L^2-Riemann-Roch for singular complex curves

Ruppenthal, Jean; Sera, Martin

doi:10.5427/jsing.2015.11d

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Cited by 3 publications

(2 citation statements)

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“…Note that any compact one dimensional complex space is projective and thus algebraic by [17,Satz 2,p. 343] (see also [27,Theorem 6.2]). We consider the following setting:…”

Section: Introductionmentioning

confidence: 99%

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Coman¹,

Marinescu²

2022

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Note that any compact one dimensional complex space is projective and thus algebraic by [17,Satz 2,p. 343] (see also [27,Theorem 6.2]). We consider the following setting:…”

Section: Introductionmentioning

confidence: 99%

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Coman¹,

Marinescu²

2022

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

View full text Add to dashboard Cite

show abstract

“…Note that any compact one dimensional complex space is projective and thus algebraic by [G,Satz 2,p. 343] (see also [RS,Theorem 6.2]). We consider the following setting:…”

Section: Introductionmentioning

confidence: 99%

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Coman¹,

Marinescu²

2020

Preprint

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We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces. CONTENTS 1. Introduction 1 2. Preliminaries 4 3. Bergman kernels and Fubini-Study measures 6 4. Proof of Theorems 1.1 and 1.2 11 5. Examples 17 References 20

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$L^2$ -Serre Duality on Singular Complex Spaces and Rational Singularities

Ruppenthal

2017

International Mathematics Research Notices

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In the present paper, we devise a version of topological L 2 -Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new L 2 -vanishing theorems for the ∂-equation on singular spaces. It is shown that complex spaces with rational singularities behave quite tame with respect to the ∂-equation in the L 2 -sense. More precisely: a singular point is rational if and only if the L 2 -∂ s -complex is exact in this point. So, we obtain an L 2 -∂-resolution of the structure sheaf in rational singular points.

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L^2-Riemann-Roch for singular complex curves

Abstract: Abstract. We present a comprehensive L 2 -theory for the ∂-operator on singular complex curves, including L 2 -versions of the Riemann-Roch theorem and some applications.

Cited by 3 publications

References 14 publications

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

Equidistribution for weakly holomorphic sections of line bundles on algebraic curves

$L^2$ -Serre Duality on Singular Complex Spaces and Rational Singularities

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