A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes

Motivic bivariant characteristic classes

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \xrightarrow {\op {id}_X} X])$ of the isomorphism class of the identity $X \xrightarrow {id_X} X$ is the Poincaré dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It "unifies" the well-known three characteristic cl…

A general construction of partial Grothendieck transformations

Fulton and MacPherson introduced the notion of bivariant theories and Grothendieck transformations related to Riemann-Roch-theorems. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation is only partially known: characteristic classes of singular spaces (e.g. Stiefel-Whitney or Chern classes), cohomology operations (e.g. singular Adams Riemann-Roch and Steenrod operations for Chow groups) or equivariant theories (e.g. Lefschetz Riemann-Roch). We introduce in this paper a simpler notion of partial (weak) bivariant theories and partial Grothe…

Chern classes of graph hypersurfaces and deletion-contraction

We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove that both these conditions hold when the edge is multiple in the graph. This leads to recursions for the Chern classes of graph hypersurfaces for graphs obtained by adding parallel edges to a given (regular) edge. Analogous results for the case of Grothendieck classes of graph hypersurfaces were obtained in previous work. Both Grothendieck classes and Chern c…

Motivic Bivariant Characteristic Classes

Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation in the algebraic context for k of characteristic zero and in the compact complex analytic context. It unifies the well-known three characteristic class transformations of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a…

On the Chern classes of singular complete intersections

We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz-MacPherson and Fulton-Johnson classes, $c^{SM}(X)$ and $c^{FJ}(X)$. Their difference (up to sign) is the total Milnor class ${\mathcal M}(X)$, a generalization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes $c^{SM}(X)$ and $c^{FJ}(X)$, and use these to prove a surprisingly simple formula for the total Milnor class when $X$ is defined by a finite number of local complete inter…

热带曲面上中等位置热带曲线的补集

The complement of tropical curves in moderate position on tropical surfaces

本文针对紧致热带流形上的热带 Cartier 除数，引入中等位置概念，提出 Riemann-Roch 数等于补集拓扑欧拉示性数的猜想，并在二维且具有 Delzant 面结构的情形下给出证明与推广。

Characteristic classes of singular toric varieties

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …

Chern class identities from tadpole matching in type IIB and F-theory

In light of Sen's weak coupling limit of F-theory as a type IIB orientifold, the compatibility of the tadpole conditions leads to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. We present the physical argument leading to the identity, and a mathematical derivation of a Chern class identity which confirms it, after taking into account singularities of the relevant loci. This identity of Chern classes holds in arbitrary dimension, and for varieties that are not necessarily Calabi-Yau. Singularities are ess…

Chern classes of blow-ups

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann-Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of the well-known formula for the normal bundle of a proper transform. We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the …

Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors

Let $X$ be a nonsingular complex projective variety and $D$ a locally quasi-homogeneous free divisor in $X$. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on $X$ with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement of $D$ in $X$. Our result confirms a conjectural formula for these classes, at least after push-forward to projective space; it proves the full form of the conjecture for locally quasi-homogeneous free divisors in $\mathbb P^n$. The result generalizes several previously known results. For example…

Hirzebruch classes and motivic Chern classes for singular (complex) algebraic varieties

In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…

Hirzebruch classes and motivic Chern classes for singular spaces

In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation …

从Schubert胞的motivic Chern类到它们的Hirzebruch类和CSM类

From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes

本文证明了旗流形中Schubert胞的motivic Chern类的多个性质与猜想，利用motivic Chern变换定义了等变Hirzebruch变换，并借助Demazure-Lusztig递归给出了Hirzebruch类及相关对偶的递归公式。