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 theor...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 12; no. 5; pp. 541 - 571
Main Author Scheiblechner, Peter
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.10.2012
Springer
Springer Nature B.V
Subjects
Algebra
Algorithms
Applications of Mathematics
Computation
Computational efficiency
Computational mathematics
Computer Science
Computing time
Economics
Foundations
Homology theory (Mathematics)
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Poles
Regularity
Smoothing (Statistics)
Unions
Variational principles
Germany
68Q25
68W30
de Rham cohomology
14Q20
Parallel polynomial time
Smooth projective variety
Cech cohomology
Hypercohomology
Algorithm
Complexity
Betti numbers
Castelnuovo–Mumford regularity
14F40
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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 ∖ V is described by differential forms with poles along V of single exponential order. By covering X with sets of this type and using a Čech process, we obtain 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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-012-9123-y