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

Characteristic classes of singular toric varieties

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.
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,, \emph{C. R. Math. Acad. Sci. Paris}, 342 (2006), 405.  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 \emph{New Perspectives in Algebraic Combinatorics} (Berkeley, (1999), 1996.   Google Scholar

[3]

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

[4]

G. Barthel, J.-P. Brasselet and K.-H. Fieseler, Classes de Chern des variétés toriques singulières,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 315 (1992), 187.   Google Scholar

[5]

J.-P. Brasselet, J. Schürmann and S. Yokura, Hirzebruch classes and motivic Chern classes of singular spaces,, \emph{J. Topol. Anal.}, 2 (2010), 1.  doi: 10.1142/S1793525310000239.  Google Scholar

[6]

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

[7]

S. E. Cappell, L. Maxim, J. Schürmann and J. L. Shaneson, Equivariant characteristic classes of complex algebraic varieties,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 1722.  doi: 10.1002/cpa.21427.  Google Scholar

[8]

S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 30 (1994), 62.  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,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 321 (1995), 885.   Google Scholar

[10]

D. Cox, The homogeneous coordinate ring of a toric variety,, \emph{J. Alg. Geom.}, 4 (1995), 17.   Google Scholar

[11]

D. Cox, J. Little and H. Schenck, Toric Varieties,, Graduate Studies in Mathematics, 124 (2011).   Google Scholar

[12]

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

[13]

D. Edidin and W. Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties,, \emph{Comm. Algebra}, 31 (2003), 3735.  doi: 10.1081/AGB-120022440.  Google Scholar

[14]

K. E. Feldman, Miraculous cancellation and Pick's theorem,, in \emph{Toric Topology}, (2008), 71.  doi: 10.1090/conm/460/09011.  Google Scholar

[15]

W. Fulton, Introduction to Toric Varieties,, Annals of Mathematics Studies, (1993).   Google Scholar

[16]

St. Garoufalidis and J. E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry,, \emph{J. Amer. Math. Soc.}, 14 (2001), 1.  doi: 10.1090/S0894-0347-00-00352-0.  Google Scholar

[17]

I. Gessel, Generating functions and generalized Dedekind sums,, \emph{Electron. J. Combin.}, 4 (1997).   Google Scholar

[18]

M.-N. Ishida, Torus embeddings and de Rham complexes,, in \emph{Commutative Algebra and Combinatorics} (Kyoto, (1987), 111.   Google Scholar

[19]

E. Materov, The Bott formula for toric varieties,, \emph{Mosc. Math. J.}, 2 (2002), 161.   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, 106 (1978).   Google Scholar

[22]

N. C. Leung and V. Reiner, The signature of a toric variety,, \emph{Duke Math. J.}, 111 (2002), 253.  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)], (1988).   Google Scholar

[24]

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

[25]

J. E. Pommersheim, Products of cycles and the Todd class of a toric variety,, \emph{J. Amer. Math. Soc.}, 9 (1996), 813.  doi: 10.1090/S0894-0347-96-00209-3.  Google Scholar

[26]

J. Shaneson, Characteristic classes, lattice points and Euler-MacLaurin formulae,, in \emph{Proceedings of the International Congress of Mathematicians, (1994), 612.   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, (1972).   Google Scholar

show all references

References:
[1]

P. Aluffi, Classes de Chern pour variétés singulières, revisitées,, \emph{C. R. Math. Acad. Sci. Paris}, 342 (2006), 405.  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 \emph{New Perspectives in Algebraic Combinatorics} (Berkeley, (1999), 1996.   Google Scholar

[3]

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

[4]

G. Barthel, J.-P. Brasselet and K.-H. Fieseler, Classes de Chern des variétés toriques singulières,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 315 (1992), 187.   Google Scholar

[5]

J.-P. Brasselet, J. Schürmann and S. Yokura, Hirzebruch classes and motivic Chern classes of singular spaces,, \emph{J. Topol. Anal.}, 2 (2010), 1.  doi: 10.1142/S1793525310000239.  Google Scholar

[6]

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

[7]

S. E. Cappell, L. Maxim, J. Schürmann and J. L. Shaneson, Equivariant characteristic classes of complex algebraic varieties,, \emph{Comm. Pure Appl. Math.}, 65 (2012), 1722.  doi: 10.1002/cpa.21427.  Google Scholar

[8]

S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 30 (1994), 62.  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,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 321 (1995), 885.   Google Scholar

[10]

D. Cox, The homogeneous coordinate ring of a toric variety,, \emph{J. Alg. Geom.}, 4 (1995), 17.   Google Scholar

[11]

D. Cox, J. Little and H. Schenck, Toric Varieties,, Graduate Studies in Mathematics, 124 (2011).   Google Scholar

[12]

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

[13]

D. Edidin and W. Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties,, \emph{Comm. Algebra}, 31 (2003), 3735.  doi: 10.1081/AGB-120022440.  Google Scholar

[14]

K. E. Feldman, Miraculous cancellation and Pick's theorem,, in \emph{Toric Topology}, (2008), 71.  doi: 10.1090/conm/460/09011.  Google Scholar

[15]

W. Fulton, Introduction to Toric Varieties,, Annals of Mathematics Studies, (1993).   Google Scholar

[16]

St. Garoufalidis and J. E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry,, \emph{J. Amer. Math. Soc.}, 14 (2001), 1.  doi: 10.1090/S0894-0347-00-00352-0.  Google Scholar

[17]

I. Gessel, Generating functions and generalized Dedekind sums,, \emph{Electron. J. Combin.}, 4 (1997).   Google Scholar

[18]

M.-N. Ishida, Torus embeddings and de Rham complexes,, in \emph{Commutative Algebra and Combinatorics} (Kyoto, (1987), 111.   Google Scholar

[19]

E. Materov, The Bott formula for toric varieties,, \emph{Mosc. Math. J.}, 2 (2002), 161.   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, 106 (1978).   Google Scholar

[22]

N. C. Leung and V. Reiner, The signature of a toric variety,, \emph{Duke Math. J.}, 111 (2002), 253.  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)], (1988).   Google Scholar

[24]

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

[25]

J. E. Pommersheim, Products of cycles and the Todd class of a toric variety,, \emph{J. Amer. Math. Soc.}, 9 (1996), 813.  doi: 10.1090/S0894-0347-96-00209-3.  Google Scholar

[26]

J. Shaneson, Characteristic classes, lattice points and Euler-MacLaurin formulae,, in \emph{Proceedings of the International Congress of Mathematicians, (1994), 612.   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, (1972).   Google Scholar

[1]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118

[2]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[3]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[4]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[5]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[6]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[7]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[8]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[9]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[10]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[11]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[12]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[13]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[14]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]