The $$\bar \partial $$ -operator on algebraic curves

Brüning, Jochen; Peyerimhoff, Norbert; Schröder, Herbert

doi:10.1007/bf02097104

Cited by 19 publications

(26 citation statements)

References 4 publications

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“…as k → ∞. Finally let e −t∆ F ∂ : L 2 (reg(X), h) → L 2 (reg(X), h) (12) be the heat operator associated to (9). Then (12) is a trace class operator and its trace satisfies the following estimates e −at Tr(e −tb∆ F ∂ ) ≤ Tr(e −t∆ F ) ≤ Tr(e −t∆ F ∂ ) (13) and, for each 0 < t ≤ 1, Tr(e −t∆ F ∂ ) ≤ r vol h (reg(X))t −v (14) where r is some positive constant.…”

Section: Introductionmentioning

confidence: 99%

On the Laplace–Beltrami operator on compact complex spaces

Bei

2019

Trans. Amer. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 99%

On the Laplace–Beltrami operator on compact complex spaces

Bei

2019

Trans. Amer. Math. Soc.

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“…Note that from the point of view of the analytic proof the contribution of the singular points of X to the Morse inequalities is caused by the fact that dom(∆ 1 ) = dom(∆ t,1 ) and thus is related to the small eigenvalues of the "transversal Laplacian" (i.e., the Laplacian on the link of the singularity). Recall that the small eigenvalues of the transversal Laplacian play an important role for L 2 -methods in the presence of singularities, namely for the lack of essential selfadjointness of ∆ |Ω0 and in the study of index theorems for regular singular operators (see [9] for the general case and [19], [8] for the case of a singular algebraic curve).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ludwig¹

2011

Ann. inst. Fourier

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“…In this case, Theorem 1.1 is much easier to prove. Questions of analysis have been treated in great detail by Nagase [27], Brüning and Lesch [11], [10] and Brüning, Peyerimhoff, and Schröder [12].…”

Section: Theorem 14 (Gauss-bonnet Theoremmentioning

confidence: 99%

Local geometry of singular real analytic surfaces

Grieser

2002

Trans. Amer. Math. Soc.

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Abstract. Let V ⊂ R N be a compact real analytic surface with isolated singularities, and assume its smooth part V 0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R N . We prove:(1) Each point of V has a neighborhood which is quasi-isometric (naturally and "almost isometrically") to a union of metric cones and horns, glued at their tips. (2) A full asymptotic expansion, for any p ∈ V , of the length of V ∩ {q : dist (q, p) = r} as r → 0. (3) A Gauss-Bonnet Theorem, saying that each singular point contributes 1 − l/(2π), where l is the coefficient of the linear term in the expansion of (2). (4) The L 2 Stokes Theorem, selfadjointness and discreteness of the LaplaceBeltrami operator on V 0 , an estimate on the heat kernel, and a GaussBonnet Theorem for the L 2 Euler characteristic. As a central tool we use resolution of singularities.

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The $$\bar \partial $$ -operator on algebraic curves

Cited by 19 publications

References 4 publications

On the Laplace–Beltrami operator on compact complex spaces

On the Laplace–Beltrami operator on compact complex spaces

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Local geometry of singular real analytic surfaces

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