[Paper Review] Stringy Chern classes of singular varieties

Tommaso de Fernex, Ernesto Lupercio+2 more|ArXiv.org|Jul 18, 2004

Homotopy and Cohomology in Algebraic Topology25 references9 citations

TL;DR

This paper introduces stringy Chern classes for singular complex varieties using motivic integration and MacPherson's transformation, providing a birationally invariant class that generalizes Chern classes to singular spaces. The key result is an explicit formula for the stringy Chern class of a Gorenstein quotient variety $X = M/G$ in terms of Chern-Schwartz-MacPherson classes of fixed-point sets $M^g/C(g)$, establishing a direct link to orbifold invariants.

ABSTRACT

Motivic integration and MacPherson's transformation are combined in this paper to construct a theory of "stringy" Chern classes for singular varieties. These classes enjoy strong birational invariance properties, and their definition encodes data coming from resolution of singularities. The singularities allowed in the theory are those typical of the minimal model program; examples are given by quotients of manifolds by finite groups. For the latter an explicit formula is proven, assuming that the canonical line bundle of the manifold descends to the quotient. This gives an expression of the stringy Chern class of the quotient in terms of Chern-Schwartz-MacPherson classes of the fixed-point set data.

Motivation & Objective

• To define a birationally invariant Chern class for singular complex varieties with log-terminal singularities.
• To extend the Chern-Schwartz-MacPherson class to singular spaces by incorporating resolution data via motivic integration.
• To provide an explicit formula for the stringy Chern class of a Gorenstein quotient of a manifold by a finite group.
• To establish a connection between stringy Chern classes and orbifold invariants such as the orbifold Euler characteristic.

Proposed method

• Construct a rational-valued constructible function $\Phi_X$ using motivic integration over the arc space of a resolution $f: Y \to X$, weighted by the order of the relative canonical $\mathbb{Q}$-divisor $K_{Y/X}$.
• Define the stringy Chern class as $c_{\text{str}}(X) := c(\Phi_X)$, where $c$ is MacPherson's natural transformation from constructible functions to Chow homology.
• Use the change-of-variables formula in motivic integration to ensure birational invariance of $\Phi_X$ and hence of $c_{\text{str}}(X)$.
• For quotient varieties $X = M/G$, express $\Phi_X$ as a sum over conjugacy classes of $G$, using the orbifold pushforward of the constant function on the stack $[M/G]$.
• Apply the MacPherson transformation to derive the formula $\widetilde{c}_{\text{str}}(X) = \sum_{g \in \mathcal{C}(G)} (\pi_g)_* c_{\text{SM}}(M^g / C(g))$, where $C(g)$ is the centralizer of $g$.
• Verify that the stringy Euler characteristic $e_{\text{str}}(X)$ matches the orbifold Euler characteristic $e(M,G) = \sum_{g \in \mathcal{C}(G)} e_c(M^g / C(g))$ via degree maps.

Experimental results

Research questions

• RQ1 Can a birationally invariant Chern class be defined for singular varieties with log-terminal singularities using motivic integration and MacPherson's transformation?
• RQ2 How does the stringy Chern class of a quotient variety $X = M/G$ relate to the fixed-point data of the group action on $M$?
• RQ3 Does the stringy Chern class recover known orbifold invariants such as the orbifold Euler characteristic?
• RQ4 Is the stringy Chern class independent of the choice of resolution, and how is this ensured by motivic integration?
• RQ5 Can the stringy Chern class be expressed as a sum of pushforwards of Chern-Schwartz-MacPherson classes of quotient fixed-point sets?

Key findings

• The stringy Chern class $c_{\text{str}}(X)$ satisfies the desired birational invariance: if $X$ and $X'$ are $K$-equivalent, then $c_{\text{str}}(X) = f_* C = f'_* C$ for some class $C$ on a common resolution.
• For a Gorenstein quotient $X = M/G$ with the canonical bundle descending, the stringy Chern class is given by $\widetilde{c}_{\text{str}}(X) = \sum_{g \in \mathcal{C}(G)} (\pi_g)_* c_{\text{SM}}(M^g / C(g))$, where $\mathcal{C}(G)$ is the set of conjugacy classes of $G$.
• The function $\Phi_X$ is independent of the choice of resolution due to the change-of-variables formula in motivic integration.
• The stringy Chern class agrees with the orbifold pushforward of the constant function on $[M/G]$, as defined by Joyce.
• The stringy Euler characteristic $e_{\text{str}}(X)$ equals the orbifold Euler characteristic $e(M,G) = \sum_{g \in \mathcal{C}(G)} e_c(M^g / C(g))$, confirming consistency with Batyrev's result.
• The construction provides a direct, manifestly birational invariant definition of a Chern class for singular varieties, extending the classical Chern-Schwartz-MacPherson class.