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.

Paoletti, R. (2008). Szegö kernels, Toeplitz operators, and equivariant fixed point formulae. JOURNAL D'ANALYSE MATHEMATIQUE, 106(1), 209-236 [10.1007/s11854-008-0048-y].

Szegö kernels, Toeplitz operators, and equivariant fixed point formulae

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.
Scheda breve Scheda completa Scheda completa (DC)
Articolo in rivista - Articolo scientifico
Szego kernel, Toeplitz operator, trace,fixed point formula
English
2008
209
236
Paoletti, R. (2008). Szegö kernels, Toeplitz operators, and equivariant fixed point formulae. JOURNAL D'ANALYSE MATHEMATIQUE, 106(1), 209-236 [10.1007/s11854-008-0048-y].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/5353
• 9
• 10