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Tommaso de Fernex's Abstract"Relative motivic integration, constructible functions, and stringy Chern classes"Previous lectures have covered the extension of certain classical invariants (such as Euler characteristic, Betti numbers and Hodge numbers) to a large class of singular varieties in a --let me say-- birationally well-behaved way. In these lectures, I will explain how motivic integration, if performed in the relative setting, can actually capture invariants such as Chern classes. This will lead to the notion of stringy Chern classes. I will start with reviewing the definition of Chern classes of complex manifolds and their extensions to singular varieties proposed by Mather and Schwartz-MacPherson. In particular, I will discuss the work of MacPherson in which he constructs a natural transformation from the functor of constructible functions to homology (or Chow groups), proving a conjecture of Deligne and Grothendieck. It is using this transformation that MacPherson proposes his theory of Chern classes for singular varieties. Then I will present a recent result of Aluffi, where it is proven that Chern classes of smooth varieties behave well under certain birational modifications, and show with explicit examples that this nice birational property is lost in the singular case if we consider Schwartz-MacPherson classes. This will be our motivation for what comes next. Aiming for a theory of Chern classes that is birationally well-behaved (i.e., with a "stringy" flavor), I will define motivic integration over a base and explain how one can extract a constructible function (defined over the base) from any given relative motivic integral. Using these functions and MacPherson transformation, we will finally be able to define "stringy Chern classes". Explicit examples will be presented to compare these classes with Schwartz-MacPherson class. Using formal properties of motivic integration, we will investigate the main properties of these classes. I will close my lectures with a discussion of stringy invariants for quotient varieties: this is a beautiful part of the story, related to a classic problem known as the "McKay correspondence". The general principle is that the stringy invariants of the quotient variety, which are defined through resolution of singularities, are already encoded (in some way) in the equivariant geometry of the manifold of which we are taking the quotient. This not only is an amazing phenomenon per se, but also provides explicit formulas to compute these invariants. It can arguably be said that stringy invariants made their first appearance in this context. |
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