## Tohoku Mathematical Journal

### Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

Michel Brion

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 61, Number 3 (2009), 365-392.

Dates
First available in Project Euclid: 16 October 2009

https://projecteuclid.org/euclid.tmj/1255700200

Digital Object Identifier
doi:10.2748/tmj/1255700200

Mathematical Reviews number (MathSciNet)
MR2568260

Zentralblatt MATH identifier
1195.14024

#### Citation

Brion, Michel. Vanishing theorems for Dolbeault cohomology of log homogeneous varieties. Tohoku Math. J. (2) 61 (2009), no. 3, 365--392. doi:10.2748/tmj/1255700200. https://projecteuclid.org/euclid.tmj/1255700200

#### References

• V. Alexeev, Complete moduli in the presence of semiabelian group actions, Ann. of Math. (2) 155 (2002), 611--708.
• V. V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), 293--338.
• A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480--497.
• F. Bien and M. Brion, Automorphisms and local rigidity of regular varieties, Compositio Math. 104 (1996), 1--26.
• C. Birkenhake and H. Lange, Complex abelian varieties, Second edition, Grundlehren Math. Wiss., 302, Springer-Verlag, Berlin, 2004.
• M. Brion, Variétés sphériques et théorie de Mori, Duke Math. J. 72 (1993), 369--404.
• 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.
• A. Broer, A vanishing theorem for Dolbeault cohomology of homogeneous vector bundles, J. Reine Angew. Math. 493 (1997), 153--169.
• J. B. Carrell and D. I. Lieberman, Holomorphic vector fields and compact Kaehler manifolds, Invent. Math. 21 (1973), 303--309.
• 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.
• C. de Concini and C. Procesi, Complete symmetric varieties, Invariant Theory (Montecatini, 1982), 1--44, Lecture Notes in Math. 996, Springer-Verlag, Berlin, 1983.
• P. Deligne, Théorie de Hodge II, Inst. Hautes Études Pub. Math. 40 (1971), 5--57.
• P. Deligne, Théorie de Hodge III, Inst. Hautes Études Pub. Math. 44 (1974), 5--78.
• S. Evens and J. H. Lu, On the variety of Lagrangian subalgebras II, Ann. Sci. École Norm. Sup. (4) 39 (2006), 347--379.
• H. Esnault and E. Viehweg, Lectures on vanishing theorems, Birkhäuser-Verlag, Basel, 1992.
• M. Franz and A. Weber, Weights in cohomology and the Eilenberg-Moore spectral sequence, Ann. Inst. Fourier (Grenoble) 55 (2005), 673--691.
• O. Fujino, Multiplication map and vanishing theorems for toric varieties, Math. Z. 257 (2007), 631--641.
• W. Fulton, Intersection theory, Second edition, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, Berlin, 1998.
• R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977.
• G. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229--239.
• V. Kiritchenko, Chern classes of reductive groups and an adjunction formula, Ann. Inst. Fourier (Grenoble) 56 (2006), 1225--1256.
• F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. (2) 140 (1994), 253--288.
• J. Kollár, Higher direct images of dualizing sheaves, Ann. of Math. (2) 123 (1986), 11--42.
• D. Luna, Variétés sphériques de type $A$, Inst. Hautes Études Pub. Math. 94 (2001), 161--226.
• A. V. Mavlyutov, Cohomology of complete intersections in toric varieties, Pacific J. Math. 191 (1999), 133--144.
• A. V. Mavlyutov, Cohomology of rational forms and a vanishing theorem for toric varieties, J. Reine Angew. Math. 615 (2008), 45--58.
• Y. Norimatsu, Kodaira vanishing theorem and Chern classes for $\partial$-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), 107--108.
• D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1986), 159--176.
• 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.
• J. Winkelmann, On manifolds with trivial logarithmic tangent bundle, Osaka J. Math. 41 (2004), 473--484.