Constant scalar curvature Kähler metrics on fibred complex surfaces
type: Item
instance of: scholarly article
title: Constant scalar curvature Kähler metrics on fibred complex surfaces
zbMATH Open document ID: 1085.53064
author: Joel Fine
published in: Journal of Differential Geometry
publication date: 2005-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0401275
review text: The main purpose of the paper is to prove the existence of constant scalar curvature Kähler metrics on certain complex surfaces. More precisely, the author proves the following theorem: ``If \(X\) is a compact connected complex surfaces admitting a holomorphic submersion onto a high genus complex curve with fibres of genus at least two, then, for all large \(r\), the Kähler class \(k_r\) contains a constant scalar curvature metric.'' The main technique used in the proof is an adiabatic limit. First, by fitting together the constant scalar curvature Kähler metrics on the fibres of \(X\) and a large multiple of a metric on the base, a family of approximate solutions is constructed. Then the analysis is developed necessary to show the existence of a genuine solution in the same cohomology class. Finally, the above theorem is extended to certain higher dimensional fibrations.
reviewed by: Constantin Călin
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 2207987
zbMATH Keywords: holomorphic submersion
zbMATH Keywords: adiabatic limit
Wikidata QID: Q115198777
MaRDI profile type: Publication
OpenAlex ID: W1519449463
arXiv ID: math/0401275
DOI: 10.4310/JDG/1115669591
description: scientific article