Intersection bundles and projective structures

2023/03/28 火 11:00 - 12:00
Gerard Freixas i Montplet
Ecole Polytechnique / CNRS

Intersection bundles have been introduced and studied by Deligne, Elkik and others, as part of a programme aimed to refining arithmetic intersection theory in a relative setting. The most popular instance of intersection bundles are Deligne pairings: for a family of algebraic varieties of dimension n, together with n+1 line bundles, it produces a line bundle on the parameter space. This line bundle represents the families intersection product of the Chern classes of the line bundles. One can then mimic the construction of characteristic classes and produce more general intersection bundles. All these constructions can be enhanced with hermitian structures, which results in a variant of the theory of arithmetic characteristic classes of Gillet-Soulé. Sometimes, though, one encounters vector bundles naturally equipped with non-unitary connections, rather than metrics. The prototypical example is the universal vector bundle over a moduli space of flat vector bundles on a compact Riemann surface. It is thus natural to explore how to incorporate such structures into the picture. In this talk, I will discuss this issue in the framework of families of compact Riemann surfaces, and for the intersection bundle which represents the fiber integral of the second Chern class. In joint work with D. Eriksson and R. Wentworth, we propose a framework in the form of a functorial theory of complex Chern-Simons bundles in the relative setting. I will present the main lines of our theory, and explain how it provides a means of classifying universal projective structures on compact Riemann surfaces.