On the Laplace–Beltrami operator on compact complex spaces

Abstract

Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension v > 1. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension 2.

Symplectic manifolds, L-cohomology and q-parabolicity

“…Then, according to [3] Prop. 4.7, we know that (A, g| A ) is q-parabolic for any q ∈ [1,2]. Hence we can conclude that Th.…”

“…As recalled above a Kähler space (X, h) is a Hermitian space such that ω, the fundamental form associated to h, satisfies dω = 0. Compact Kähler spaces (and more generally compact Hermitian spaces) have finite volume and are q-parabolic for any q ∈ [1,2], see [3] and [31]. Therefore we can conclude that Th.…”

“…Clearly the latter is the cohomology of the L 2 -maximal Dolbeault complex whose definition is omitted since it is completely analogous to that of the L 2 -maximal de Rham complex. We refer to [1] for more details. Proposition 3.3.…”

“…[35]. For various reasons, besides the usual de Rham cohomology, in the non-compact setting it is also interesting to consider the L p -de Rham cohomology which, roughly speaking, is defined as the quotient between L p -closed forms modulo L p -exact forms 1 . Thus, looking at (1), it is natural to wonder whether something similar holds true also for the L p -cohomology of a non-compact almost Kähler manifold.…”

“…For various reasons, besides the usual de Rham cohomology, in the non-compact setting it is also interesting to consider the L p -de Rham cohomology which, roughly speaking, is defined as the quotient between L p -closed forms modulo L p -exact forms 1 . Thus, looking at (1), it is natural to wonder whether something similar holds true also for the L p -cohomology of a non-compact almost Kähler manifold. Certainly we cannot expect that a generalization of (1) holds true for an arbitrary non-compact almost Kähler manifold without any further assumption on (M, ω, J, g) or p. First, as ω is parallel with respect to some Hermitian connection, it is necessary to assume that vol g (M ) < ∞ in order to have ω k ∈ L p Ω 2k (M, g) and p ∈ [1, ∞).…”