作者:Gavril Farkas,Ian Morrison
版权页： 插图： is a projective equivariant completion of G/NG(H), called the Demazure embedding of that homogeneous space. In fact, the variety X is wonderful by a result of Losev (see ) based on earlier results of several mathematicians, including Demazure and Knop (see [33, Corollary 7.2]). Moreover, by embedding theory of sphericalhomogeneous spaces, the log homogeneous embeddings of G/H are exactly those smoothequivariant embeddings that admit a morphism to X； then the logarithmic tangent bundleis the pull-back of the tautological quotient bundle on Gr(g). Also, by embedding theoryagain, a complete log homogeneous variety X′ is wonderful if and only if the morphismX′→X is finite. It follows that every spherical homogeneous space G/H such that H = NG(H) admits a wonderful equivariant completion； in the converse direction, if G/H admits such a completion X, then X is unique, and the quotient NG(H)/H is finite. In particular, the center of G acts on X via a finite quotient； thus, one can assume that G is semi-simple when considering wonderful G-varieties. Since the G-variety Cr(g) contains only finitely many isomorphism classes of spherical G-orbits, and any G-homogeneous space admits only finitely many finite equivariant coverings, we see that the number of isomorphism classes of wonderful G-varieties is finite (for a given group G). Also, note that the wonderful varieties are exactly those log homogeneous varieties that are log Fano, i.e., the determinant of the logarithmic tangent sheaf is ample. To classify wonderful G-varieties, it suffices to characterize those triples (A,V,D) that occur as combinatorial invariants of their open G-orbits, in view of Losev's uniqueness result. In fact, part of the information contained in such triples is more conveniently encoded by abstract combinatorial objects called spherical systems.
作者：（德国）法卡斯（Gavril Farkas） （美国）莫里森（Ian Morrison）
Volume Ⅰ Preface Gavril Farkas and Ian Morrison Logarithmic geometry and moduli Dan Abramovich, Qile Chen, Danny Cillam, Yuhao Huang, Martin Olsson, Matthew Satriano and Shenghao Sun Invariant Hilbert schemes Michel Brion Algebraic and tropical curves： comparing their moduli spaces Lucia Caporaso A superfiaal working guide to deformations and moduli F. Catanese Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes Norman Do Equivariant geometry and the cohomology of the moduli space of curves Dan Edidin Tautological and non-tautological cohomology of the moduli space of curves C. Faber and R. Pandharipande Alternate compaaifications of moduli spaces of curves Maksym Fedorchuk and David Ishii Smyth The cohomology of the moduli space ofAbelian varieties Gerard van der Geer Moduli of K3 surfaces and irreduable symplectic manifolds V Gritsenko, K. Hulek and C.K. Sankaran Normal functions and the geometry of moduli spaces of curves Richard Hain Volume Ⅱ Parameter spaces of curves Joe Harris Global topology of the Hitchin system Tamas Hausel Differential forms on singular spaces, the minimal model program, and hyperboliaty of moduli stacks Stefan Kebekus Contractible extremal rays on Mo,n Sean Keel and James McKernan Moduli of varieties of general type Janos Kollar Singularities of stable varieties Sandor J Kovacs Soliton equations and the Riemann-Schottky problem I. Krichever and T. Shiota GIT and moduli with a twist Radu Laza Good degenerations of moduli spaces Jun Li Localization in Gromov-Witten theory and Orbifold Gromov-Witten theory Chiu-Chu Melissa Liu From WZW models to modular functors Eduard Looijenga Shimura varieties and moduli J.s. Milne The Torelli locus and special subvarieties Ben Moonen and Frans Oort Volume Ⅲ Birational geometry for nilpotent orbits Yoshinori Namikawa Cell decompositions of moduli space, lattice points and Hurwitz problems Paul Norbury Moduli of abelian varieties in mixed and in positive characteristic Frans Oort Local models of Shimura varieties, I. Geometry and combinatorics Georgios Pappas, Michael Rapoport and Brian Smithling Generalized theta linear series on moduli spaces of vector bundles on curves Mihnea Popa Computer aided unirationality proofs of moduli spaces Frank-Olaf Schreyer Deformation theory from the point of view of fibered categories Mattia Talpo and Angelo Vistoli Mumford's conjecture-a topological outlook Ulrike Tillmann Rational parametrizations of moduli spaces of curves Alessandro Verra Hodge loci Claire Voisin Homological stability for mapping class groups of surfaces Nathalie Wahl
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