Vanishing theorems for Dolbeault cohomology of log homogeneous varieties
Michel Brion
Source: Tohoku Math. J. (2) Volume 61, Number 3
(2009), 365-392.
Abstract
We consider a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections. We obtain an optimal vanishing theorem for logarithmic Dolbeault cohomology of nef line bundles in that setting. This implies a vanishing theorem for ordinary Dolbeault cohomology which generalizes results of Broer for flag varieties, and of Mavlyutov for toric varieties.
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Permanent link to this document: http://projecteuclid.org/euclid.tmj/1255700200
Digital Object Identifier: doi:10.2748/tmj/1255700200
Zentralblatt MATH identifier: 05650401
Mathematical Reviews number (MathSciNet): MR2568260
References
V. Alexeev, Complete moduli in the presence of semiabelian group actions, Ann. of Math. (2) 155 (2002), 611--708.
Mathematical Reviews (MathSciNet): MR1923963
Zentralblatt MATH: 1052.14017
Digital Object Identifier: doi:10.2307/3062130
JSTOR: links.jstor.org
V. V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293--338.
Mathematical Reviews (MathSciNet): MR1290195
Zentralblatt MATH: 0851.14021
Digital Object Identifier: doi:10.1215/S0012-7094-94-07509-1
Project Euclid: euclid.dmj/1077287614
A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480--497.
Mathematical Reviews (MathSciNet): MR366940
Digital Object Identifier: doi:10.2307/1970915
JSTOR: links.jstor.org
F. Bien and M. Brion, Automorphisms and local rigidity of regular varieties, Compositio Math. 104 (1996), 1--26.
Mathematical Reviews (MathSciNet): MR1420707
Zentralblatt MATH: 0910.14004
C. Birkenhake and H. Lange, Complex abelian varieties, Second edition, Grundlehren Math. Wiss., 302, Springer-Verlag, Berlin, 2004.
Mathematical Reviews (MathSciNet): MR2062673
M. Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), 369--404.
Mathematical Reviews (MathSciNet): MR1248677
Digital Object Identifier: doi:10.1215/S0012-7094-93-07213-4
Project Euclid: euclid.dmj/1077289424
M. Brion, Log homogeneous varieties, Actas del XVI Coloquio Latinoamericano de Álgebra, 1--39, Biblioteca de la Revista Matemática Iberoamericana, Madrid, 2007.
A. Broer, Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 1--20.
Mathematical Reviews (MathSciNet): MR1223221
Zentralblatt MATH: 0807.14043
Digital Object Identifier: doi:10.1007/BF01244299
A. Broer, A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles, J. Reine Angew. Math. 493 (1997), 153--169.
Mathematical Reviews (MathSciNet): MR1491811
J. B. Carrell and D. I. Lieberman, Holomorphic vector fields and compact Kaehler manifolds, Invent. Math. 21 (1973), 303--309.
Mathematical Reviews (MathSciNet): MR326010
Zentralblatt MATH: 0253.32017
Digital Object Identifier: doi:10.1007/BF01418791
V. I. Danilov and A. G. Khovanskiǐ, Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 925--945.
Mathematical Reviews (MathSciNet): MR873655
C. de Concini and C. Procesi, Complete symmetric varieties, Invariant Theory (Montecatini, 1982), 1--44, Lecture Notes in Math. 996, Springer-Verlag, Berlin, 1983.
Mathematical Reviews (MathSciNet): MR718125
P. Deligne, Théorie de Hodge II, Inst. Hautes Études Pub. Math. 40 (1971), 5--57.
Mathematical Reviews (MathSciNet): MR498551
Digital Object Identifier: doi:10.1007/BF02684692
P. Deligne, Théorie de Hodge III, Inst. Hautes Études Pub. Math. 44 (1974), 5--78.
Mathematical Reviews (MathSciNet): MR498552
Digital Object Identifier: doi:10.1007/BF02685881
S. Evens and J. H. Lu, On the variety of Lagrangian subalgebras II, Ann. Sci. École Norm. Sup. (4) 39 (2006), 347--379.
Mathematical Reviews (MathSciNet): MR2245536
Zentralblatt MATH: 1162.17020
Digital Object Identifier: doi:10.1016/j.ansens.2005.11.004
H. Esnault and E. Viehweg, Lectures on vanishing theorems, Birkhäuser-Verlag, Basel, 1992.
Mathematical Reviews (MathSciNet): MR1193913
Zentralblatt MATH: 0779.14003
M. Franz and A. Weber, Weights in cohomology and the Eilenberg-Moore spectral sequence, Ann. Inst. Fourier (Grenoble) 55 (2005), 673--691.
Mathematical Reviews (MathSciNet): MR2147902
O. Fujino, Multiplication map and vanishing theorems for toric varieties, Math. Z. 257 (2007), 631--641.
Mathematical Reviews (MathSciNet): MR2328817
Zentralblatt MATH: 1129.14029
Digital Object Identifier: doi:10.1007/s00209-007-0140-5
W. Fulton, Intersection theory, Second edition, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin, 1998.
Mathematical Reviews (MathSciNet): MR1644323
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.
Mathematical Reviews (MathSciNet): MR463157
G. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229--239.
Mathematical Reviews (MathSciNet): MR424841
Zentralblatt MATH: 0338.14015
Digital Object Identifier: doi:10.1007/BF01390321
V. Kiritchenko, Chern classes of reductive groups and an adjunction formula, Ann. Inst. Fourier (Grenoble) 56 (2006), 1225--1256.
Mathematical Reviews (MathSciNet): MR2266889
F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. (2) 140 (1994), 253--288.
Mathematical Reviews (MathSciNet): MR1298713
Zentralblatt MATH: 0828.22017
Digital Object Identifier: doi:10.2307/2118600
JSTOR: links.jstor.org
J. Kollár, Higher direct images of dualizing sheaves, Ann. of Math. (2) 123 (1986), 11--42.
Mathematical Reviews (MathSciNet): MR825838
Digital Object Identifier: doi:10.2307/1971351
JSTOR: links.jstor.org
D. Luna, Variétés sphériques de type $A$, Inst. Hautes Études Pub. Math. 94 (2001), 161--226.
Mathematical Reviews (MathSciNet): MR1896179
Digital Object Identifier: doi:10.1007/s10240-001-8194-0
A. V. Mavlyutov, Cohomology of complete intersections in toric varieties, Pacific J. Math. 191 (1999), 133--144.
Mathematical Reviews (MathSciNet): MR1725467
Zentralblatt MATH: 1032.14013
A. V. Mavlyutov, Cohomology of rational forms and a vanishing theorem for toric varieties, J. Reine Angew. Math. 615 (2008), 45--58.
Mathematical Reviews (MathSciNet): MR2384331
Y. Norimatsu, Kodaira vanishing theorem and Chern classes for $\partial$-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), 107--108.
Mathematical Reviews (MathSciNet): MR494655
Digital Object Identifier: doi:10.3792/pjaa.54.107
Project Euclid: euclid.pja/1195517743
D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1986), 159--176.
Mathematical Reviews (MathSciNet): MR863714
Zentralblatt MATH: 0596.32016
Digital Object Identifier: doi:10.1007/BF01450932
B. Totaro, Chow groups, Chow cohomology, and linear varieties, preprint available at www.dpmms. cam.ac.uk/~bt219/papers.html, to appear in Journal of Algebraic Geometry.
J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Math. 149, Cambridge University Press, Cambridge, 2003.
Mathematical Reviews (MathSciNet): MR1988690
Zentralblatt MATH: 1075.13007
J. Winkelmann, On manifolds with trivial logarithmic tangent bundle, Osaka J. Math. 41 (2004), 473--484.
Mathematical Reviews (MathSciNet): MR2069097
Zentralblatt MATH: 1058.32011
Project Euclid: euclid.ojm/1153493523
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