Szego kernels, Toeplitz operators, and equivariant fixed point formulae

TL;DR

This paper investigates the asymptotic behavior of traces of automorphisms and Toeplitz operators on sections of line bundles over complex projective manifolds, extending fixed point formulas in the presence of group actions.

Contribution

It extends the Lefschetz fixed point formula to include Toeplitz operators and equivariant settings, providing asymptotic expansions and leading term computations.

Findings

Asymptotic expansion of traces as tensor power grows large

Leading term of the trace expansion computed explicitly

Extension of fixed point formulas to equivariant Toeplitz operators

Abstract

Let $\gamma$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $\gamma$ induces an automorphism $\gamma_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\gg 0$, the Lefschetz fixed point formula expresses the trace of $\gamma_k$ in terms of fixed point data. More generally, one may consider the composition of $\gamma_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,\gamma)$, one may consider induced linear maps on the equivariant summands associated to the irreducible representations of $G$. In this paper, under familiar assumptions in the theory of symplectic reductions, we show that the traces of these maps admit an asymptotic expansion as $k\to +\infty$, and compute its leading term.