Lecture series: Hirzebruch-Riemann-Roch as a categorical trace.

Professor Dennis Gaitsgory (Harvard University)

Abstract

Let \(X\) be a smooth proper scheme over a field of characteristic 0, and let \(E\) be a vector bundle on \(X\). The classical Hirzebruch-Riemann-Roch says that the Euler characteristic of the cohomology \(H^*(X,E)\) equals \(\int_X \text{ch}(E) \; \text{Td}(X)\). Thus, HRR is an equality of numbers, i.e., elements of a set. In these talks, we will explain a proof of HRR that uses the hierarchy \[\{2-\text{categories}\} \rightarrow \{1-\text{categories}\} \rightarrow \{\text{Vector spaces}\} \rightarrow \{\text{Numbers}\}.\] I.e., the origin of HRR will be 2-categorical. The procedure by which we go down from 2-categories to numbers is that of *categorical trace*.

However, in order to carry out our program, we will need to venture into the world of higher categories: the 2-category we will be working with consists of DG-categories, the latter being higher categorical objects. And the process of calculation of the categorical trace will involve derived algebraic geometry: the key geometric player will be the self-intersection of the diagonal of \(X\), a.k.a. the inertia (derived) scheme of \(X\).

So, this series of talks can be regarded as providing a motivation for studying higher category theory and derived algebraic geometry: we will use them in order to prove an equality of numbers. That said, we will try to make these talks self-contained, and so some necessary background will be supplied.

Video recordings are available at https://www.mpim-bonn.mpg.de/node/7032