Characteristic classes of singular toric varieties

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2013, 20: 109-120. doi: 10.3934/era.2013.20.109

Characteristic classes of singular toric varieties

Laurenţiu Maxim ^1, and Jörg Schürmann ^2,

1.	Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, United States
2.	Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

Received October 2013 Revised November 2013 Published December 2013

Abstract
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We introduce a new approach for the computation of characteristic classes of singular toric varieties and, as an application, we obtain generalized Pick-type formulae for lattice polytopes. Many of our results (e.g., lattice point counting formulae) hold even more generally, for closed algebraic torus-invariant subspaces of toric varieties. In the simplicial case, by combining this new computation method with the Lefschetz-Riemann-Roch theorem, we give new proofs of several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.

Keywords: characteristic classes, weighted projective space., Toric varieties, Hirzebruch- and Lefschetz-Riemann-Roch, lattice points, lattice polytopes, Pick formula, Ehrhart polynomial.

Mathematics Subject Classification: 14M25, 52B20, 14C17, 14C4.

Citation: Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109

References:

[1]	P. Aluffi, Classes de Chern pour variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris, 342 (2006), 405-410. doi: 10.1016/j.crma.2006.01.002.
[2]	A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-97), Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999, 91-147.
[3]	P. Baum, W. Fulton and R. MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math., 45 (1975), 101-145.
[4]	G. Barthel, J.-P. Brasselet and K.-H. Fieseler, Classes de Chern des variétés toriques singulières, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 187-192.
[5]	J.-P. Brasselet, J. Schürmann and S. Yokura, Hirzebruch classes and motivic Chern classes of singular spaces, J. Topol. Anal., 2 (2010), 1-55. doi: 10.1142/S1793525310000239.
[6]	M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math., 482 (1997), 67-92.
[7]	S. E. Cappell, L. Maxim, J. Schürmann and J. L. Shaneson, Equivariant characteristic classes of complex algebraic varieties, Comm. Pure Appl. Math., 65 (2012), 1722-1769. doi: 10.1002/cpa.21427.
[8]	S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.), 30 (1994), 62-69. doi: 10.1090/S0273-0979-1994-00436-7.
[9]	S. E. Cappell and J. L. Shaneson, Euler-MacLaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 885-890.
[10]	D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom., 4 (1995), 17-50.
[11]	D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.
[12]	V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154. doi: 10.1070/RM1978v033n02ABEH002305.
[13]	D. Edidin and W. Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties, Comm. Algebra, 31 (2003), 3735-3752. doi: 10.1081/AGB-120022440.
[14]	K. E. Feldman, Miraculous cancellation and Pick's theorem, in Toric Topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 71-86. doi: 10.1090/conm/460/09011.
[15]	W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993.
[16]	St. Garoufalidis and J. E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry, J. Amer. Math. Soc., 14 (2001), 1-23. doi: 10.1090/S0894-0347-00-00352-0.
[17]	I. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin., 4 (1997), 17 pp.
[18]	M.-N. Ishida, Torus embeddings and de Rham complexes, in Commutative Algebra and Combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, (1987), 111-145.
[19]	E. Materov, The Bott formula for toric varieties, Mosc. Math. J., 2 (2002), 161-182, 200.
[20]	L. Maxim and J. Schürmann, Characteristic classes of singular toric varieties, arXiv:1303.4454.
[21]	B. Moonen, Das Lefschetz-Riemann-Roch-Theorem für Singuläre Varietäten, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn. Bonner Mathematische Schriften, 106, Universität Bonn, Mathematisches Institut, Bonn, 1978.
[22]	N. C. Leung and V. Reiner, The signature of a toric variety, Duke Math. J., 111 (2002), 253-286. doi: 10.1215/S0012-7094-02-11123-5.
[23]	T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988.
[24]	J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann., 295 (1993), 1-24. doi: 10.1007/BF01444874.
[25]	J. E. Pommersheim, Products of cycles and the Todd class of a toric variety, J. Amer. Math. Soc., 9 (1996), 813-826. doi: 10.1090/S0894-0347-96-00209-3.
[26]	J. Shaneson, Characteristic classes, lattice points and Euler-MacLaurin formulae, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 612-624.
[27]	D. Zagier, Equivariant Pontrjagin Classes and Applications to Orbit Spaces. Applications of the G-Signature Theorem to Transformation Groups, Symmetric Products and Number Theory, Lecture Notes in Mathematics, Vol. 290, Springer-Verlag, Berlin-New York, 1972.

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References:

[1]	P. Aluffi, Classes de Chern pour variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris, 342 (2006), 405-410. doi: 10.1016/j.crma.2006.01.002.
[2]	A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-97), Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999, 91-147.
[3]	P. Baum, W. Fulton and R. MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math., 45 (1975), 101-145.
[4]	G. Barthel, J.-P. Brasselet and K.-H. Fieseler, Classes de Chern des variétés toriques singulières, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 187-192.
[5]	J.-P. Brasselet, J. Schürmann and S. Yokura, Hirzebruch classes and motivic Chern classes of singular spaces, J. Topol. Anal., 2 (2010), 1-55. doi: 10.1142/S1793525310000239.
[6]	M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math., 482 (1997), 67-92.
[7]	S. E. Cappell, L. Maxim, J. Schürmann and J. L. Shaneson, Equivariant characteristic classes of complex algebraic varieties, Comm. Pure Appl. Math., 65 (2012), 1722-1769. doi: 10.1002/cpa.21427.
[8]	S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.), 30 (1994), 62-69. doi: 10.1090/S0273-0979-1994-00436-7.
[9]	S. E. Cappell and J. L. Shaneson, Euler-MacLaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 885-890.
[10]	D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom., 4 (1995), 17-50.
[11]	D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.
[12]	V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154. doi: 10.1070/RM1978v033n02ABEH002305.
[13]	D. Edidin and W. Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties, Comm. Algebra, 31 (2003), 3735-3752. doi: 10.1081/AGB-120022440.
[14]	K. E. Feldman, Miraculous cancellation and Pick's theorem, in Toric Topology, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008, 71-86. doi: 10.1090/conm/460/09011.
[15]	W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993.
[16]	St. Garoufalidis and J. E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry, J. Amer. Math. Soc., 14 (2001), 1-23. doi: 10.1090/S0894-0347-00-00352-0.
[17]	I. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin., 4 (1997), 17 pp.
[18]	M.-N. Ishida, Torus embeddings and de Rham complexes, in Commutative Algebra and Combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, (1987), 111-145.
[19]	E. Materov, The Bott formula for toric varieties, Mosc. Math. J., 2 (2002), 161-182, 200.
[20]	L. Maxim and J. Schürmann, Characteristic classes of singular toric varieties, arXiv:1303.4454.
[21]	B. Moonen, Das Lefschetz-Riemann-Roch-Theorem für Singuläre Varietäten, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn. Bonner Mathematische Schriften, 106, Universität Bonn, Mathematisches Institut, Bonn, 1978.
[22]	N. C. Leung and V. Reiner, The signature of a toric variety, Duke Math. J., 111 (2002), 253-286. doi: 10.1215/S0012-7094-02-11123-5.
[23]	T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988.
[24]	J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann., 295 (1993), 1-24. doi: 10.1007/BF01444874.
[25]	J. E. Pommersheim, Products of cycles and the Todd class of a toric variety, J. Amer. Math. Soc., 9 (1996), 813-826. doi: 10.1090/S0894-0347-96-00209-3.
[26]	J. Shaneson, Characteristic classes, lattice points and Euler-MacLaurin formulae, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 612-624.
[27]	D. Zagier, Equivariant Pontrjagin Classes and Applications to Orbit Spaces. Applications of the G-Signature Theorem to Transformation Groups, Symmetric Products and Number Theory, Lecture Notes in Mathematics, Vol. 290, Springer-Verlag, Berlin-New York, 1972.

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