Unobstructedness of deformations of weak Fano manifolds
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.
scientific article; zbMATH DE number 6365886