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Register a new userOn Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties
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We explicate the combinatorial/geometric ingredients of Arthurs proof of the convergence and polynomiality, in a truncation parameter, of his non-invariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthurs results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthurs work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence-Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.
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We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y-{0})/C* such that the toric residue becomes the local residue at 0 in Y.
In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billeys formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of root systems (generalizing
Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.
As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of $SL_2(Z)$, Hijikata defines and characterizes the notion of a split order in $M_2(k)$, where $k$ is a local field. In this paper, we generalize the notion of a split order to $M_n(k)$ for $n>2$ and give a natural geometric characterization in terms of the affine building for $SL_n(k)$. In particular, we show that there is a one-to-one correspondence between split orders in $M_n(k)$ and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the $n=2$ case in which split orders correspond to geodesics in the tree for $SL_2(k)$ with the split order given as the intersection of the endpoints of the geodesic.
We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.
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