S. Cupit-Foutou
A geometrical realization of wonderful varieties by means of a suitable class of invariant Hilbert schemes is given. As a consequence, Luna's conjecture asserting that wonderful varieties are classified by spherical systems, triples of combinatorial invariants, is proved.
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In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits or model spaces. To make the paper self-contained as much as possible, we shall gather some known results on these families and more generally on wonderful varieties.
Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample.
Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group of X and we calculate Luna's invariants of X under its action.
The invariant Hilbert schemes considered in \cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of these invariant Hilbert scheme by means of deformation theoretical arguments. As a consequence we recover in a shorter way the classification of strict wonderful varieties. This provides an alternative and new approach to answer Luna's conjecture.
We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and report the references where some of them have already been studied. Finally, we analyze the rest case-by-case.
We show that the proof of Luna's conjecture about the classification of general wonderful G-varieties can be reduced to the analysis of finitely many families of primitive cases. We work out all primitive cases arising with any classical group G.
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.
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。
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.
This work is a PhD thesis. First we provide some general context on wonderful varieties and moduli spaces of rational curves. Working over complex numbers we prove that the moduli space of rational curves with no marked points on the wonderful compactification of a symmetric space is not irreducible in general. Lastly we show that in the case of wonderful group compactifications the set of rational curves with no marked points and irreducible source is irreducible.
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。
Curves of best approximation on wonderful varieties
本文给出了 $A_n$ 型半单李群伴随型奇妙紧化的 Coba 猜想的无条件证明，并证明了任意李群伴随型奇妙紧化的一个稍弱猜想。
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.
We study equivariant real structures on spherical varieties. We call such a structure canonical if it is equivariant with respect to the involution defining the split real form of the acting reductive group G. We prove the existence and uniqueness of a canonical structure for homogeneous spherical varieties G/H with H self-normalizing and for their wonderful embeddings. For a strict wonderful variety we give an estimate of the number of real form orbits on the set of real points.
We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.
Vanishing theorems on wonderful varieties
本文研究了奇妙簇上重言丛的消没定理，证明了Brieskorn结果的类似物及Orlik--Solomon代数的比较定理，扩展了Borel--Weil--Bott型消没定理，并将White基猜想的弱版本归约为上同调消没问题。
Let $\mathbf{G}$ be a connected reductive complex algebraic group with split real form $(G,σ)$. Consider a strict wonderful $\mathbf{G}$-variety $\bf{X}$ equipped with its $σ$-equivariant real structure, and let $X$ be the corresponding real locus. Further, let $E$ be a real differentiable $G$-vector bundle over $X$. In this paper, we introduce a distribution character for the regular representation of $G$ on the space of smooth sections of $E$, and show that on a certain open subset of $G$ of transversal elements it is locally integrable and given by a sum over fixed points.
$K$-rings of wonderful varieties and matroids
本文通过超平面构形奇妙簇的$K$-环给出仅依赖底层拟阵的组合表示，并构造到拟阵Chow环的例外同构及HRR型公式，推广了置换多面体簇的已有结果，并应用于线丛欧拉示性数及增广奇妙簇和稳定曲线模空间。
We describe the equivariant Chow ring of the wonderful compactification $X$ of a symmetric space of minimal rank, via restriction to the associated toric variety $Y$. Also, we show that the restrictions to $Y$ of the tangent bundle $T_X$ and its logarithmic analogue $S_X$ decompose into a direct sum of line bundles. This yields closed formulae for the equivariant Chern classes of $T_X$ and $S_X$, and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of the results of Can-Joyce-Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.
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。
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…
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…
Algebraic cobordism rings of wonderful varieties and matroids
本文给出了无环拟阵的Bergman扇的环面簇的代数配边环的两个组合表示，并证明了其与拟阵的Chow环和张量积的同构，推广了Chow环与K-环之间的积分同构。
In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger group).
Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so.
In \cite{MW}, B. Moonen and the author defined a new invariant, called $F$-Zips, of certain varieties in positive characteristics. We showed that the isomorphism classes of these invariants can be interpreted as orbits of a certain variety $Z$ with an action of a reductive group $G$. In loc. cit. we gave a combinatorial description of the set of these orbits. In this manuscript we give an explicit combinatorial recipe to decide which orbits are in the closure of a given orbit. We do this by relating $Z$ to a semi-linear variant of the wonderful compactification of $G$ constructed by de Concini…
We construct wonderful compactifications of the spaces of linear maps, and symmetric linear maps of a given rank as blow-ups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps.
Grothendieck Weights on Permutohedral Varieties and Matroids
本文研究排列胞菌簇上的Grothendieck权重，证明了K-平衡条件和环结构乘法规则，并应用于Matroids，计算了超平面排列补集的motivic Chern类，证明其仅依赖于Matroid而非具体实现。
Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII
本文利用Delcroix的K-多稳定性组合判据，证明了AIII型对称齐性空间的绝妙紧化 $X_m$ 对所有 $m \geq 4$ 均容许Kähler-Einstein度量，而其沿唯一闭轨道的blowup $Y_m$ 仅在 $m=4,5$ 时容许该度量。
We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
Multiprojective Seshadri stratifications for Schubert varieties and standard monomial theory
本文借助 Seshadri 分层，为 Lakshmibai-Seshadri 表及标准单项式基建立几何解释，并通过 Schubert 簇多齐次坐标环的滤过得到由标准表索引的一维子商。
Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $δ$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a…
Let $X$ be a $\Q$-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group ${\rm Pic}(X)$ in the group ${\rm Cl}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of ${\rm Pic}(X)$ in ${\rm Cl}(X)$ is contained in a free part of the latter group.
Principal polarizations on products of abelian varieties over finite fields
本文改进并推广有限域上阿贝尔簇乘积主极化的已有结果，分析通常与超奇异阿贝尔簇乘积同源类中不可约主极化缺失的原因，并给出几何单阿贝尔曲面与椭圆曲线情形的存在条件，证明相应阿贝尔三维簇或其二次扭曲为 Jacobian。
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …
Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$ its generic fiber. We show, under some assumptions on $\mathcal X_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}(\mathcal X_K, \mathbb Z(d)) \simeq π_1^{ab}(\mathcal X_K)_{W}$ from a new cohomology theory to an integral model $π_1^{ab}(\mathcal X_K)_{W}$ of the abelianized geometric fundamental groups $π_1^{ab}(\mathcal X_…
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