2009/05/13 by Peter Scheiblechner, Scheiblechner, Peter
Computer Science · Mathematics · #14Q15 #14Q20 #68W30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.SC #math.AG #msc:14Q15 #msc:14Q20 #msc:68W30
paper · pdf · doi:10.48550/arxiv.0905.2212
32 pages - filled a gap in Section 4.2, specific example added, minor improvements
openalex publication_date 2009/05/13 · arxiv created 2011/12/12 · arxiv updated 2011/12/13 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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∖ 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.