Topology of Kempf–Ness sets for algebraic torus actions
نویسنده
چکیده
In the theory of algebraic group actions on affine varieties, the concept of a Kempf–Ness set is used to replace the geometric quotient by the quotient with respect to a maximal compact subgroup. By making use of the recent achievements of “toric topology” we show that an appropriate notion of a Kempf–Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the “geometric invariant theory” approach to toric varieties. We proceed by studying the cohomology of these Kempf–Ness sets.
منابع مشابه
Toric Kempf–Ness sets
In the theory of algebraic group actions on affine varieties, the concept of a Kempf– Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology,” we show that an appropriate notion of a Kempf–Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arra...
متن کاملCategorically-algebraic topology and its applications
This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...
متن کاملToric Topology in the Work of Taras Panov
Since the 1970s, the study of torus actions has become increasingly important in various areas of pure mathematics, and has stimulated the formation of interdisciplinary links between algebraic geometry, combinatorial and convex geometry, commutative and homological algebra, differential topology and homotopy theory. As their net has spread wider, and the literature grown, a field of activity h...
متن کاملMoment Maps and Geometric Invariant Theory
These are expanded notes from a set of lectures given at the school “Actions Hamiltoniennes: leurs invariants et classification” at Luminy in April 2009. The topics center around the theorem of Kempf and Ness [58], which describes the equivalence between the notion of quotient in geometric invariant theory introduced by Mumford in the 1960’s [80], and the notion of symplectic quotient introduce...
متن کاملThe Kempf-ness Theorem and Invariant Theory
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
متن کامل