Szegő kernels and Poincaré series

Zhiqin Lu; Steve Zelditch

doi:10.1007/s11854-016-0033-9

Szegő kernels and Poincaré series

Zhiqin Lu^*, Steve Zelditch

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Let M= M˜ / Γ be a Kähler manifold, where Γ ~ π₁(M) and M˜ is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L² Szegő projector Π˜ _N for L²-holomorphic sections on the lifted bundle L˜ ^N is related to the Szegő projector for H⁰(M, L^N) by Π^ _N(x, y) = ∑ _γ _∈ _ΓΠ^ ˜ _N(γ⋅ x, y). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of M˜ with respect to L˜ ^N and to surjectivity of Poincaré series.

Original language	English (US)
Pages (from-to)	167-184
Number of pages	18
Journal	Journal d'Analyse Mathematique
Volume	130
Issue number	1
DOIs	https://doi.org/10.1007/s11854-016-0033-9
State	Published - Nov 1 2016

ASJC Scopus subject areas

Analysis
General Mathematics

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abstract = "Let M= M˜ / Γ be a K{\"a}hler manifold, where Γ ~ π1(M) and M˜ is the universal K{\"a}hler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szeg{\H o} projector Π˜ N for L2-holomorphic sections on the lifted bundle L˜ N is related to the Szeg{\H o} projector for H0(M, LN) by Π^ N(x, y) = ∑ γ ∈ ΓΠ^ ˜ N(γ⋅ x, y). We then apply this result to give a simple proof of Napier{\textquoteright}s theorem on the holomorphic convexity of M˜ with respect to L˜ N and to surjectivity of Poincar{\'e} series.",

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N2 - Let M= M˜ / Γ be a Kähler manifold, where Γ ~ π1(M) and M˜ is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szegő projector Π˜ N for L2-holomorphic sections on the lifted bundle L˜ N is related to the Szegő projector for H0(M, LN) by Π^ N(x, y) = ∑ γ ∈ ΓΠ^ ˜ N(γ⋅ x, y). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of M˜ with respect to L˜ N and to surjectivity of Poincaré series.

AB - Let M= M˜ / Γ be a Kähler manifold, where Γ ~ π1(M) and M˜ is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L2 Szegő projector Π˜ N for L2-holomorphic sections on the lifted bundle L˜ N is related to the Szegő projector for H0(M, LN) by Π^ N(x, y) = ∑ γ ∈ ΓΠ^ ˜ N(γ⋅ x, y). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of M˜ with respect to L˜ N and to surjectivity of Poincaré series.

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Szegő kernels and Poincaré series

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