Derived Kodaira Spencer map, Cosection lemma, and semiregularity

TL;DR

This paper extends the cosection lemma to derived settings without perfect obstruction theories and offers a concise proof of semiregularity using derived Kodaira Spencer maps.

Contribution

It introduces a derived version of the cosection lemma and applies it to prove semiregularity under broader conditions than previously possible.

Findings

Derived cosection lemma holds without perfect obstruction theory.

Short proof of Kodaira's semiregularity principle.

Application of derived Kodaira Spencer map in deformation theory.

Abstract

The cosection lemma proved by J. Li and Y.H. Kiem said the intrinsic normal cone lies inside the kernel of any cosection of the obstruction sheaf when the moduli has a perfect obstruction theory. With a definition of higher tangent vectors of a scheme at a point, and a construction of the derived Kodaira Spencer map by K. Behrend and B. Fantechi, we prove a derived version of cosection lemma without perfect obstruction theory condition. As an application we give a short proof of the Kodaira's Principle \textit{ambient cohomology annihilates obstruction} (semiregularity), assuming the existence of locall universal family.