American Institute of Mathematical Sciences

Advanced
  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

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.  Google Scholar

[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.  Google Scholar

[3]

P. Baum, W. Fulton and R. MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math., 45 (1975), 101-145.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math., 482 (1997), 67-92.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom., 4 (1995), 17-50.  Google Scholar

[11]

D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[12]

V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154. doi: 10.1070/RM1978v033n02ABEH002305.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[17]

I. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin., 4 (1997), 17 pp.  Google Scholar

[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.  Google Scholar

[19]

E. Materov, The Bott formula for toric varieties, Mosc. Math. J., 2 (2002), 161-182, 200.  Google Scholar

[20]

L. Maxim and J. Schürmann, Characteristic classes of singular toric varieties,, \arXiv{1303.4454}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann., 295 (1993), 1-24. doi: 10.1007/BF01444874.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

P. Baum, W. Fulton and R. MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math., 45 (1975), 101-145.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math., 482 (1997), 67-92.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. Cox, The homogeneous coordinate ring of a toric variety, J. Alg. Geom., 4 (1995), 17-50.  Google Scholar

[11]

D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[12]

V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154. doi: 10.1070/RM1978v033n02ABEH002305.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[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.  Google Scholar

[17]

I. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin., 4 (1997), 17 pp.  Google Scholar

[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.  Google Scholar

[19]

E. Materov, The Bott formula for toric varieties, Mosc. Math. J., 2 (2002), 161-182, 200.  Google Scholar

[20]

L. Maxim and J. Schürmann, Characteristic classes of singular toric varieties,, \arXiv{1303.4454}., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann., 295 (1993), 1-24. doi: 10.1007/BF01444874.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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