$K$-equivalence in birational geometry and characterizations of complex elliptic genera
Author:
Chin-Lung Wang
Journal:
J. Algebraic Geom. 12 (2003), 285-306
DOI:
https://doi.org/10.1090/S1056-3911-02-00312-0
Published electronically:
September 26, 2002
MathSciNet review:
1949645
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Abstract |
References |
Additional Information
Abstract: We show that for smooth complex projective varieties the most general combinations of Chern numbers that are invariant under the $K$-equivalence relation consist of the complex elliptic genera.
AKMW D. Abramovich, K.Karu, K. Matsuki and J. Wlodarsczyk; Torifications and factorizations of birational maps, preprint 1999, arXiv:mathAG/9904135.
Ahlfors L. Ahlfors; Complex Analysis, McGraw-Hill 1979.
Batyrev V. Batyrev; Stringy Hodge numbers of varieties with Gorenstein canonical singularities. arXiv:math.AG/9711008.
BoLi L. Borisov and A. Libgober; Elliptic genera of singular varieties, preprint 2000, arXiv:math.AG/0007108.
DeLo J. Denef and F. Loeser; Germs of arcs on singular algebraic varieties and motivic integration, Inv. Math. 135 (1999), 201-232.
Fulton W. Fulton; Intersection Theory, Erge. Math. ihr. Gren.;3. Folge, Bd 2, Springer-Verlag 1984.
Hirzebruch F. Hirzebruch; Topological Methods in Algebraic Geometry, Grund. der Math. Wissen. 131, Springer 1966.
HBJ F. Hirzebruch, T. Berger and R. Jung; Manifolds and Modular Forms, Max-Plank-Institut für Math. 1992.
Huybrechts D. Huybrechts; Birational symplectic manifolds and their deformations, J. Diff. Geom. 45 (1997).
Huybrechts2 D. Huybrechts; Compact hyperkähler manifolds: basic results, Invent. Math. 135 (1999), 63–113. Erratum math.AG/0106014.
Kollar J. Kollár; Flops, Nagoya Math. J. 113 (1989), 15–36.
KoMo J. Kollár and S. Mori; Classification of three dimensional flips, J. AMS 5 No.3 (1992), 533-703.
Krichever I. Krichever; Generalized elliptic genera and Baker-Akhiezer functions, Math. Notes 47 (1990), 132-142.
milnor J. Milnor; On the cobordism ring $\Omega ^\ast$ and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
Ruan Y. Ruan; Surgery, quantum cohomology and birational geometry, preprint 1998, arXiv:math.AG/9810039.
Totaro B. Totaro; Chern numbers for singular varieties and elliptic homology, to appear in Annals of Math. 2000.
Wang C.-L. Wang; On the topology of birational minimal models, J. Diff. Geom. 50 (1998), 129-146.
Wlodarsczyk J. Wlodarsczyk; Combinatorial structures on toroidal varieties and a proof of the weak factorization theorems, preprint 1999, arXiv:math.AG/9904076.
AKMW D. Abramovich, K.Karu, K. Matsuki and J. Wlodarsczyk; Torifications and factorizations of birational maps, preprint 1999, arXiv:mathAG/9904135.
Ahlfors L. Ahlfors; Complex Analysis, McGraw-Hill 1979.
Batyrev V. Batyrev; Stringy Hodge numbers of varieties with Gorenstein canonical singularities. arXiv:math.AG/9711008.
BoLi L. Borisov and A. Libgober; Elliptic genera of singular varieties, preprint 2000, arXiv:math.AG/0007108.
DeLo J. Denef and F. Loeser; Germs of arcs on singular algebraic varieties and motivic integration, Inv. Math. 135 (1999), 201-232.
Fulton W. Fulton; Intersection Theory, Erge. Math. ihr. Gren.;3. Folge, Bd 2, Springer-Verlag 1984.
Hirzebruch F. Hirzebruch; Topological Methods in Algebraic Geometry, Grund. der Math. Wissen. 131, Springer 1966.
HBJ F. Hirzebruch, T. Berger and R. Jung; Manifolds and Modular Forms, Max-Plank-Institut für Math. 1992.
Huybrechts D. Huybrechts; Birational symplectic manifolds and their deformations, J. Diff. Geom. 45 (1997).
Huybrechts2 D. Huybrechts; Compact hyperkähler manifolds: basic results, Invent. Math. 135 (1999), 63–113. Erratum math.AG/0106014.
Kollar J. Kollár; Flops, Nagoya Math. J. 113 (1989), 15–36.
KoMo J. Kollár and S. Mori; Classification of three dimensional flips, J. AMS 5 No.3 (1992), 533-703.
Krichever I. Krichever; Generalized elliptic genera and Baker-Akhiezer functions, Math. Notes 47 (1990), 132-142.
milnor J. Milnor; On the cobordism ring $\Omega ^\ast$ and a complex analogue, Amer. J. Math. 82 (1960), 505–521.
Ruan Y. Ruan; Surgery, quantum cohomology and birational geometry, preprint 1998, arXiv:math.AG/9810039.
Totaro B. Totaro; Chern numbers for singular varieties and elliptic homology, to appear in Annals of Math. 2000.
Wang C.-L. Wang; On the topology of birational minimal models, J. Diff. Geom. 50 (1998), 129-146.
Wlodarsczyk J. Wlodarsczyk; Combinatorial structures on toroidal varieties and a proof of the weak factorization theorems, preprint 1999, arXiv:math.AG/9904076.
Additional Information
Chin-Lung Wang
Affiliation:
Department of Mathematics, National Tsing-Hua University, Hsinchu, Taiwan
MR Author ID:
615969
ORCID:
0000-0003-2873-8472
Email:
dragon@math2.math.nthu.edu.tw
Received by editor(s):
October 30, 2000
Published electronically:
September 26, 2002
Additional Notes:
Supported in part by NSC project 89-2115-M-007-045
