Let (M, J) be a compact complex manifold of complex dimension m and let gs be a one-parameter family of Hermitian forms on M that are smooth and positive definite for each fixed s ∈ (0, 1] and that somehow degenerates to a Hermitian pseudometric h for s tending to 0. In this paper under rather general assumptions on gs we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians ∆∂,m,0,s associated to gs and acting on the canonical bundle of M. In particular we show that, as s tends to zero, the eigenvalues, the heat operators and the heat kernels corresponding to the family ∆∂,m,0,s converge to the eigenvalues, the heat operator and the heat kernel of ∆∂,m,0,abs, a suitable self-adjoint operator with entirely discrete spectrum defined on the limit space (A, h|A).

Degenerating Hermitian metrics, canonical bundle and spectral convergence / Bei, F.. - In: COMMUNICATIONS IN ANALYSIS AND GEOMETRY. - ISSN 1019-8385. - 32:1(2024), pp. 65-117. [10.4310/CAG.240905214047]

Degenerating Hermitian metrics, canonical bundle and spectral convergence

Bei F.
2024

Abstract

Let (M, J) be a compact complex manifold of complex dimension m and let gs be a one-parameter family of Hermitian forms on M that are smooth and positive definite for each fixed s ∈ (0, 1] and that somehow degenerates to a Hermitian pseudometric h for s tending to 0. In this paper under rather general assumptions on gs we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians ∆∂,m,0,s associated to gs and acting on the canonical bundle of M. In particular we show that, as s tends to zero, the eigenvalues, the heat operators and the heat kernels corresponding to the family ∆∂,m,0,s converge to the eigenvalues, the heat operator and the heat kernel of ∆∂,m,0,abs, a suitable self-adjoint operator with entirely discrete spectrum defined on the limit space (A, h|A).
2024
hermitian pseudometric; hodge-kodaira laplacian; canonical bundle; spectral convergence; mosco convergence
01 Pubblicazione su rivista::01a Articolo in rivista
Degenerating Hermitian metrics, canonical bundle and spectral convergence / Bei, F.. - In: COMMUNICATIONS IN ANALYSIS AND GEOMETRY. - ISSN 1019-8385. - 32:1(2024), pp. 65-117. [10.4310/CAG.240905214047]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1722911
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