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

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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∖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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Acknowledgements

The author is very grateful to Saugata Basu for being his host, for his many important and interesting discussions, and for recommending the book [38]. Without him this work would not have been possible. The author also thanks Manoj Kummini for fruitful discussions about the Castelnuovo–Mumford regularity, Christian Schnell for a discussion on the cohomology of hypersurfaces, Nicolas Perrin and Martí Lahoz for useful discussions about exterior powers of sheaves, and the anonymous referees for valuable comments and suggestions.

Research partially supported by DFG grant SCHE 1639/1-1.

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    Peter Scheiblechner

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Communicated by Elizabeth Mansfield.

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Scheiblechner, P. Castelnuovo–Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties. Found Comput Math 12, 541–571 (2012). https://doi.org/10.1007/s10208-012-9123-y

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