We study various geometrical quantities for CalabiYau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the SasakiEinstein base of the corresponding CalabiYau cone are calculated. By doing so, we conjecture new bounds for the SasakiEinstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.

In this paper, we study the <i></i>-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that <i></i>-ordinary locus is a union of certain maximal EkedahlKottwitzOortRapoport strata introduced in [12] and we give criteria on the density of the <i></i>-ordinary locus.

In this paper, we describe all (2,3)-torus structures of a highly symmetric 39-cuspidal degree 12 curve. A direct computer-aided determination of these torus structures seems to be out of reach. We use various quotients by automorphisms to find torus structures. We use a height pairing argument to show that there are no further structures.

Let X be a compact K\" ahler manifold and X be a simple normal crossing divisor. If X is the support of some effective X -ample divisor, we show <div class="gsh_dspfr"> X

By a case-free approach we give a precise description of the closure of a Steinberg fiber within a twisted wonderful compactification of a simple linear algebraic group. In the nontwisted case this description was earlier obtained by the first author.