An introduction to motivic integration

TL;DR

This paper introduces motivic integration, illustrating its use in algebraic geometry to relate singularities, resolutions, and Hodge numbers, and demonstrates calculations for quotient singularities within the cohomological McKay correspondence.

Contribution

It provides an elementary introduction to motivic integration and applies it to prove invariance of Hodge numbers under crepant resolutions, with explicit calculations for quotient singularities.

Findings

Motivic integrals can be explicitly calculated for certain quotient singularities.

Hodge numbers of crepant resolutions are independent of the resolution choice.

Connections between motivic integration and the cohomological McKay correspondence are discussed.

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.