Castelnuovo-Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

TL;DR

This paper presents a parallel polynomial time algorithm for computing the Betti numbers of smooth complex projective varieties, leveraging bounds on Castelnuovo-Mumford regularity to efficiently compute de Rham cohomology.

Contribution

It introduces the first single exponential time algorithm for Betti numbers of a broad class of complex varieties and bounds Castelnuovo-Mumford regularity for differential sheaves.

Findings

Parallel polynomial time algorithm for Betti numbers

Bound on Castelnuovo-Mumford regularity of differential sheaves

Efficient computation of de Rham cohomology

Abstract

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety $X$. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential $p$-forms on $X$ is bounded by $p(em+1)D$, where $e$, $m$, and $D$ are the maximal codimension, dimension, and degree, respectively, of all irreducible components of $X$. It follows that, for a union $V$ of generic hyperplane sections in $X$, the algebraic de Rham cohomology of $X\setminus V$ is described by differential forms with poles along $V$ of single exponential order. This yields a similar description of the de Rham cohomology of $X$, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth.