P. Bravi, D. Luna
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.
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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.
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.
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.
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 this paper, we will look at the algebra of global differential operators $D_X$ on wonderful compactifications $X$ of symmetric spaces $G/H$ of type $A_1$ and $A_2$. We will first construct a global differential operator on these varieties that does not come from the infinitesimal action of $\mathfrak{g}$. We will then focus on type $A_2$, where we will show that $D_X$ is an algebra of finite type, and that for any invertible sheaf ${\cal L}$ on $X$, $H^{0}(X,{\cal L})$ is either 0 or a simple left $D_{X,{\cal L}}$-module. Finally, we will show with the help of local cohomology that this is …
An infinite type toric variety is a normal toric variety given by a fan with infinitely many cones. We construct examples in this paper coming from representation theory of loop groups. The fans that appear are cones on Voronoi tilings on a vector space equipped with an inner product. We also explain the affine analogue of the connection between a generic torus orbit closure in a flag variety and the closure a maximal torus in the wonderful compactification.
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.
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 give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
We prove the existence of defective secant varieties of three-factor and four-factor Segre-Veronese varieties embedded in certain multi-degree. These defective secant varieties were previously unknown and are of importance in the classification of defective secant varieties of Segre-Veronese varieties with three or more factors.
We will give several descriptions of some basic examples of wild character varieties, including a discussion of links to work of Sibuya, Calabi and Euler, amongst others.
In this article we will introduce, among others, the variety of subcomplexes and the variety of maps between complexes of given rank. Also, varieties of $\mathfrak{g}$-structure like $\mathfrak{g}$-Grassmannian, $\mathfrak{g}$-determinantal varieties and finally the variety $\mathcal{DGLA}(E)$ of differential graded Lie algebra structures on $E$. We will compute the dimensions of these varieties and also some relevant properties. The motivation of this article is to give examples of moduli spaces relevant to deformation theory.
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.
We give various examples of Q-factorial projective toric varieties such that the sum of the squared torus invariant prime divisors is positive. We also determine the generators for the cone of effective $2$-cycles on a toric variety of Picard number two. This result is convenient to explain our examples.
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
These are notes from talks of the authors on some explicit examples of families of Calabi-Yau threefolds which are parametrised by a Shimura variety. We briefly review the periods of Calabi-Yau threefolds and we discuss a recent result on Picard-Fuchs equations for threefolds which are hypersurfaces with many automorphisms. Next various examples of families parametrised by Shimura varieties are given. Most of these are due to J.C. Rohde. The examples with an automorphism of order three are given in some detail. We recall that such families do not have maximally unipotent monodromy and that the…
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin's conjecture.
We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.
We give examples of smooth quasi-projective varieties over complex numbers, in the context of connected Shimura varieties, which are not homeomorphic to a conjugate of itself by an automorphism of the complex numbers.
Let $K$ be the fraction field of a Henselian discrete valuation ring with algebraically closed residue field $k$. In this article we give a sufficient criterion for a projective variety over such a field to have index $1$.
We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in di…
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
Generalize Kobayashi's example for the Noether inequality in dimension three, we provide examples of n-folds of general type with small volumes.
We find new examples of the $P=W$ identity of de Cataldo-Hausel-Migliorini by studying cluster varieties. We prove that the weight filtration of 2D cluster varieties correspond to the perverse filtration of elliptic fibrations which are deformation equivalent to the elliptic fibrations of types $I_b$ or $2I_b$. These examples do not arise from character varieties or moduli of Higgs bundles.
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anticanonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $α$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dim…
Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau varieties with the largest known index in high dimensions. In our examples, the index grows doubly exponentially with dimension. We conjecture that our examples have the largest possible index, with supporting evidence in low dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau varieties with an ample Weil divisor of small volume. We also g…
The purpose of this largely expository note is to give some examples of smooth projective varieties with zero first Betti number but nontrivial families of local systems of higher ranks. The most interesting examples are due to Simpson.
New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups
本文验证椭圆K3曲面和代数群是Javanpeykar-Rousseau意义下的几何特殊簇，并证明附加假设下移除余维≥2闭子集后该特殊性仍保持，提供了几何特殊簇的新例子。
Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a "differential valuative criterion" and using them to give new examples of complete differential varieties. W…
Characterizing subadjoint varieties among Legendrian varieties
本文利用迷向表示以及射影第三基本形式和勒让德簇上直线的特殊性质，刻画了具有正维自同构群的非奇异勒让德簇中的次伴随簇。
We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
We construct examples to show that having nef cotangent bundle is not preserved under finite ramified covers. Our examples also show that a projective manifold with Stein universal cover may not have nef cotangent bundle, disproving a conjecture of Liu-Maxim-Wang.
We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.
A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least four and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli space of stable, rank two bundles with fixed determinant of odd degree on a curve of genus at least three, is not birationally rigid.
Fano varieties with conjecturally largest Fano index
本文构造了具有典范和末端奇点的 Fano 簇中大 Fano 指标的示例，并基于低维证据猜想这些示例在所有维度上达到最大 Fano 指标。
On the integral Hodge conjecture for varieties with trivial Chow group
本文构造了全周群有限秩却违反整 Hodge 猜想的复光滑射影簇，并证明此类反例也可定义在数域上。
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