Holomorphic Morse Inequalities and Bergman Kernels

Semiclassical analysis, geometric representation and quantum ergodicity

“…The Berezin-Toeplitz quantization. In this subsection, we describe the Berezin-Toeplitz quantization introduced by Berezin [4] and Boutet de Monvel-Guillemin [15], and further developed by Bordemann-Meinrenken-Schlichenmaier [11], Schlichenmaier [37] and Ma-Marinescu [29].…”

“…By the proof of [29,Theorem 7.4.1], we have the following product formula for the Berezin-Toeplitz quantization with an estimation for the remainder.…”

The asymptotics of the holomorphic analytic torsion forms

“…As explained above and in Section 3, we will use the formalism of Toeplitz operators to prove this theorem. We now recall the definition given in [33,Def. 7…”

“…In [18], Bismut and Vasserot computed the asymptotics of the analytic torsion associated with increasing powers of a positive line bundle, using the heat kernel method of [5] (see also [33,Sect. 5.5]).…”

“…In particular, we cannot hope to have a limiting operator for the rescaled operator, nor limiting coefficients in the development of the heat kernel, and in all our proofs, we have to make uniform estimates on spaces that change. To overcome these issues, we will draw inspiration from [16,17] and use the formalism of Toeplitz operators of [33]. The idea is to use the operator norm on matrices to have uniform boundedness properties for Toeplitz operators, and to replace the convergence to limiting objects by an approximation by objects with Toeplitz coefficients.…”