Upper bounds for regularized determinants

TL;DR

This paper investigates upper bounds for the regularized determinants of Laplace operators on holomorphic vector bundles over compact Kähler manifolds, proving boundedness in specific cases related to Riemann surfaces and the projective line.

Contribution

It proves the boundedness of the regularized determinant of Laplace operators in two special cases, advancing understanding of spectral invariants in complex geometry.

Findings

Determinant remains bounded on Riemann surfaces with low-dimensional cohomology.

Determinant is bounded for rotationally invariant metrics on the projective line.

Supports the conjecture that determinants are generally bounded when varying metrics.

Abstract

Let $E$ be a holomorphic vector bundle on a compact K\"ahler 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 $det'(\Delta)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(\Delta)$ 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 $dim(H^0 (X,E)) + dim(H^1 (X,E)) \leq 2$, 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.