Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Building blocks of amplified endomorphisms of normal projective varieties

Abstract: Let X be a normal projective variety. A surjective endomorphism $$f{:}X\rightarrow X$$ is int-amplified if $$f^*L - L =H$$ for some ample Cartier divisors L and H. This is a generalization of the so-called polarized endomorphism which requires that $$f^*H\sim qH$$ for some ample Cartier divisor H and $$q>1$$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.

Equivariant K-theory

Abstract: The equivariant K-theory was developed by R. Thomason in [21]. Let an algebraic group G act on a variety X over a field F . We consider G-modules, i.e., OX -modules over X that are equipped with an G-action compatible with one on X. As in the non-equivariant case there are two categories: the abelian category M(G;X) of coherent G-modules and the full subcategory P(G;X) consisting of locally free OX -modules. The groups K ′ n(G;X) and Kn(G;X) are defined as the K-groups of these two categories respectively. In the second section we present definitions and formulate basic theorems in the equivariant K-theory such as the localization theorem, projective bundle theorem, strong homotopy invariance property and duality theorem for regular varieties. In the following section we define an additive category C(G) ofG-equivariant K-correspondences that was introduced by I. Panin in [15]. This category is analogous to the category of Chow correspondences presented in [9]. Many interesting functors in the equivariant K-theory of algebraic varieties factor through C(G). The category C(G) has more objects (for example, separable F algebras are also the objects of C(G)) and has much more morphisms than the category of G-varieties. For instance, every projective homogeneous variety is isomorphic to a separable algebra (Theorem 4.1). In section 4, we consider the equivariant K-theory of projective homogeneous varieties developed by I. Panin in [15]. The following section is devoted to the computation of the K-groups of toric models and toric varieties (see [12]). In sections 6 and 7, we construct a spectral sequence

Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence

Abstract: We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasi-transitive graphs, and operators on percolation graphs.