Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

Mehdi Lejmi; Markus Upmeier

doi:10.2748/tmj.20191025

Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

Mehdi Lejmi, Markus Upmeier^* (Corresponding Author)

^*Corresponding author for this work

Mathematical Science

City University of New York

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

Abstract

We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

Original language	English
Pages (from-to)	581-594
Number of pages	14
Journal	Tohoku Mathematical Journal
Volume	72
Issue number	4
Early online date	22 Dec 2020
DOIs	https://doi.org/10.2748/tmj.20191025
Publication status	Published - 31 Dec 2020

Bibliographical note

The first author was supported in part by a PSC-CUNY award #60053-00 48, jointly funded by The Professional Staff Congress and The City University of New York. The second author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –UP 85/2-1, UP 85/3-1.

Keywords

Holomorphic sectional curvature
almost-Kähler geometry
canonical Hermitian connection

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title = "Closed almost K{\"a}hler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are K{\"a}hler",

abstract = "We show that a closed almost K{\"a}hler 4-manifold of pointwise constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically K{\"a}hler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.",

keywords = "Holomorphic sectional curvature, almost-K{\"a}hler geometry, canonical Hermitian connection",

author = "Mehdi Lejmi and Markus Upmeier",

note = "The first author was supported in part by a PSC-CUNY award #60053-00 48, jointly funded by The Professional Staff Congress and The City University of New York. The second author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –UP 85/2-1, UP 85/3-1.",

year = "2020",

month = dec,

day = "31",

doi = "10.2748/tmj.20191025",

language = "English",

volume = "72",

pages = "581--594",

journal = "Tohoku Mathematical Journal",

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T1 - Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

AU - Lejmi, Mehdi

AU - Upmeier, Markus

N1 - The first author was supported in part by a PSC-CUNY award #60053-00 48, jointly funded by The Professional Staff Congress and The City University of New York. The second author was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –UP 85/2-1, UP 85/3-1.

PY - 2020/12/31

Y1 - 2020/12/31

N2 - We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

AB - We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature k≥0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k<0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

KW - Holomorphic sectional curvature

KW - almost-Kähler geometry

KW - canonical Hermitian connection

U2 - 10.2748/tmj.20191025

DO - 10.2748/tmj.20191025

M3 - Article

SN - 0040-8735

VL - 72

SP - 581

EP - 594

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

IS - 4

ER -

Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

Abstract

Bibliographical note

Keywords

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