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Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

  • Daniel Greb (a1), Stefan Kebekus (a2) and Sándor J. Kovács (a3)

Abstract

Given a normal variety Z, a p-form σ defined on the smooth locus of Z and a resolution of singularities , we study the problem of extending the pull-back π*(σ) over the π-exceptional set . For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along E. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov–Sommese vanishing theorem to log canonical threefold pairs follows.

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References

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[1]Beltrametti, M. C. and Sommese, A. J., The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16 (Walter de Gruyter, Berlin, 1995).
[2]Brieskorn, E., Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1967–1968), 336358.
[3]Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).
[4]Esnault, H. and Viehweg, E., Lectures on vanishing theorems, DMV Seminar, vol. 20 (Birkhäuser, Basel, 1992).
[5]Flenner, H., Extendability of differential forms on nonisolated singularities, Invent. Math. 94 (1988), 317326.
[6]Hartshorne, R., Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961 (Springer, Berlin, 1967).
[7]Hartshorne, R., Ample subvarieties of algebraic varieties, Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, vol. 156 (Springer, Berlin, 1970).
[8]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).
[9]Hassett, B. and Kovács, S. J., Reflexive pull-backs and base extension, J. Algebraic Geom. 13 (2004), 233247.
[10]Iitaka, S., Algebraic geometry, Graduate Texts in Mathematics, vol. 76 (Springer, New York, 1982); An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, vol. 24.
[11]Kaup, W., Infinitesimale Transformationsgruppen komplexer Räume, Math. Ann. 160 (1965), 7292.
[12]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283360.
[13]Kebekus, S. and Kovács, S. J., The structure of surfaces mapping to the moduli stack of canonically polarized varieties, Preprint (2007), arXiv:0707.2054.
[14]Kebekus, S. and Kovács, S. J., Families of canonically polarized varieties over surfaces, Invent. Math. 172 (2008), 657682; DOI: 10.1007/s00222-008-0128-8.
[15]Kebekus, S. and Kovács, S. J., The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Preprint (2008), arXiv:0812.2305.
[16]Kebekus, S. and Solá Conde, L., Existence of rational curves on algebraic varieties, minimal rational tangents, and applications, in Global aspects of complex geometry (Springer, Berlin, 2006), 359416.
[17]Keel, S. and McKernan, J., Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), viii+153.
[18]Kollár, J., Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, NJ, 2007).
[19]Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998), with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original.
[20]Langer, A., The Bogomolov–Miyaoka–Yau inequality for log canonical surfaces, J. London Math. Soc. (2) 64 (2001), 327343.
[21]Langer, A., Logarithmic orbifold Euler numbers of surfaces with applications, Proc. London Math. Soc. (3) 86 (2003), 358396.
[22]Mumford, D., The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. Inst. Hautes Études Sci. 9 (1961), 522.
[23]Namikawa, Y., Extension of 2-forms and symplectic varieties, J. Reine Angew. Math. 539 (2001), 123147.
[24]Okonek, C., Schneider, M. and Spindler, H., Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3 (Birkhäuser, Boston, MA, 1980).
[25]Reid, M., Young person’s guide to canonical singularities, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 345414.
[26]Steenbrink, J. H. M., Vanishing theorems on singular spaces, Astérisque 130 (1985), 330341.
[27]van Straten, D. and Steenbrink, J., Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 (1985), 97110.
[28]Wahl, J. M., A characterization of quasihomogeneous Gorenstein surface singularities, Compositio Math. 55 (1985), 269288.
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Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties

  • Daniel Greb (a1), Stefan Kebekus (a2) and Sándor J. Kovács (a3)

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