Contractions of symplectic varieties
Author:
Jan Wierzba
Journal:
J. Algebraic Geom. 12 (2003), 507-534
DOI:
https://doi.org/10.1090/S1056-3911-03-00318-7
Published electronically:
March 6, 2003
MathSciNet review:
1966025
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Abstract |
References |
Additional Information
Abstract: We consider singularities that arise from birational contractions of smooth symplectic varieties. We establish some general properties of these singularities and study the case of divisorial contractions in more detail.
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CMSB Cho, Y.; Miyaoka, Y. and Shepherd-Barron, N. I.: Characterisations of Projective Spaces and Applications, in preparation.
GH Griffiths, P. and Harris, J.: Principles of Algebraic Geometry, John Wiley & Sons Inc., New York (1978).
GR Grauert, H. and Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Inv. Math. 11 (1970), 263–292.
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Hin Hinich, V.: On Brieskorn’s Theorem, Israel J. Math. 76 (1991), 153–160.
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Kal Kaledin, D.: Symplectic resolutions: Deformations and birational maps, math.AG/ 0012008.
Kaw Kawamata, Y.: On the length of an extremal rational curve, Invent. Math. 105 (1991), 609–611.
KKMSD Kempf, G.; Knudsen, F.; Mumford, D. and Saint-Donat, B.: Toroidal Embeddings I, Springer-Verlag, Berlin-New York (1973).
KM Kollár, J. and Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge University Press, Cambridge (1998).
KMM Kawamata, Y.; Matsuda, K. and Matsuki, K.: Introduction to the Minimal Model Problem. In: Oda, T. (ed.), Algebraic Geometry. Sendai Adv. Stud. Pure Math. 10, Kinokuniya–North-Holland (1987), 283–360.
Kol Kollár, J.: Rational Curves on Algebraic Varieties, Springer-Verlag, Berlin (1996).
KV Kaledin, D. and Verbitsky, M.: Period map for non-compact holomorphically symplectic manifolds, alg-geom/0005007.
Ran Ran, Z.: Hodge theory and deformation of maps, Comp. Math. 97 (1995), 309–328.
Sei Seidenberg, A.: Derivations and integral closure, Pacific J. Math. 16 (1966), 167–173.
Slo Slodowy, P.: Simple Singularities and Simple Algebraic Groups, Springer Verlag, Berlin (1980).
Wil Wilson, P.M.H.: Symplectic deformations of Calabi-Yau Threefolds, J. Diff. Geom. 45 (1997), 611–637.
Wis Wiśniewski, J.A.: Cohomological invariants of complex manifolds coming from extremal rays, Asian J. Math. 2 (1998), 289–301.
Bea Beauville, A.: Symplectic Singularities, Invent. Math. 139 (2000), 541–549.
CMSB Cho, Y.; Miyaoka, Y. and Shepherd-Barron, N. I.: Characterisations of Projective Spaces and Applications, in preparation.
GH Griffiths, P. and Harris, J.: Principles of Algebraic Geometry, John Wiley & Sons Inc., New York (1978).
GR Grauert, H. and Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Inv. Math. 11 (1970), 263–292.
Gro Grothendieck, A.: Eléments de géométrie algébrique, Publ. Math. IHES, 4 (1960), 8 (1961), 17 (1961) et 28 (1966).
Hin Hinich, V.: On Brieskorn’s Theorem, Israel J. Math. 76 (1991), 153–160.
Hum Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Springer Verlag, New York-Berlin (1972).
Kal Kaledin, D.: Symplectic resolutions: Deformations and birational maps, math.AG/ 0012008.
Kaw Kawamata, Y.: On the length of an extremal rational curve, Invent. Math. 105 (1991), 609–611.
KKMSD Kempf, G.; Knudsen, F.; Mumford, D. and Saint-Donat, B.: Toroidal Embeddings I, Springer-Verlag, Berlin-New York (1973).
KM Kollár, J. and Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge University Press, Cambridge (1998).
KMM Kawamata, Y.; Matsuda, K. and Matsuki, K.: Introduction to the Minimal Model Problem. In: Oda, T. (ed.), Algebraic Geometry. Sendai Adv. Stud. Pure Math. 10, Kinokuniya–North-Holland (1987), 283–360.
Kol Kollár, J.: Rational Curves on Algebraic Varieties, Springer-Verlag, Berlin (1996).
KV Kaledin, D. and Verbitsky, M.: Period map for non-compact holomorphically symplectic manifolds, alg-geom/0005007.
Ran Ran, Z.: Hodge theory and deformation of maps, Comp. Math. 97 (1995), 309–328.
Sei Seidenberg, A.: Derivations and integral closure, Pacific J. Math. 16 (1966), 167–173.
Slo Slodowy, P.: Simple Singularities and Simple Algebraic Groups, Springer Verlag, Berlin (1980).
Wil Wilson, P.M.H.: Symplectic deformations of Calabi-Yau Threefolds, J. Diff. Geom. 45 (1997), 611–637.
Wis Wiśniewski, J.A.: Cohomological invariants of complex manifolds coming from extremal rays, Asian J. Math. 2 (1998), 289–301.
Additional Information
Jan Wierzba
Affiliation:
Lettestrasse 6a, 10437 Berlin, Germany
Email:
jwierzba@mi.uni-koeln.de
Received by editor(s):
November 9, 2000
Published electronically:
March 6, 2003
Additional Notes:
The author was supported by Trinity College, Cambridge and EPSRC
