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Springer Science+Business Media New York (2020)

TANNAKIAN CLASSIFICATION OF EQUIVARIANT PRINCIPAL BUNDLES ON TORIC VARIETIES INDRANIL BISWAS

ARIJIT DEY

School of Mathematics Tata Institute of Fundamental Research Mumbai, India

Department of Mathematics Indian Institute of Technology-Madras Chennai, India

[email protected]

[email protected]

MAINAK PODDAR Mathematics Group Middle East Technical University Northern Cyprus Campus Guzelyurt, Mersin 10, Turkey and Department of Mathematics Indian Institute of Science Education and Research (IISER) Pune, India [email protected]

Abstract. Let X be a complete toric variety equipped with the action of a torus T , and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible Σ-filtered algebra associated to X, generalizing the notion of a compatible Σ-filtered vector space due to Klyachko, where Σ denotes the fan of X. We combine Klyachko’s classification of T -equivariant vector bundles on X with Nori’s Tannakian approach to principal G-bundles, to give an equivalence of categories between T -equivariant principal G-bundles on X and certain compatible Σfiltered algebras associated to X, when the characteristic of K is 0.

1. Introduction Let X be a toric variety, with fan Σ, under the action of a torus T , and let G be a reductive algebraic group; all are defined over an algebraically closed field K. A T -equivariant vector bundle E on X is a vector bundle on X endowed DOI: 10.1007/S00031-020-09557-5 Received June 8, 2018. Accepted May 8, 2019. Corresponding Author: Mainak Poddar, e-mail: [email protected]

INDRANIL BISWAS, ARIJIT DEY, MAINAK PODDAR

with a lift of the T -action which is linear on fibers. The T -equivariant vector bundles over a nonsingular toric variety were first classified by Kaneyama [21]. This classification result for toric vector bundles is up to isomorphism and it involves both combinatorial and linear algebraic data modulo an equivalence relation. Recently this work has been generalized for T -equivariant principal G-bundles [2], [3]; also see [4], [8], when K is the field C of complex numbers. In a foundational paper Klyachko gave an alternative description of equivariant vector bundles on arbitrary toric varieties (possibly non-smooth) defined over any algebraically closed field [13]. His correspondence gives an equivalence between the category VecT (X) of equivariant vector bundles on X and the category Cvec(Σ) of finite-dimensional vector spaces with collection of decreasing Z-graded filtrations, indexed by the rays of Σ, satisfying a certain compatibility condition. Klyachko used his classification theorem to compute the Chern characters and sheaf cohomology of equivariant vector bundles. As a major application, later he used his classification of equivariant vector bundles over P2 to prove Horn’s conjecture on eigenvalues of sums of Hermitian matrices [14]. Another interesting and more recent application of Klyachko’s work is a theorem of Payne [27] that the modul