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Toric varieties, lattice points and Dedekind sums. (English) Zbl 0789.14043

In this paper, we prove a formula for the Todd class of toric variety, which we use to obtain a formula for the number of lattice points in an arbitrary lattice tetrahedron, and a generalization of Rademacher’s three-term reciprocity formula for Dedekind sums. It is well known that the Chern classes of a nonsingular toric variety are expressed nicely as the sum of the classes of certain special subvarieties. For simplicial but possibly singular toric varieties, we use this same sum to define the mock Chern class, and then define the mock Todd class via the Todd polynomials. We prove that in the dimension of the singular locus, the difference between the actual Todd class and the mock Todd class has a local expression. The codimension two part of this difference is expressed explicitly in terms of Dedekind sums. In this way, we obtain an expression for the codimenson two part of the Todd class of an arbitrary toric variety given in terms of Dedekind sums.
A consequence of this formula is a formula for the number of lattice points in a general lattice tetrahedron \(\Delta\) given in terms of:
(1) The volume of \(\Delta\), the lattice areas of its faces, and the lattice lengths of its edges, and
(2) certain functions of the dihedral angles formed at each edge, computed in terms of Dedekind sums.
As a special case, we give a formula for the number of lattice points in the tetrahedron in \(\mathbb{Z}^ 3 \) with vertices at (0,0,0), \((a,0,0)\), \((0,b,0)\), and \((0,0,c)\) for arbitrary positive integers \(a,b,c\). This generalizes the lattice point formula of Mordell.
As another application, we prove a formula expressing the sum of two arbitrary Dedekind sums in terms of a third one. This formula is seen to be a generalization of Rademacher’s three-term reciprocity law for Dedekind sums. A consequence of our formula is an \(n\)-term reciprocity law for classical Dedekind sums.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
11F20 Dedekind eta function, Dedekind sums
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11H06 Lattices and convex bodies (number-theoretic aspects)
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References:

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