Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Abstract

Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension m. In this paper we are interested in the Dolbeault operator acting on the space of L 2 sections of the canonical bundle of reg(X), the regular part of X. More precisely let dm,0 : L 2 Ω m,0 (reg(X), h) → L 2 Ω m,1 (reg(X), h) be an arbitrarily fixed closed extension of ∂m,0 : L 2 Ω m,0 (reg(X), h) → L 2 Ω m,1 (reg(X), h) where the domain of the latter operator is Ω m,0 c (reg(X)). We establish various properties such as closed range of dm,0, compactness of the inclusion D(dm,0) → L 2 Ω m,0 (reg(X), h) where D(dm,0), the domain of dm,0, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian d * m,0 • dm,0 with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to d * m,0 • dm,0, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces. IntroductionConsider a complex projective variety V ⊂ CP n . The regular part of V , reg(V ), comes equipped with a natural Kähler metric g, which is the one induced by the Fubini-Study metric of CP n . In particular, whenever V has a nonempty singular set, we get an incomplete Kähler manifold of finite volume. In the seminal papers [10] and [22], given a singular projective variety V , many questions with a rich interaction of topology and analysis, for instance intersection cohomology, L 2 -cohomology and Hodge theory, have been raised for the incomplete Kähler manifold (reg(V ), g). Some of the most important among them are the Cheeger-Goresky-MacPherson's conjecture and the MacPherson's conjecture. The former, which is still open, says that the maximal L 2 -de Rham cohomology groups of (reg(V ), g) are isomorphic to the middle perversity intersection cohomology groups of V while the latter, proved in [29], asks whether the L 2 -∂-cohomology groups in bidegree (0, q) of (reg(V ), g) are 1 arXiv:1607.00289v3 [math.DG] 17 Feb 2018 * E,m,0 )}, has discrete spectrum.We point out explicitly that in the previous theorem the assumption concerning the parabolicity of (A h , g| A h ) does not depend on the particular Hermitian metric g that we fix on M . Indeed if g is another Hermitian metric on M then, since g and g are quasi-isometric on M , we have that (A h , g| A h ) is parabolic if and only if (A h , g | A h ) is parabolic. Consider again the setting of Theorem 0.1 and let ∂ t E,m,0 be the formal adjoint of ∂ E,m,0 with respect to g. Let ∆ ∂,E,m,0 : Ω m,0 (M, E) → Ω m,0 (M, E), ∆ ∂,E,m,0 = ∂ t E,m,0 • ∂ E,m,0 be the Hodge-Kodaira Laplacian in bidegree (m, 0). Since M is compact and ∆ ∂,E,m,0 is elliptic and formally self-adjoint we have that ∆ ∂,E,m,0 , acting on L 2 Ω m,0 (M, E, g) with domain Ω m,0 (M, E), is essentially self-adjoint. With * m,0 )}, has ...

On the Laplace–Beltrami operator on compact complex spaces

“…0.2 had been already proved in [20] and [28] in the setting of projective varieties with isolated singularities and complex dimension 2 and 3 respectively. Concerning (2) we prove similar results that we can summarize in the next two theorems: Theorem 0.3. Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension v. There exist positive constants a and b such that for each f ∈ C ∞ c (reg(X)) we have df 2 L 2 Ω 1 (reg(X),h) ≤ a f 2 L 2 (reg(X),h) + b ∂f 2 L 2 Ω 0,1 (reg(X),h) .…”

“…where both the operators above are viewed as unbounded, densely defined and closable operators acting on L 2 (reg(X), h). Our main results show the existence of a self-adjoint extension with discrete spectrum for both (1) and (2) with an estimate for the growth of the corresponding eigenvalues. Let us go into some more details by describing how the paper is sort out.…”

“…In this section we extend to the setting of compact and irreducible Hermitian complex spaces of complex dimension 2 some of the results proved in [2] in the case of complex projective surfaces endowed with the Fubini-Study metric. Since many of the following results follow by the same strategy used in [2] we will omit the details in the proofs and we will refer directly to the corresponding results in [2]. For each q = 0, 1, 2 we have the following properties:…”

Von Neumann dimension, Hodge index theorem and geometric applications

“…M ). As b(2) 2 (M ) = 2h 2,0(2) (M ) + h 1,1 (2) (M ) and 2h 2,0 (2) (M ) − h 1,1 (2) (M ) = 0 we can conclude that b(2) 2 (M ) > 0 and therefore χ(M ) > 0 as desired. Finally, according to Th.…”

“…The first aim of this note is to reformulate the above two theorems by replacing the terms on the right hand sides of (1) and (2) with the corresponding L 2 -versions. More precisely one of the main result of this paper can be summarized in this way:…”