the same information about U -manifold M n . We will use the following convenient class of U -manifolds from [7].Let M n be a smooth real manifold of dimension 2 n . A complex framing of M n is a choice of complex line bundle L on M n , such that the directsum T M n ⊕ L admits a structure of trivial complex vector bundle. Thuscomplex framing is a U -structure of very special type. The examples of suchstructures is given by the following construction.Let X be a complex manifold of complex dimension n + 1 with holomor-phically 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 onsider M n = { x ∈ X : S ( x ) = 0 } ⊂ X , which is a smooth real-analyticsubmanifold of X . Then the line bundle L = i ∗ ( L ), where i : M n → X isthe embedding, determines the complex framing on M. An explicit example of such submanifold for X being a product of n + 1elliptic curves will be discussed in section 5.3. Theta divisors of abelian varieties
Consider now our main example, when X = A n +1 is principally polarisedabelian variety A n +1 = C n +1 / Γ with lattice Γ generated by the columns ofthe ( n + 1) × n + 1) matrix ( I τ ) with complex symmetric ( n + 1) × ( n + 1)matrix τ having positive imaginary part [8]. It has a canonical line bundle L with one-dimensional space of sections generated by the classical Riemann θ -function θ ( z, τ ) = X l ∈ Z n +1 exp[ πi ( l, τ l ) + 2 πi ( l, z )] , z ∈ C n +1 . (9)The corresponding theta divisor Θ n ⊂ A n +1 given by θ ( z, τ ) = 0 is known(after Andreotti and Mayer [1]) to be smooth for general principally po-larised abelian variety A n +1 . In particular, for n = 1 a generic abelian surface is Jacobi variety ofa smooth genus 2 curve C with theta divisor Θ ∼ = C , for n = 2 this isJacobi variety of genus 3 curve C with Θ ∼ = S ( C ) being smooth for allnon-hyperelliptic C . For n ≥ L is ample with L known (after Lefschetz [3]) to be veryample in the sense that the sections of L determine the embedding of A n +1 into corresponding projective space P N , N = 3 n +1 − . The correspondingquadratic and cubic equations, defining the image in P N , were described byBirkenhake and Lange [2] (see also Ch. 7 in [3]). For the elliptic curves thisreduces to the Hasse cubic equation x + y + z = 3 λxyz. Note that the line bundle L is very ample only on the quotient X/ Z byinvolution z → − z , which is known as Kummer variety.
It has the sectionsgiven by θ ǫ,δ ( z ), where the theta-functions with characteristics ( ǫ, δ ) ∈ Z n are defined by θ ǫ,δ ( z ) = X l ∈ Z n +1 exp πi ( l + ǫ/ , τ l + ǫ/
2) + 2 πi ( l + ǫ/ , z + δ/ , z ∈ C n +1 . In that case its image in P N , N = 2 n +1 − D = c ( L ) ∈ H (Θ n , Z ) be the first Chern class of line bundle L . Then the total Chern class c (Θ n ) = 1 + c (Θ n ) + · · · + c n (Θ n ) of Θ n satisfies c (Θ n )(1 + D ) = 1 (10) ince the tangent bundle of an abelian variety is trivial. This means thatthe Euler characteristic χ (Θ n ) = c n (Θ n ) = ( − n D n = ( − n ( n + 1)! , (11)since D n = ( n + 1)! (see next section).The Betti numbers of the theta divisors Θ n are not difficult to compute,see e.g. [13]. Indeed, by Lefschetz theorem the embedding i : Θ n → A n +1 induces the isomorphisms i ∗ : H k ( A n +1 , Z ) → H k (Θ n , Z )for k < n . This means that for k < n the Betti numbers are b k (Θ n ) = (cid:18) n + 2 k (cid:19) = b n − k (Θ n ) . The remaining middle Betti number can be found then using the formula(11) for the Euler characteristic: b n (Θ n ) = ( n + 1)! + nn + 2 (cid:18) n + 2 n + 1 (cid:19) . Since the cohomology groups of Θ n have no torsion [13], this defines themuniquely, but multiplication structure seems still to be understood. Notethat the signature τ (Θ n ) for even n has been computed in [7]: τ (Θ n ) = 2 n +2 (2 n +2 − n + 2 B n +2 , (12)where B n are Bernoulli numbers.In the simplest case n = 1 the general abelian variety is Jacobi variety ofsome genus 2 curve Γ, so by Riemann theorem in this case the theta divisorΘ = Γ is genus two curve.For all n the smooth theta divisor Θ n is a projective variety of generaltype. Indeed, by the adjunction formula the canonical class K Θ n = L := i ∗ ( L ) is ample. In particular, L is known to have n -dimensional space ofsections generated by the partial derivatives ∂ ξ θ ( z, τ ) of the theta function.By Bertini theorem the system of equations θ ( z, τ ) = 0 , ∂ ξ θ ( z, τ ) = 0 , . . . , ∂ ξ k θ ( z, τ ) = 0 , z ∈ A n +1 (13)with generic ξ , . . . , ξ k ∈ C n +1 determine smooth complete intersectionsΘ n − kk ⊂ Θ n ⊂ A n +1 mentioned above.The canonical class of S = Θ n − kk ⊂ A n +1 is K S = O (( k + 1) D ) | S , where D is the principal polarisation divisor of A n +1 . It is ample, so S is of generaltype as well. In particular, Θ n consists of ( n + 1)! points and Θ n − is a curvewith Euler characteristic χ = n ( n + 1)! . Note that the varieties Θ n − kk with k = n are irreducible, since they aresmooth and (by Lefschetz theorem) their Betti number b = 1.4. Proofs
Now we are ready to prove our results. The proofs are actually quitesimple, but based on the deep results from the theory of complex cobordisms.In particular, the following result from [4] is crucial for us. heorem 4.1. (Buchstaber [4] ) The coefficient [ B n ] of the Chern-Dold char-acter in complex cobordisms (1) is the cobordism class of a U -manifold M n ,which is uniquely determined by the following conditions: c νω ( M n ) = 0 (14) for any partition ω of n different from one-part partition ω = ( n ) , and c ν ( n ) ( M n ) = ( n + 1)! , (15) where c νω ( M n ) are Chern numbers defined by (8). To prove our Theorem 1.1 we need now only to check that the thetadivisor Θ n of a general principally polarised abelian variety A n +1 satisfiesthese conditions.The normal bundle ν Θ n can be identified with L = i ∗ ( L ), where i : Θ n → A n +1 is natural embedding and L is the principle polarisation line bundleon A n +1 . This immediately implies the condition (14).To prove condition (15) we need only to use the well-known fact that D g = g ! ∈ H g ( X g , Z ) = Z where D ∈ H ( X g , Z ) is the Poincare dual cohomology class of the thetadivisor Θ ⊂ X g of any principally polarised abelian variety (see e.g. [3]).Geometrically, this means that the intersection of g shifts of theta divisor Θof abelian variety X g consists of g ! points. To see this one can consider thedegenerate case when X g = E g is the product of g elliptic curves. In thatcase the theta divisor is a singular union of g coordinate hypersurfaces, butthe calculation of the intersection number is still valid and simple, giving g !.This completes the proof of Theorem 1.1.The formula (4) now follows from comparison of the Chern numbers ofboth sides.To prove Theorem 1.2 recall that the cobordism class α ∈ U ( X ) iscalled geometric if it belongs to the image of the natural homomorphism H ( X, Z ) → U ( X ) (see [17]).Novikov proved that for any geometric cobordism class α the Landweber-Novikov operation S ω ( α ) = 0 if ω = ( k ) for some k ∈ N , and S ( k ) ( α ) = α k +1 for all k ∈ N (see Lemma 5.6 in [17]).Since Θ n realises geometric cobordism class in U ( A n +1 ) correspondingto the principal polarisation D = c ( L ) ∈ H ( A n +1 , Z ) we conclude that S ω [Θ n ] = 0 if ω = ( k ) for some k ∈ N . When ω = ( k ) the cobordism class S ( k ) [Θ n ] is Poincare dual to D k +1 , which is represented by the intersectiontheta divisor Θ n − kk . This proves Theorem 1.2.5.
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 / L , where L is alattice with the periods of 2 ω , ω . Let L be the natural line bundle on it with the holomorphic section S ( u ) = σ ( u ) . . . σ ( u n +1 ) , u = ( u , . . . , u n +1 ) ∈ E n +1 , here σ is the classical Weierstrass elliptic function [22]. The zero set ofthis section given by the union of the coordinate hypersurfaces u i = 0 , i =1 , . . . , n + 1 is singular. One can actually show that there are no smoothalgebraic representatives in the same cohomology class. We will show now that one can find a smooth real-analytic representativeof the same cohomology class, which can be determined in terms of classicalelliptic functions.Consider the classical Weierstrass elliptic function η ( z ) with the simplepole at the lattice points and the transformation properties ζ ( z + 2 ω ) = ζ ( z ) + 2 η , ζ ( z + 2 ω ) = ζ ( z ) + 2 η , where η i = ζ ( ω i ) , i = 1 , . Let us introduce the following non-holomorphicfunction (cf. [12, 21]) ξ ( z ) = ζ ( z ) + az + b ¯ z, (16)where a, b is the (unique) solution of the following linear system aω + b ¯ ω + η = 0 , aω + b ¯ ω + η = 0 , (17)or, explicitly a = − η ¯ ω − η ¯ ω ω ¯ ω − ω ¯ ω = − η ¯ ω − η ¯ ω ℑ ( ω ¯ ω ) , b = η ω − η ω ω ¯ ω − ω ¯ ω = π ℑ ( ω ¯ ω ) , (18)where we have used the Legendre identity [22] η ω − η ω = πi . Lemma 5.1.
The function ξ is an odd doubly-periodic real-analytic functionand with the asymptotic behaviour ξ ∼ /z at zero and with zeroes at all 3half periods ω , ω , ω = ω + ω . In the lemniscatic case with ω = ω ∈ R , ω = iω we have ξ ( z ) = ζ ( z ) − π ω ¯ z, (19) which has zeros precisely at 3 half-periods.Proof. The property ξ ( − z ) = − ξ ( z ) follows from the same property of ζ ( z ) . The double-periodicity follows from the transformation properties of func-tion ζ : ξ ( z + 2 ω ) = ξ ( z ) + 2 η + 2 aω + 2 b ¯ ω = ξ ( z ) ,ξ ( z + 2 ω ) = ξ ( z ) + 2 η + 2 aω + 2 b ¯ ω = ξ ( z )due to (17). Combining these two properties we have ξ ( ω i ) = − ξ ( − ω i ) = − ξ ( ω i ), so ξ ( ω i ) = 0 for all half-periods.In the lemniscatic case with ω = ω ∈ R , ω = iω we have η ( iz ) = − iη ( z )and thus η = − iη , η ∈ R . Corresponding ℘ -function satisfies equation( ℘ ′ ) = 4 ℘ ( ℘ − e ) , e = ℘ ( ω ) = Γ (1 / πω where Γ is the classical Euler’s Γ-function. We are grateful to Ivan Cheltsov and Artie Prendergast-Smith for explaining how todo this rigorously. rom Legendre identity we have iωη − ωη = 2 iωη = πi , so in this case η = π ω , η = − i π ω , which gives a = 0 , b = − π ω and the relation (19).Note that the function ξ ( z ) satisfies the equation ∂ ¯ ∂ξ ( z, ¯ z ) = 0 and thusis a complex-valued harmonic function. The zeros of such functions wereextensively studied, see [19] and references therein.The number of the zeros of such functions depends on the position of zeroin relation with the caustic defined as the image ξ (Σ) ⊂ C of the critical setΣ := { z ∈ C : J ξ ( z, ¯ z ) = 0 } , where J ξ is the Jacobian of the map ξ : R → R . The real Jacobian of theharmonic function f ( z, ¯ z ) = g ( z ) + h (¯ z ) with holomorphic g, h is J f ( z, ¯ z ) = | g ′ ( z ) | − | h ′ ( z ) | . Thus in our case the critical set isΣ := { z ∈ C : | ℘ ( z ) | = π ω } . Note that this level of the real function F ( z, ¯ z ) = | ℘ ( z ) | is non-singular.Indeed, if ℘ ′ ( z ) = 0 then z is a half-period and thus ℘ ( z ) = 0, or ℘ ( z ) = ± e. Since 3 . < Γ(1 / < . e = | ℘ ( z ) | = Γ (1 / πω > π ω . Thus by general theory [19] the equation ξ ( z ) = c has one solution if c liesoutside the caustic and 3 solutions if c is inside the caustic. Since c = 0is inside the equation ξ ( z ) = 0 has 3 solutions, which are precisely thehalf-periods.Note that the corresponding Jacobians at the half-periods ω = ω, ω = iω are J ( z, ¯ z ) = | ℘ ( ω ) | − (cid:16) π ω (cid:17) = (cid:18) Γ (1 / πω (cid:19) − (cid:16) π ω (cid:17) > , while at z = ω + ı ω we have J ( z, ¯ z ) = | ℘ ( z ) | − (cid:16) π ω (cid:17) = − (cid:16) π ω (cid:17) < , so the sum of the indices is 1, as it is expected since the degree of the map ξ : E → C P is 1. (cid:3) Let I ⊂ [ n + 1] := { , , . . . , n + 1 } be a finite subset and denote ξ I ( u ) := Y i ∈ I ξ ( u i ) , u ∈ E n +1 . Consider now the following family of real-analytic (but non-holomorphic)sections of the line bundle L given by S ( u, a ) = S + S X I,J ⊂ [ n +1] ,I ∩ J = ∅ a IJ ( ξ I ( u ) + ξ J ( u )) (20)with arbitrary coefficients a IJ = a JI ∈ C . heorem 5.2. For generic coefficients a IJ the zero set of this section V na = { u ∈ E n +1 : S ( u, a ) = 0 } ⊂ E n +1 is a smooth connected real-analytic U -manifold, which can be used as a rep-resentative of the cobordism class [ B n ] from the Chern-Dold character.Proof. Consider the set
M ⊂ E n × C N ( a ) defined by S ( u, a ) = 0. We claimthat this is a smooth submanifold of this product. Indeed, assume that ∂S∂a IJ = S ( ξ I ( u ) + ξ J ( u )) = 0for all pairs of non-intersecting subsets I, J ⊂ [ n ] . Then, in particular, wehave n Y i =1 σ ( u i ) = 0 , n Y i =1 σ ( u i ) ξ ( u i ) = 0 , so some of the coordinates of the potential singularities equal to 0 and someto a half-period. Let I = { i ∈ [ n ] : σ ( u i ) = 0 } , J = { j ∈ [ n ] : ξ ( u i ) = 0 } , I ∩ J = ∅ , then the corresponding S ( ξ I ( u ) + ξ J ( u )) = 0, which means that the subset S ( u, a ) = 0 is non-singular. Now the claim follows from Sard’s Lemma,saying that the set of critical values of the natural projection π : M → C N ( a )has measure zero. (cid:3) Discussion: Milnor-Hirzebruch problem
The Milnor-Hirzebruch problem was first posed by Hirzebruch in his ICM-1958 talk [11]. Its algebraic version can be formulated in our notations asfollows:
Which sets of π ( n ) integers c ω can be realised as the Chern numbers c ω ( M n ) of some smooth irreducible complex algebraic variety? In this version it still remains largely open, although some restrictions areknown since the work of Milnor and Hirzebruch. In particular, in (complex)dimension n = 1 , , n = 1 : c ≡ ,n = 2 : c + c ≡ ,n = 3 : c c ≡ , c ≡ c ≡ .n = 4 : − c + c c +3 c +4 c c − c ≡ , c c +2 c ≡ , − c + c c ≡ . Since the total Chern class of Θ n satisfies the relation (10) with D n = ( n +1)!all the characteristic numbers of Θ n equal ± ( n + 1)! . Remark 6.1.
It is interesting to note that Chern numbers c ω of the nor-mal bundle ν Θ n is delta-like function of ω (see (14),(15)) while the Chernnumbers of the tangent bundle T Θ n are equidistributed. The analogy withthe Fourier transform might be worthy to explore. n particular, for n = 1 we have c = − n = 2 : c = c = 6 ,n = 3 : c = − c c = c = − ,n = 4 : c = c c = c c = c = c = 120 . We see that the first Hirzebruch congruence is sharp for the theta divisors,which means that it cannot be improved in the algebraic setting. This isrelated to the fact that the Todd genus of the theta divisor
T d (Θ n ) = ( − n , which follows from formula (5) (see also [4]).Recall that the divisibility conditions in terms of characteristic classes in K -theory were described by Hattori [10] and Stong [20]. Applying to formula(4) the Landweber-Novikov operations S ω and taking the Todd genus we getall divisibility conditions on the Chern numbers c νω ( M n ) of U -manifolds ina different, more effective way. It is natural to ask if they can be improvedfor irreducible algebraic varieties.We are planning to discuss this problem in a separate publication.7. Acknowledgements
We are very grateful to S.P. Novikov for his interest and encouragement,and to I. Cheltsov, S. Grushevsky, A. Prendergast-Smith and Yu. Prokhorovfor useful discussion of algebro-geometric aspects of this work.
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Steklov Mathematical Institute and Moscow State University, Russia
E-mail address : [email protected] Department of Mathematical Sciences, Loughborough University, Lough-borough LE11 3TU, UK, Moscow State University and Steklov MathematicalInstitute, Russia