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

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 th...

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Bibliographic Details
Published in:arXiv.org 2011-12
Main Author: Scheiblechner, Peter
Format: Article
Language:English
Subjects:
Algorithms
Computing time
Graph algorithms
Homology
Hyperplanes
Polynomials
Regularity
Traveling salesman problem
Online Access:Get full text
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Summary: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.
ISSN:2331-8422