An equivariant Riemann-Roch theorem for complete, simplicial toric varieties.

Introduction. The theory of toric varieties establishes a now classical connection between algebraic geometry and convex polytopes. In particular, äs observed by Danilov in the seventies, finding a closed formula for the Todd class of complete toric varieties would have important consequences for enumeration of lattice points in convex lattice polytopes. Since then, a number of such formulas have been proposed; see [M], [Pl], [P2] The Todd class of complete simplicial toric varieties is computed in [G-G-K], using the Riemann-Roch formula of T. Kawasaki [Ka].

[1]  Don Zagier,et al.  The Atiyah-Singer Theorem and Elementary Number Theory , 1974 .

[2]  J.-M. Kantor,et al.  Une application du Théorème de Riemann-Roch combinatoire au polynôme d'Ehrhart des polytopes entiers de Rd , 1993 .

[3]  R. Morelli,et al.  Pick's theorem and the Todd class of a toric variety , 1993 .

[4]  David A. Cox The homogeneous coordinate ring of a toric variety , 2013 .

[5]  A. Klyachko EQUIVARIANT BUNDLES ON TORAL VARIETIES , 1990 .

[6]  Michèle Vergne,et al.  Lattice points in simple polytopes , 1997 .

[7]  C. Procesi,et al.  Cohomology of regular embeddings , 1990 .

[8]  M. Audin The Topology of Torus Actions on Symplectic Manifolds , 1991 .

[9]  W. Fulton,et al.  Intersection Theory: Ergebnisse Der Mathematik Und Ihrer Grenzgebiete; 3 Folge; Band 2 , 1984 .

[10]  Don Zagier,et al.  Higher dimensional dedekind sums , 1973 .

[11]  James Pommersheim,et al.  Toric varieties, lattice points and Dedekind sums , 1993 .

[12]  Sylvain E. Cappell,et al.  Genera of algebraic varieties and counting of lattice points , 1994, math/9401219.

[13]  R. Thomason XX. Algebraic K-Theory of Group Scheme Actions , 1988 .

[14]  W. Fulton Introduction to Toric Varieties. , 1993 .

[15]  T. Kawasaki The Riemann-Roch theorem for complex V -manifolds , 1979 .

[16]  Michael Atiyah,et al.  The moment map and equivariant cohomology , 1984 .

[17]  W. Graham,et al.  Localization in equivariant intersection theory and the Bott residue formula , 1995, alg-geom/9508001.

[18]  H. Bass,et al.  Linearizing certain reductive group actions , 1985 .

[19]  George Quart Localization theorem inK-theory for singular varieties , 1979 .

[20]  S. Cappell,et al.  Euler-MacLaurin expansions for lattices above dimension one , 1995 .

[21]  James Pommersheim Products of Cycles and the Todd Class of a Toric Variety , 1996 .

[22]  M. Brion Piecewise polynomial functions, convex polytopes and enumerative geometry , 1996 .

[23]  L. Billera The algebra of continuous piecewise polynomials , 1989 .