Unobstructedness of deformations of weak Fano manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1310.14019
author: Taro Sano
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.0705
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 6365886
zbMATH Keywords: Fano manifold
zbMATH Keywords: weak Fano manifold
zbMATH Keywords: deformation theory
zbMATH Keywords: unobstructed deformations
MaRDI profile type: Publication
OpenAlex ID: W1982221672
arXiv ID: 1302.0705
DOI: 10.1093/IMRN/RNT116
published in: IMRN. International Mathematics Research Notices
title: Unobstructedness of deformations of weak Fano manifolds
review text: Let \(X\) be a projective manifold defined over an algebraically closed field of characteristic zero. We say that \(X\) has unobstructed deformations if the Kuranishi space of \(X\) is smooth. Basic deformation theory shows that the Kuranishi space is smooth if the cohomology group \(H^2(X, T_X)\) vanishes. In particular if \(X\) is a Fano manifold, i.e. the anticanonical divisor \(-K_X\) is ample, then \(H^2(X, T_X)\) vanishes by the Nakano vanishing theorem and \(X\) has unobstructed deformations. If \(X\) is a weak Fano manifold, i.e. the anticanonical divisor \(-K_X\) is nef and big, it is in general not true that \(H^2(X, T_X)=0\). However, the author proves in this paper that weak Fano manifolds have unobstructed deformations. In fact the result is more general: let \(X\) be a projective manifold such that \(H^1(X, \mathcal O_X)=0\). If for some \(m \in \mathbb N\) there exists a smooth divisor \(D \in | - m K_X |\) such that \(H^1(D, N_{D/X})=0\), then \(X\) has unobstructed deformations. By the basepoint free theorem this immediately implies the main statement. Apart from the surprisingly short proof the paper also contains some interesting examples of (weak) Fano manifolds showing the optimality of the results.
reviewed by: Andreas Höring
description: scientific article; zbMATH DE number 6365886