Vladimir L. Popov
It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found.
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In this paper we prove that the universal cover of a smooth projective variety with nilpotent fundamental group is holomorphically convex.
We give a classification of the principal and distinguished nilpotent pairs in all classical Lie algebras. As a classification of the principal pairs in the exceptional simple Lie algebras was obtained earlier (see Appendix to Ginzburg's preprint math.AG/9903059), we thus have a complete classification of principal nilpotent pairs.
We develop techniques to split the principal parts on the projective line over an arbitrary ring and apply these techniques to give a complete classification of the principal parts on the projective line over any field of characteristic zero.
Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a complex projective curve and constructed a projective moduli space of such objects. We generalize Ramanathan's notion and construction to the case of higher dimension, allowing also objects which we call semistable principal G-sheaves.
The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map
本文研究拟 Albanese 映射为真的正规拟射影簇及光滑拟 Kähler 簇，证明其基本群的无挠幂零商由拟 Albanese 像控制，并推出 special 簇的相应商为 Abel。
We prove that for any two projective symplectic resolutions $Z_1$ and $Z_2$ for a nilpotent orbit closure in a complex simple Lie algebra of classical type, then $Z_1$ is deformation equivalen to $Z_2$.
We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmon…
Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2$-bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.
Let X be a smooth projective complex variety, and let G be an algebraic reductive complex group. We define the notion of principal G-sheaf, that generalises the notion of principal G-bundle. Then we define a notion of semistability, and construct the projective moduli space of semistable principal G-sheaves on X. This is a natural compactification of the moduli space of principal G-bundles. This is the announcement note presented by the second author in the conference held at Catania (11-13 April 2001), dedicated to the 60th birthday of Silvio Greco. Detailed proofs will appear elsewhere.
Lectures on the construction on the moduli space of principal bundles, given in the Mini-School on Moduli Spaces at the Banach center (Warsaw) 26-30 April 2005.
The non-abelian Hodge correspondence for extensions on quasi-projective curves
本文为拟射影曲线上的对数平坦丛与对数希格斯丛建立精确等价，推广了温和辛普森-望月对应至幂零留数情形。
Two-step nilpotent monodromy of local systems on special varieties
研究显示，特殊变种上的局部系统单值群是虚拟 nilpotent 的，且阶不超过2，改进了Cadorel等人的定理，并发展了局部系统的变形理论。
Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e_1,e_2) consists of two commuting elements in \g that can independently be contracted to the origin. A principal nilpotent pair is a double counterpart of a regular nilpotent element. Consequently, the theory of nilpotent pairs should stand out as double counterpart of the theory of nilpotent orbits. We show that any nilpotent pair has a characteristic (h_1,h_2), whic…
Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this article, we give criterion for existence of a logarithmic connection on $E_G$ singular along $D$.
We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit $\bar O_{\mathrm{min}}$ of $\mathfrak{sp}_{2n}$, intersected with the Borel subalgebra $\mathfrak n_+$ of $\mathfrak{sp}_{2n}$, using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of $\mathfrak{sp}_{2n}$, which contains the maximal parabolic subalgebra $\mathfrak p$ determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algeb…
We study the splitting-type of the bi-modules of principal parts (Grothendiecks analogue of jet-bundles in algebraic geometry) as left and right O-module on the projective line in positive characteristic, and obtain explicit examples where the splitting-type as left O-module differs from the splitting-type as right O-module. Hence the principal parts are the first examples of a sheaf of abelian groups equipped with two non-isomorphic structures as locally free O-module. We also give explicit examples showing that in some cases the splitting-type changes with the characteristic of the base-fiel…
Counting Parabolic Principal G-bundles with Nilpotent Sections over $\mathbb{P}^{1}$
本文研究了抛物主G-丛在$\mathbb{P}^{1}$上的计数问题，通过显式公式和生成函数推导了相关结果。
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), …
A co-Higgs sheaf is a pair of a torsion-free coherent sheaf $\mathcal{E}$ and a global section of $\mathcal{E}nd(\mathcal{E})\otimes T_X$ with $T_X$ the tangent bundle. We construct $2$-nilpotent co-Higgs sheaves of rank two for some rational surfaces and of rank three for $\mathbb{P}^3$, using the Hartshorne-Serre correspondence. Then we investigate the non-existence, specially over projective spaces.
Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last sectio…
On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven
研究了光滑拟射影簇中 torsion-free nilpotent 基本群的结构，证明了其在特定秩条件下与某些 nilpotent Lie 群的同构关系，并回答了关于 nilpotent (quasi-)Kähler 群的问题。
Thick subcategories on weighted projective curves and nilpotent representations of quivers
本文研究加权射影曲线的导出范畴中厚子类别的结构，证明任何厚子类别要么等价于某些旗形的 nilpotent 表示导出范畴，要么是 torsion sheaves 的正交补，同时探讨了加权射影线的 admissible 子类别的生成条件及 Jordan-Holder 性质。
Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction
本文通过哈密顿约化方法，构造了单连通半单代数群幂零锥的收缩射影部分解，并实现了其通用泊松形变的基变换。
Let (X, ω) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, where G is a connected reductive linear algebraic group defined over C. Let Z (G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerica…
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal o…
Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξon M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomor…
Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $μ$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck compactification of the moduli space of $μ$--semistable principal $H$--bundles with fixed characteristic classes and describe its points. For large characteristic classes we show that the moduli space of $μ$--stable principal $H$--bundles is non--empty.
Let $Y_{1},\dots,Y_{l}$ be smooth irreducible projective curves and let $Y$ be its disjoint union. Given a semisimple reductive algebraic group $G$ and a faithful representation $ρ:G\hookrightarrow \textrm{SL}(V)$ we construct a projective moduli space of $(\underlineκ,δ)$-(semi)stable singular principal $G$-bundles with generalized parabolic structure of type $\underline{e}$. In case $Y$ is the normalization of a connected and reducible projective nodal curve $X$, there is a closed subscheme coarsely representing the subfunctor corresponding to descending bundles. We prove that the descent op…
In this article, we give a proof for a geometric presentation theorem for any irreducible scheme $X$ smooth projective over a discrete valuation ring $R$. As a consequence, for any reductive $R$-group scheme $\mathbf{G}$, we prove that any generically trivial principal $\mathbf{G}$-bundle over $X$ glued to a principal $\mathbf{G}_U$-bundle over the affine line $\mathbb{A}^1_U$ for a semilocal affine scheme $U$.
Let H be a closed subgroup of a linear algebraic group G defined over a field F. There is an equivalence of categories between the category of linear finite-dimensional representations of H and the category of finite rank G-homogeneous vectorbundles on G/H. In this short note we study this correspondence for the sheaves of principal parts (=jetbundles) of homogeneous linebundles on projective space in characteristic zero. We describe the representation corresponding to the principal parts of a homogeneous linebundle and as a corollary we obtain known formulas on the splitting type of the princ…
Let k be an algebraically closed field of characteristic zero. Let f:X-->S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable singular principal bundles on the fibres of f. This generalizes the result of A. Schmitt who studied the case when X is a nodal curve.
Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the complexification of its maximal compact subgroup $H\subset G$. We consider classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus $g\geq2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles over $X$, in contrast with the situation when $g=1$, and to show reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$ c…
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by χis numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he…
Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting nilpotent elements is called a pn-pair, if the dimension of their simultaneous centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition is satisfied. Ginzburg proved that many familiar results of the `ordinary' theory have analogues for pn-pairs. The aim of this article is to develop the theory of nilpotent pairs a bit further and to present some a…
Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that th…
Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.
Let M be an irreducible smooth projective variety defined over \bar{{\mathbb F}_p}. Let π(M, x_0) be the fundamental group scheme of M with respect to a base point x_0. Let G be a connected semisimple linear algebraic group over \bar{{\mathbb F}_p}. Fix a parabolic subgroup P \subsetneq G, and also fix a strictly anti-dominant character χof P. Let E_G \to M be a principal G-bundle such that the associated line bundle E_G(χ) \to E_G/P is numerically effective. We prove that E_G is given by a homomorphism π(M, x_0)\to G. As a consequence, there is no principal G-bundle E_G \to M such that degree…
Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$--bundles on $X$ determine uniquely the isomorphism class of $X$.
We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit closures $X$ such that $X=\check X$ where $\check X$ is the projective dual of $X$. We give algebraic criteria of projective self-duality for the considered orbit closures.
Let $X$ be a smooth irreducible projective curve. Recently, Pauly and Peón-Nieto shows that a vector bundle over $X$ is very stable if and only if the Hitchin map on the vector space of Higgs field on that vector bundle is proper. In this notes, we generalize this result to principal $G-$bundles for any semisimple linear algebraic group $G$. We also study the relation between very stability and other stability conditions in the case of $\text{SL}_2-$bundles.
Lie Algebroid Connections on Principal Bundles
本文定义了主丛上的李代数胚联络，研究了其在结构群扩张与约化下的性质，并给出了射影曲线上存在此类联络的判据。
Let B be a commutative $\mathbb{Z}$-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that $B_{(f)}$ is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d > 0, we give sufficient conditions that guarantee that every derivation of $B^{(d)} = \oplus_i B_{di}$ can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaiden…
Extremality of principal quiver Grassmannians
研究箭图格拉斯曼流形族，通过明确描述其平坦不可约轨迹，证明该族成员不可约且具预期维度的充要条件是周围箭图表示退化为特定直和。
Diamonds: Homology and the Central Series of Groups
研究通过群同调条件，证明非同构数域和超曲面的基本群可有同构的universal nilpotent quotients，揭示了伽罗瓦作用与nilpotent表示论的关系。
We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principa…
Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are…
PrIncipal quiver Grassmannians: conjectures
本文通过计算机实验提出关于引物格拉斯曼流形代数几何性质的猜想，探讨子表示的结构与维度关系。
For a simple complex Lie group G the connected components of the moduli space of G-bundles over an elliptic curve are weighted projective spaces. In this note we will provide a new proof of this result using the invariant theory of Kac-Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G.
Let $F$ be a local non-archimedian field and let $G$ be a group of points of a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set $X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component $M$ we construct an explicit unitary isomorphism $L^2(X_P)\to L^2(X_Q)$ (which depends on a choice of an additive character of $F$). The formula for the above isomorphism involves the action of the principal nilpotent element in the Langlands dual group of $M$ on the unipotent radicals of the corresponding dual parabolics. We use the above isomorphisms to define a …
The image of the principal minor map for n x n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n=4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in $\PP^{15}$. The latter is the main component in the singular locus of the 2 x 2 x 2 x 2-hyperdeterminant.
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