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abstract: 'Consider a compact K[ä]{}hler manifold endowed with a prequantum bundle. Following the geometric quantization scheme, the associated quantum spaces are the spaces of holomorphic sections of the tensor powers of the prequantum bundle. In this paper we construct an asymptotic representation of the prequantum bundle automorphism group in these quantum spaces. We estimate the characters of these representations under some transversality assumption. The formula obtained generalizes in some sense the Lefschetz fixed point formula for the automorphisms of the prequantum bundle preserving its holomorphic structure. Our results will be applied in two forthcoming papers to the quantum representation of the mapping class group.'
author:
- 'Laurent CHARLES [^1]'
bibliography:
- 'biblio.bib'
title: A Lefschetz fixed point formula for symplectomorphisms
---

Consider a compact K[ä]{}hler manifold $M$ endowed with a Hermitian holomorphic bundle $L \rightarrow M$ whose curvature is the fundamental two-form. In the point of view of geometric quantization, $M$ is the classical phase space and the space $H^0 (M, L)$ of holomorphic sections of $L$ is the quantum space.

The group of holomorphic automorphisms of $L$ acts naturally on the quantum space. Furthermore, if the higher cohomology groups ($H^q(M,
L),$ $q \geqslant 1$) of the sheaf of 