Simple Immersions of Wonderful Varieties

An introduction to wonderful varieties with many examples of type F4

We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms between them, and on their relations with the representation theory of G.

绝妙对称簇上极小有理切线簇的图表示

Diagrams for varieties of minimal rational tangents on the wonderful symmetric varieties

本研究针对绝妙对称簇上的极小有理切线簇的表示问题，通过带标记Dynkin图进行刻画，证明当限制根系非A型时，由极小有理切线簇不可约分支得到的带标记Dynkin图与由带标记Kac图标记相邻节点得到的图一致。

Standard monomial theory for wonderful varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.

从 Poisson 几何视角看 wonderful 簇

Wonderful varieties with a view towards Poisson geometry

本文是一篇面向初学者的扩展讲义，介绍 wonderful 簇与半伴随群的 wonderful 紧化，并阐述其与 Poisson 几何及 Lagrange 子代数簇的联系。

Wonderful varieties of type D

Let G be a complex connected semisimple group, whose simple components have type A or D. We prove that wonderful G-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type A; thanks to another result of Luna, this implies also the classification of all spherical G-varieties for the groups G we are considering. For these G we also prove the smoothness of the embedding of Demazure.

The total coordinate ring of a wonderful variety

We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings.

完备对称簇上的极小有理曲线

Minimal rational curves on complete symmetric varieties

本研究刻画完备对称簇上的极小有理曲线族与对应VMRT，证明其齐性，明确非例外不可解wonderful簇的唯一性，并结合限制根系否定Hwang的相关问题。

Wonderful subgroups of reductive groups and spherical systems

Let G be a semisimple complex algebraic group, and H a wonderful subgroup of G. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to consider a spherical system S as a datum defined by purely combinatorial axioms, and under certain circumstances our results prove the existence of a wonderful subgroup H associated to S. As a byproduct, we reduce for any group G the proof of the classification of wonderful G-varieties, known as the Luna conjecture, to its verification on a small family of cas…

Wonderful Compactification of Character Varieties

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the étale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these condition…

Projective normality of model varieties and related results

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spa…

全非负奇妙紧化的正则性

Regularity of Totally Nonnegative Wonderful Compactification

针对伴随型分裂实半单群的奇妙紧化全非负部分，证明其胞腔分解为正则CW复形，证实了Galashin等人的猜想，所用方法结合了Bruhat atlas与CL-标号。

Frobenius splitting and geometry of $G$-Schubert varieties

Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, proj…

Normal projectivity of complete symmetric varieties

Chiriv\`ı and Maffei \cite{CM II} have proved that the multiplication of sections of any two ample spherical line bundles on the wonderful symmetric variety $X=\bar{G/H}$ is surjective. We have proved two criterions that allows ourselves to reduce the same problem on a (smooth) complete symmetric variety to the corresponding problem on the complete toric variety (respectively to the open toric variety). We have also studied in details some family of complete symmetric varieties, in particular those of rank 2.

Primitive spherical systems

A spherical system is a combinatorial object, arising in the theory of wonderful varieties, defined in terms of a root system. All spherical systems can be obtained by means of some general combinatorial procedures (such as parabolic induction and wonderful fiber product) from the so-called primitive spherical systems. Here we classify primitive spherical systems. As an application, we prove that the quotients of a spherical system are in correspondence with the so-called distinguished subsets of colors.

Construction of equivariant vector bundles

Let $X$ be the wonderful compactification of a complex adjoint symmetric space $G/K$ such that $rk(G/K)=rk(G)-rk(K)$. We show how to extend equivariant vector bundles on $G/K$ to equivariant vector bundles on $X$, generated by their global sections and having trivial higher cohomology groups. This relies on a geometric construction of equivariant vector bundles in the setting of varieties with reductive group action and ``multiplicity-free'' subvarieties.

The ring of sections of a complete symmetric variety

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators for Pic(X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit closures in X. This gives a degeneration result and the rational singularityness for A(X).

齐次纤维丛在 Schubert 簇中的应用

Applications of Homogeneous Fiber Bundles to the Schubert Varieties

本文利用旗簇上齐次纤维丛框架研究 Schubert 簇与等变嵌入的关系，证明了相关簇的环面性质，识别了奇妙簇，并给出了光滑球面 Schubert 簇的刻画及其局部刚性等应用。