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Foundations of Computational Mathematics
Erschienen in: Foundations of Computational Mathematics 5/2012

01.10.2012

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

verfasst von: Peter Scheiblechner

Erschienen in: Foundations of Computational Mathematics | Ausgabe 5/2012

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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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Literatur
1.
D. Arapura, Frobenius amplitude and strong vanishing theorems for vector bundles, Duke Math. J. 121(2), 231–267 (2004). With an appendix by Dennis S. Keeler. MathSciNetMATHCrossRef
2.
M.F. Atiyah, R. Bott, L. Gȧrding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math. 131, 145–206 (1973). MathSciNetMATHCrossRef
3.
S. Basu, Computing the first few Betti numbers of semi-algebraic sets in single exponential time, J. Symb. Comput. 41(10), 1125–1154 (2006). MATHCrossRef
4.
S. Basu, Algorithmic semi-algebraic geometry and topology—recent progress and open problems, in Surveys on Discrete and Computational Geometry. Contemp. Math., vol. 453 (Am. Math. Soc., Providence, 2008), pp. 139–212. CrossRef
5.
D. Bayer, D. Mumford, What can be computed in algebraic geometry, in Computational Algebraic Geometry and Commutative Algebra, Cortona, 1991. Sympos. Math., vol. XXXIV (Cambridge Univ. Press, Cambridge, 1993), pp. 1–48.
6.
7.
S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, Inf. Process. Lett. 18(3), 147–150 (1984). MathSciNetMATHCrossRef
8.
A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Am. Math. Soc. 4(3), 587–602 (1991). MathSciNetMATHCrossRef
9.
N. Bourbaki, Elements of Mathematics. Algebra, Part I: Chapters 1–3 (Hermann, Paris, 1974). Translated from the French.
10.
11.
P. Bürgisser, F. Cucker, Variations by complexity theorists on three themes of Euler, Bézout, Betti, and Poincaré, in Complexity of Computations and Proofs, ed. by J. Krajíček. Quaderni di Matematica [Mathematics Series], vol. 13 (Department of Mathematics, Seconda Università di Napoli, Caserta, 2004), pp. 73–152.
12.
P. Bürgisser, P. Scheiblechner, On the complexity of counting components of algebraic varieties, J. Symb. Comput. 44(9), 1114–1136 (2009). MATHCrossRef
13.
P. Bürgisser, P. Scheiblechner, Counting irreducible components of complex algebraic varieties, Comput. Complex. 19(1), 1–35 (2010). MATHCrossRef
14.
L. Caniglia, A. Galligo, J. Heintz, Equations for the projective closure and effective Nullstellensatz, Discrete Appl. Math. 33(1–3), 11–23 (1991). MathSciNetMATHCrossRef
15.
P. Deligne, Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970). MATH
16.
P. Deligne, A. Dimca, Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. Ec. Norm. Super. 23(4), 645–656 (1990). MathSciNetMATH
17.
A. Dimca, Singularities and Topology of Hypersurfaces. Universitext (Springer, Berlin, 1992). MATHCrossRef
18.
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995). MATH
19.
20.
D. Eisenbud, D. Grayson, M. Stillman, B. Sturmfels (eds.), Computations in Algebraic Geometry with Macaulay 2. Algorithms and Computations in Mathematics, vol. 8 (Springer, Berlin, 2001).
21.
D. Eisenbud, G. Fløystad, F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Trans. Am. Math. Soc. 355(11), 4397–4426 (2003) (electronic). MATHCrossRef
22.
N. Fitchas, A. Galligo, Nullstellensatz effectif et conjecture de Serre (théorème de Quillen-Suslin) pour le calcul formel, Math. Nachr. 149, 231–253 (1990). MathSciNetMATHCrossRef
23.
N. Fitchas, A. Galligo, J. Morgenstern, Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields, J. Pure Appl. Algebra 67, 1–14 (1990). MathSciNetMATHCrossRef
24.
A. Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble) 29(2), 107–184 (1979). MathSciNetMATHCrossRef
25.
D. Giaimo, On the Castelnuovo–Mumford regularity of connected curves, Trans. Am. Math. Soc. 358(1), 267–284 (2006) (electronic). MathSciNetMATHCrossRef
26.
M. Giusti, Some effectivity problems in polynomial ideal theory, in EUROSAM’84: Proceedings of the International Symposium on Symbolic and Algebraic Computation, London, UK (Springer, Berlin, 1984), pp. 159–171.
27.
M. Giusti, J. Heintz, Algorithmes – disons rapides – pour la décomposition d’une variété algébrique en composantes irréductibles et équidimensionnelles, in Effective Methods in Algebraic Geometry (Proceedings of MEGA’90), ed. by T.C. Mora Traverso. Progress in Math., vol. 94 (Birkhäuser, New York, 1991), pp. 169–193. CrossRef
28.
29.
P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978). MATH
30.
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. IHES 39, 93–103 (1966).
31.
L. Gruson, R. Lazarsfeld, C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72(3), 491–506 (1983). MathSciNetMATHCrossRef
32.
R. Hartshorne, Algebraic Geometry (Springer, New York, 1977). MATH
33.
34.
J. Kollár, Sharp effective Nullstellensatz, J. Am. Math. Soc. 1(4), 963–975 (1988). MATHCrossRef
35.
S. Kwak, Generic projections, the equations defining projective varieties and Castelnuovo regularity, Math. Z. 234(3), 413–434 (2000). MathSciNetMATHCrossRef
36.
37.
38.
R. Lazarsfeld, Positivity in Algebraic Geometry. I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48 (Springer, Berlin, 2004). Classical setting: line bundles and linear series. CrossRef
39.
40.
E.W. Mayr, A.R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46(3), 305–329 (1982). MathSciNetMATHCrossRef
41.
J. McCleary, A User’s Guide to Spectral Sequences. Mathematics Lecture Series, vol. 12 (Springer, Berlin, 1985).
42.
K. Mulmuley, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field, Combinatorica 7(1), 101–104 (1987). MathSciNetMATHCrossRef
43.
D. Mumford, Lectures on Curves on an Algebraic Surface. Annals of Mathematics Studies, vol. 59 (Princeton University Press, Princeton, 1966). With a section by G.M. Bergman. MATH
44.
D. Mumford, Varieties defined by quadratic equations, in Questions on Algebraic Varieties, C.I.M.E., III Ciclo, Varenna, 1969 (Edizioni Cremonese, Rome, 1970), pp. 29–100.
45.
D. Mumford, Algebraic Geometry I: Complex Projective Varieties. Grundlehren der mathematischen Wissenschaften, vol. 221 (Springer, Berlin, 1976). MATH
46.
T. Oaku, N. Takayama, An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation, J. Pure Appl. Algebra 139, 201–233 (1999). MathSciNetMATHCrossRef
47.
48.
49.
Z. Ran, Local differential geometry and generic projections of threefolds, J. Differ. Geom. 32(1), 131–137 (1990). MathSciNetMATH
50.
P. Scheiblechner, On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety, J. Complex. 23(3), 359–379 (2007). MathSciNetMATHCrossRef
51.
P. Scheiblechner, On a generalization of Stickelberger’s theorem, J. Symb. Comput. 45(12), 1459–1470 (2010). MEGA’2009. MathSciNetMATHCrossRef
52.
P. Scheiblechner, On the complexity of counting irreducible components and computing Betti numbers of complex algebraic varieties. Ph.D. Thesis, 2007.
53.
J.P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6, 1–42 (1955–1956). CrossRef
54.
G. Smith, Computing global extension modules, J. Symb. Comput. 29, 729–746 (2000). MATHCrossRef
55.
J. Stückrad, W. Vogel, Castelnuovo’s regularity and multiplicity, Math. Ann. 281(3), 355–368 (1998). CrossRef
56.
Á. Szántó, Complexity of the Wu-Ritt decomposition, in PASCO’97: Proceedings of the Second International Symposium on Parallel Symbolic Computation (ACM Press, New York, 1997), pp. 139–149. CrossRef
57.
Á. Szántó, Computation with polynomial systems. Ph.D. Thesis, 1999.
58.
W.V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 2 (Springer, Berlin, 1998). CrossRef
59.
C. Voisin, Hodge Theory and Complex Algebraic Geometry. I. Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002). Translated from the French original by Leila Schneps. MATHCrossRef
60.
J. von zur Gathen, Parallel arithmetic computations: a survey, in MFOCS86. LNCS, vol. 233 (1986), pp. 93–112 SV.
61.
U. Walther, Algorithmic computation of de Rham cohomology of complements of complex affine varieties, J. Symb. Comput. 29(4–5), 795–839 (2000). MathSciNetMATHCrossRef
62.
U. Walther, Algorithmic determination of the rational cohomology of complex varieties via differential forms, in Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, South Hadley, MA, 2000. Contemp. Math., vol. 286 (Am. Math. Soc., Providence, 2001), pp. 185–206. CrossRef
Metadaten
Titel
Castelnuovo–Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties
verfasst von
Peter Scheiblechner
Publikationsdatum
01.10.2012
Verlag
Springer-Verlag
Erschienen in
Foundations of Computational Mathematics / Ausgabe 5/2012
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-012-9123-y