Todd Class Research Articles

Riemann--Roch theorem for operations in cohomology of algebraic varieties

The Riemann-Roch theorem for multiplicative operations in oriented cohomology theories for algebraic varieties is proved and an explicit formula for the corresponding Todd classes is given. The formula obtained can also be applied in the topological situation, and the theorem can be regarded as a change-of-variables formula for the integration of cohomology classes. The classical Riemann-Roch theorem (1), stated and proved by Hirzebruch, calculates the Euler characteristic of a vector bundle on a smooth projective algebraic variety X/C in terms of its rank and Chern classes. This theorem states that χ(E )c oincides with the 2nth component of the characteristic class ch(E )t d(X) under the identification H 2n (X(C), Q )= Q ,w heren =d imX ,c h(E) is the Chern character of E ,a nd td(X) is the Todd class of the tangent bundle TX . By the splitting principle, the calculation of the Todd classes reduces to the case of a line bundle L ,w here td(L) is obtained by substituting z = c1(L )i n the series (1) td(z )= z

Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces

This article can be seen as a sequel to the first author's article “Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane,” where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of “universality” of certain formulas about the cohomology of the Hilbert schemes by defining a universal algebra of creation operators. This helps us to reformulate and extend a lot of the first author's previous results in a very precise manner. Secondly, we are able to extend the previously found results by showing how to calculate any characteristic class of the Hilbert scheme of points on the affine plane in terms of the creation operators. In particular, we have included the calculation of the total Segre class and the square root of the Todd class. Using this methods, we have also found a way to calculate any characteristic class of any tautological sheaf on the Hilbert scheme of points on the affine plane. This in fact gives another complete description of the ring structure of the cohomology spaces of the Hilbert schemes of points on the affine plane.

The equivariant Todd genus of a complete toric variety, with Danilov condition

We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torus which satisfy Danilov's requirement.

Classes de Hirzebruch et classes de Chern motiviques

Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, T y is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Higher Todd Classes and Holomorphic Group Actions

This paper attempts to provide an analogue of the Novikov conjecture for algebraic (or Kahler) manifolds. Inter alia, we prove a conjecture of Rosenberg’s on the birational invariance of higher Todd genera. We argue that in the algebraic geometric setting the Novikov philosophy naturally includes non-birational mappings.

Nonabelian localization in equivariant [formula omitted]-theory and Riemann–Roch for quotients

We prove a localization formula in equivariant algebraic K-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to nondiagonalizable groups the localization formulas of Nielsen [Diagonalizably linearized coherent sheaves, Bull. Soc. Math. France 102 (1974) 85–97] and Thomason [Une formule de Lefschetz en K-théorie équivariante algébrique, Duke Math. J. 68(3) (1992) 447–462]. As an application we give a Riemann–Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of Toen [Théorèmes de Riemann–Roch pour les champs de Deligne-Mumford, K-Theory 18(1) (1999) 33–76] and the authors [Riemann–Roch for quotients and Todd classes of simplicial toric varieties, Comm. Algebra 31 (2003) 3735–3752].

Cycles representing the Todd class of a toric variety

In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number μ ( σ ) \mu (\sigma ) to each rational polyhedral cone σ \sigma in the lattice, such that for any toric variety X X with fan Σ \Sigma in the lattice, we have \[ Td ⁡ ( X ) = ∑ σ ∈ Σ μ ( σ ) [ V ( σ ) ] . \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )]. \] This constitutes an improved answer to an old question of Danilov. In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.

Order-preserving maps from a poset to a chain, the order polytope, and the Todd class of the associated toric variety

We give a new formula for the number of order-preserving maps from a finite poset Q to a finite chain, which sums a contribution from each face of the order polytope of Q. The formula we derive could also be obtained if we knew that a certain cycle represented the Todd class of the toric variety associated to Q. We conjecture that, for toric varieties arising from posets, this cycle does represent the Todd class; we prove this conjecture in the case where Q is a chain.

Algebraic unimodular counting

We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Grobner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function.

Riemann-Roch for Quotients and Todd Classes of Simplicial Toric Varieties#

In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne (Brion M. and Vergne M. ([1997]). An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math.482:67–92) using different techniques. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Riemann–Roch Theorems via Deformation Quantization, I

We deduce the Riemann–Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann–Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.

Riemann–Roch Theorems via Deformation Quantization, II

Abstract We deduce the Riemann–Roch type formula expressing the microlocal Euler class of a perfect complex of D -modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann–Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.

Chern classes of automorphic vector bundles

1.1. Suppose X is a compact n-dimensional complex manifold. Each partition I = {i1, i2, . . . , ir} of n corresponds to a Chern number c(X) = (ci1(X)∪ci2(X)∪. . .∪cir(X)∩[X]) ∈ Z where c(X) ∈ H(X;Z) are the Chern classes of the tangent bundle, [X] ∈ H2n(X;Z) is the fundamental class, and : H0(X;Z) → Z is the augmentation. Many invariants of X (such as its complex cobordism class) may be expressed in terms of its Chern numbers ([Mi], [St]). During the last 25 years, characteristic classes of singular spaces have been defined in a variety of contexts: Whitney classes of Euler spaces [Su], [H-T], [Ak], Todd classes of singular varieties [BFM], Chern classes of singular algebraic varieties [Mac], L-classes of stratified spaces with even codimension strata [GM1], Wu classes of singular spaces [Go2], [GP] (to name a few). However, these characteristic classes are invariably homology classes and as such, they cannot be multiplied with each other. In some cases it has been found possible to “lift” these classes from homology to intersection homology, where (some) characteristic numbers may be formed ([BBF], [BW], [Go2],[GP], [T]). The case of locally symmetric spaces is particularly interesting. Suppose Γ is a torsionfree arithmetic group acting on a complex n-dimensional Hermitian symmetric domain D = G/K, where G is the group of real points of a semisimple algebraic group G defined over Q with Γ ⊂ G(Q), and where K ⊂ G is a maximal compact subgroup. Then X = Γ\D is a Hermitian locally symmetric space. When X is compact, Hirzebruch’s proportionality theorem [Hr1] says that there is a number v(Γ) ∈ Q so that for every partition I = {i1, i2, . . . , ir} of n, the Chern number satisfies c(X) = v(Γ)c(Ď), where Ď = Gu/K is the compact dual symmetric space (and Gu is a compact real form of G(C) containing K). If X = Γ\D is noncompact, it has a canonical Baily-Borel (Satake) compactification, X. This is a (usually highly singular) complex projective algebraic variety. To formulate a proportionality theorem in the noncompact case, one might hope that the tangent bundle TX extends as a complex vector bundle over X, but this is false. In [Mu1], D. Mumford showed that TX has a particular extension EΣ → XΣ over any toroidal resolution τ : XΣ → X of the Baily-Borel compactification and that for any partition I of n, the resulting Chern numbers c(EΣ) = (c (EΣ) ∪ c(EΣ) ∪ . . . ∪ c(EΣ) ∩ [XΣ]) (1.1.1)

Todd Classes of Affine Cones of Grassmannians

A ring A which is a homomorphic image of a regular local ring S is said to be a Roberts ring if τA/S([A]) = [Spec A]dim A, where τA/S is the Riemann-Roch map for Spec A. Such rings satisfy a vanishing theorem for intersection multiplicities, as was proved by P. Roberts. It is known that complete intersections are Roberts rings, and the first author showed that a determinantal ring is a Roberts ring precisely if it is a complete intersection. Let Ad(n) denote the affine cone of the Grassmann variety Gd(n) under the Plucker embedding. In this paper, we determine precisely when Ad(n) is a Roberts ring.

Values of zeta functions at negative integers, Dedekind sums and toric geometry

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.

Perturbed Dolbeault operators and the homology Todd class

This paper discusses the role played by perturbed Dolbeault operators in relating the coherent sheaf and elliptic operator perspectives on the K K homology of projective varieties. Among the consequences are index formulas for perturbed Dolbeault operators.

Intersection multiplicities over Gorenstein rings

Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality \(\lim_{n\to\infty}\frac{\ell(F^n_R(M))}{p^{nd}} = \ell(M)\) holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for which this equality fails to hold. This then provides an example of a nonzero Todd class \(\tau_3(R)\), and of a bounded free complex whose local Chern character does not vanish on this class.

Barvinok's algorithm and the Todd class of a toric variety

In this paper we prove that the Todd class of a simplicial toric variety has a canonical expression as a power series in the torus-invariant divisors. Given a resolution of singularities corresponding to a nonsingular subdivision of the fan, we give an explicit formula for this power series which yields the Todd class. The computational feasibility of this procedure is implied by the additional fact that the above formula is compatible with Barvinok decompositions (virtual subdivisions) of the cones in the fan. In particular, this gives an algorithm for determining the coefficients of the Todd class in polynomial time for fixed dimension. We use this to give a polynomial-time algorithm for computing the number of lattice points in a simple lattice polytope of fixed dimension, a result first achieved by Barvinok.

Localization and the quantization conjecture

Let ( M, ω) be a compact symplectic manifold with a Hamiltonian action of a compact Lie group K. Suppose that 0 is a regular value of the moment map μ: M → Lie(K) ∗ , so that the Marsden-Weinstein reduction M red = μ −(0) K is a symplectic orbifold. In our earlier paper ( Quart. J. Math., 47, 1996) we proved a formula (the residue formula) for η 0 e ω0 [ M red ] for any η 0 ϵ H ∗(M red) , where ω 0 is the induced symplectic form on M red . This formula is given in terms of the restrictions of classes in the equivariant cohomology H ∗ T(M) of M to the components of the fixed point set of a maximal torus T in M. In this paper, we consider a line bundle L on Mfor which c 1( L) = ω . If M is given a K-invariant complex structure compatible with ω we may apply the residue formula when η 0 is the Todd class of M red to obtain a formula for the Riemann-Roch number RR( L red) of the induced line bundle L red on M red when K acts freely on μ −1(0). More generally when 0 is a regular value of μ, so that M red is an orbifold and L red is an orbifold bundle, Kawasaki's Riemann-Roch theorem for orbifolds can be applied, in combination with the residue formula. Using the holomorphic Lefschetz formula we similarly obtain a formula for the K-invariant Riemann-Roch number RR K( L) of L . We show that the formulae obtained for RR( L red) and RR K( L) are almost identical and in many circumstances (including when K is a torus) are the same. Thus in these circumstances a special case of the residue formula is equivalent to the conjecture of Guillemin and Sternberg ( Invent. Math. 67 (1982), 515–538) (proved in various degrees of generality by Guillemin and Sternberg themselves and others including Sjamaar, Guillemin, Vergne and Meinrenken) that RR( L red) = RR K( L) .

Intersection theory on toric varieties

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullen's polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.