The purpose of this paper is to calculate the asymptotics of the Ray-Singer analytic torsion associated with the $p$-th symmetric power of a holomorphic Hermitian positive vector bundle when $p$ tends to $+\infty$. We thus extend our previous results on positive line bundles.

A new object is introduced - the "Fischer bundle". It is, formally speaking, an Hermitean bundle of infinite rank over a bounded symmetric domain whose fibers are Hilbert spaces whose elements can be realized as entire analytic functions square integrable with respect to a Gaussian measure ("Fischer spaces"). The definition was inspired by our previous work on the "Fock bundle". An even more general framework is indicated, which allows one to look upon the two concepts from a unified point of view....

We study the evolution of pluri-anticanonical line bundles $K_M^{-\nu }$ along the Kähler Ricci flow on a Fano manifold $M$. Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of $M$. For example, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface satisfying $c_1^2\left(M\right)=1$ or $c_1^2\left(M\right)=3$. Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups....