# American Institute of Mathematical Sciences

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
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##### References:
 [1] Ricardo Diaz and Sinai Robins. The Ehrhart polynomial of a lattice n -simplex. Electronic Research Announcements, 1996, 2: 1-6. [2] Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1 [3] Maksim Maydanskiy, Benjamin P. Mirabelli. Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes. Electronic Research Announcements, 2011, 18: 131-143. doi: 10.3934/era.2011.18.131 [4] Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure & Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1 [5] Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445 [6] Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007 [7] Dmitry Kleinbock, Xi Zhao. An application of lattice points counting to shrinking target problems. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 155-168. doi: 10.3934/dcds.2018007 [8] Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81 [9] Xiaoying Han, Peter E. Kloeden. Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021143 [10] Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051 [11] Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001 [12] Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005 [13] William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307 [14] Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511 [15] Kang-Ling Liao, Chih-Wen Shih. A Lattice model on somitogenesis of zebrafish. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2789-2814. doi: 10.3934/dcdsb.2012.17.2789 [16] Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134 [17] Thomas Espitau, Antoine Joux. Certified lattice reduction. Advances in Mathematics of Communications, 2020, 14 (1) : 137-159. doi: 10.3934/amc.2020011 [18] Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497 [19] Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019 [20] Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233

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