Publication

Relative Langlands duality of toric periods

2026

Abstract

The relative Langlands program introduced by Ben-Zvi–Sakellaridis–Venkatesh [5] posits a duality structure exchanging automorphic periods and L-functions, which can be encoded by pairs of dual Hamiltonian actions. In [7] an extension of the definitions to certain singular spaces was made with the objective of restoring duality in some well-known automorphic integrals. In this companion article we apply the definitions of loc. cit to establish duality in the context of affine toric varieties, and study finer structures regarding regularization that are instructive for the general case.

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2026

Abstract

Official source

Related concepts (28)

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.

Singularity theory

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string.

Toric variety

In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable.

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Relative Langlands duality of toric periods

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