Upper bounds for regularized determinants

Abstract

Let $E$ be a holomorphic vector bundle on a compact Kähler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its regularized determinant $de{t}^{\prime }(\Delta )$. We conjectured elsewhere that, when $h$ varies, this determinant $de{t}^{\prime }(\Delta )$ remains bounded from above. In this paper we prove this in two special cases. The first case is when $X$ is a Riemann surface, $E$ is a line bundle and $dim(H^0(X,E))+dim(H^1(X,E))\le 2$, and the second case is when $X$ is the projective line, $E$ is a line bundle, and all metrics under consideration are invariant under rotation around a fixed axis.

22 pages, plain TeX

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