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

Ruppenthal, Jean

doi:10.1093/imrn/rnx097

Cited by 6 publications

(2 citation statements)

References 31 publications

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“…By [R6,Theorem 1.2], the kernel of the∂ s -operator on Dom∂ s ⊆ L 2 (Z) is exactly O(Z), the ring of weakly holomorphic functions on Z. Thus, if ϕ ∈ Dom∂ s , we would thus get that ϕ ∈ O(Z) sincē ∂ϕ = 0.…”

Section: Introductionmentioning

confidence: 99%

Koppelman formulas on the A1-singularity

Lärkäng

Ruppenthal

2016

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

Koppelman formulas on the A1-singularity

Lärkäng

Ruppenthal

2016

Journal of Mathematical Analysis and Applications

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“…To the best of our knowledge, the only known cases of Theorem 1.1 for general surfaces with canonical singularities are the following: Part (i) for p = 2 was proven in [26,Corollary 1.3]. Part (ii) for p = 2 and (0, 2)-forms was proven in [17,Theorem 4.3], which builds on the vanishing result from [28].…”

Section: Introductionmentioning

confidence: 97%

Estimates for the $${\bar{\partial }}$$-Equation on Canonical Surfaces

et al. 2019

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We study the solvability in L p of the∂-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case p = 2 for two natural closed extensions∂ s and∂ w of∂. For∂ s we have solvability, whereas for∂ w there is solvability if and only if a certain boundary condition ( * ) is fulfilled at the singularity. Our main tool is certain integral operators for solving∂ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.

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On Analytic Todd Classes of Singular Varieties

Bei

Piazza

2019

International Mathematics Research Notices

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Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

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

Cited by 6 publications

References 31 publications

Koppelman formulas on the A1-singularity

Koppelman formulas on the A1-singularity

Estimates for the $${\bar{\partial }}$$-Equation on Canonical Surfaces

On Analytic Todd Classes of Singular Varieties

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