Twisted Kodaira–Spencer Classes and the Inequality of Severi †

Abstract

Let X be a smooth minimal complex surface of general type subject to the inequality KX2<4χ(OX). The old problem of Severi states that the image of the Albanese map of such a surface is at most a curve. The problem was completely settled by Rita Pardini using the works of Xiao on fibered surfaces. The paper suggests a different approach to Severi type inequalities through the study of the cohomology group H1(ΘX(−KX)). For X irregular, with no irrational pencil, it is shown that the cohomology classes of the group carry geometric data: P(H1(ΘX(−KX)))∋[ξ]↦(Fξ,σξ,Aξ) where Fξ is a rank 2 bundle, σξ:Fξ⟶ΩX is a modification of ΩX along an effective, nonzero divisor Eξ inducing the isomorphism on the level of global sections: H0(σξ):H0(Fξ)≅H0(ΩX);Aξ is at most 0-dimensional subscheme of Eξ. The paper establishes many geometrical properties of Eξ and Aξ. As a consequence, for an irregular surface X without irrational pencil, sufficient conditions are given for a Severi type inequality 2KX2≥c2+q−2, where (KX2,c2) and q=h0(ΩX) are the Chern numbers and the irregularity of X, respectively. The dependence of the geometric data (Fξ,σξ,Aξ) on [ξ] is explored: it is shown that there is a finite collection of projective subspaces {P(WE)} of P(H1(ΘX(−KX))), labeled by effective nonzero divisors E; the data are constant on the complement of the collection and undergoes a change on each P(WE). One attaches a quiver to this collection which is an interesting object to study.