Stabilization of the cohomology of thickenings

TL;DR

This paper proves that the cohomology of vector bundles on the formal completion of a projective variety along a subvariety stabilizes for high thickenings, and extends Kodaira vanishing to these thickenings using positivity and vanishing theorems.

Contribution

It establishes cohomology stabilization and Kodaira vanishing for all sufficiently high thickenings of a local complete intersection in projective space.

Findings

Cohomology on high thickenings can be computed from any sufficiently high thickening.

Kodaira vanishing extends to all high thickenings of the subvariety.

Positivity of the normal bundle is key to the stabilization results.

Abstract

For a local complete intersection subvariety $X=V({\mathcal I})$ in ${\mathbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of ${\mathbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t=V({\mathcal I^t})$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira-Akizuki-Nakano vanishing theorem for $X$, formulated in terms of the cotangent complex.