An introduction to motivic integration

Abstract

By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of singularities Y of X, the Hodge numbers of Y do not depend upon the choice of the resolution. In this article we provide an elementary introduction to the theory of motivic integration, leading to a proof of the result described above. We calculate the motivic integral of several quotient singularities and discuss these calculations in the context of the cohomological McKay correspondence.

32 pages, 1 figure. Stringy E-function redefined and examples given in more detail. Also we present a proof of the cohomological McKay correspondence for a finite Abelian subgroup of SL(n,C) to illustrate the simplicity of Batyrev's approach in this case

References in corpus (3)

• Motivic Igusa zeta functions
• Stringy Hodge numbers of varieties with Gorenstein canonical singularities
• Motivic integration, quotient singularities and the McKay correspondence

Cited by in corpus (30)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Jet Schemes of Locally Complete Intersection Canonical Singularities
• Twisted jets, motivic measure and orbifold cohomology
• Jet schemes, log discrepancies and Inversion of Adjunction
• Modification systems and integration in their Chow groups
• The $p$-cyclic McKay correspondence via motivic integration
• Jet schemes, arc spaces and the Nash problem
• Jet schemes and invariant theory
• Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial
• Celestial integration, stringy invariants, and Chern-Schwartz-MacPherson classes
• Stringy Chern classes of singular varieties
• Configurations in abelian categories. IV. Invariants and changing stability conditions
• Chern classes of proalgebraic varieties and motivic measures
• Dimensions of jet schemes of log singularities
• Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties
• Virtual Betti numbers of real algebraic varieties
• Motivic integration and the birational invariance of BCOV invariants
• Integration with respect to Euler characteristic over the projectivization of the space of functions and the Alexander polynomial of a plane curve singularity
• Standard monomials and invariant theory for arc spaces III: special linear group
• Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups
• Multiplicity of jet schemes of monomial schemes
• On the algebraic stringy Euler number
• Constructible functions on Artin stacks
• Singularities of Pairs via Jet Schemes
• Zeta functions and 'Kontsevich invariants' on singular varieties
• What is Motivic Measure?
• The global McKay-Ruan correspondence via motivic integration
• Vertex Algebras and Commutative Algebras
• The motivic zeta function and its smallest poles
• The stringy E function of the moduli space of rank 2 bundles over a Riemann surface of genus 3