Prakash Belkale
We give a geometric proof of a conjecture of W. Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).
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We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.
We give a new geometric proof of a conjecture of Fulton on the Littlewood-Richardson coefficients. This conjecture was firstly proved by Knutson, Tao and Woodward using the Honeycomb theory. A geometric proof was given by Belkale. Our proof is based on the geometry of Horn's cones.
We present an alternate proof, much quicker and more straightforward than the original one, of a celebrated Fulton's conjecture on the ample cone of the moduli space of stable rational curves with n marked points in the case n=7.
Fulton's conjecture for the moduli space of stable pointed rational curves, \bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves is linearly equivalent to an effective sum of boundary divisors. Our main result is a proof of Fulton's conjecture for n=7. A key ingredient in the proof is an \binom{n}{4} dimensional-subspace of the Neron-Severi space of \bar{M}_{0,n}, defined by averages of Keel relations, for which we prove Fulton's conjecture for all n.
A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by considering the relative Fulton-MacPherson configuration space of the universal curve. The resulting compactification differs from the Deligne-Mumford space $\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and the Deligne-Mumford compactification are essentially the distinct minimal resolutions of the fiber product over $\overline{M}_g$ of the universal curve with itself.
The Littlewood-Richardson coefficients $c^ν_{λ,μ}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the triples of partitions $(λ, μ, ν)$ of length at most $n$. By the so-called Fulton conjecture, if $c^ν_{λ,μ}=1$ then $c^{kν}_{kλ,kμ}= 1$, for any $k \geq 0$. Similarly, as proved by Ikenmeyer or Sherman, if $c^ν_{λ,μ}=2$ then $c^{kν}_{kλ,kμ} = k + 1$, for any $k\geq 0$. Here, given a partition $λ$, we set $λ(p, q) = p(qλ')'$ , where prime denotes the conjugate partition. We o…
We give a new residual intersection decomposition for the refined intersection products of Fulton-MacPherson. Our formula refines the celebrated residual intersection formula of Fulton, Kleiman, Laksov, and MacPherson. The new decomposition is more likely to be compatible with the canonical decomposition of the intersection products and each term in the decomposition thus has simple geometric meaning. Our study is motivated by its applications to some geometric problems. In particular, we use the decomposition to find the distribution of limiting linear subspaces in degenerations of hypersurfa…
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
The (small) quantum cohomology ring of a flag manifold F encodes enumerative geometry of rational curves on F. We give a proof of the presentation of the ring and of a quantum Giambelli formula, which is more direct and geometric than the previously known proof. Furthermore, this proof gives a relationship between quantum Schubert polynomials and universal Schubert polynomials, which arise in a degeneracy locus formula of Fulton.
We provide a geometric proof of the Schubert calculus interpretation of the Horn conjecture, and show how the saturation conjecture follows from it. The geometric proof gives a strengthening of Horn and saturation conjectures. We also establish transversality theorems for Schubert calculus in non-zero characteristic. Some parts of the version posted in Nov 2002 (concerning explicit invariants constructed from Schubert calculus) have been removed from this version. They have appeared separately in IMRN 2004, no. 69, pages 3709--3721 " Invariant theory of GL(n) and Intersection theory of Grassma…
In their 2004 paper, Stretched Littlewood-Richardson and Kostka Coefficients, King, Tollu, and Toumazet conjectured that if a Littlewood-Richardson coefficient of value 2 is stretched by a factor of N, the resulting coefficient has value N+1. A generalization of this is proven using geometric methods and Schubert calculus.
We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotient…
This is a survey paper on Geometric Manin's conjecture which was proposed by Brian Lehmann and the author. We introduce Geometric Manin's conjecture (GMC) and review some recent progress on this conjecture.
Let X be a smooth, complete, geometrically connected curve over a field of characteristic p. The geometric Langlands conjecture states that to each irreducible rank n local system E on X one can attach a perverse sheaf on the moduli stack of rank n bundles on X (irreducible on each connected component), which is a Hecke eigensheaf with respect to E. In this paper we derive the geometric Langlands conjecture from a certain vanishing conjecture. Furthermore, using recent results of Lafforgue, we prove this vanishing conjecture, and hence the geometric Langlands conjecture, in the case when the g…
We use abelianization of Higgs bundles near infinity to prove the homotopy commutativity assertion of Simpson's geometric P=W conjecture in the Painlevé VI case.
We develop the notion of singular support of a coherent sheaf on a quasi-smooth DG scheme or stack and use it to formulate the Geometric Langlands Conjecture.
We outline a proof of the categorical geometric Langlands conjecture for GL(2), as formulated in reference [AG], modulo a number of more tractable statements that we call Quasi-Theorems.
The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.
Inspired by the Bloch-Beilinson conjectures, Voisin has formulated a conjecture concerning the behaviour of 0-cycles on self-products of varieties of geometric genus one. This note presents some new examples of surfaces for which Voisin's conjecture is verified.
Geometric Shafarevich boundedness conjecture for families of polarized varieties
本文研究极小稳定模型模栈上的几何沙法列维奇有界猜想，主要贡献是证明了该猜想的成立。
A geometric perspective on Amitsur's conjecture
本文从几何角度重新证明了罗quette关于塞维-布拉uer变种与阿米图尔猜想关系的结果，通过简单明确的birational同构提供新的证明方法。
We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric P=W conjecture in all Painlevé cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.
We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and degeneration techniques to study the topology of the dual boundary complex of certain character varieties. We also clarify the relation between the geometric and the cohomological P=W conjectures.
We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2-sphere.
We prove that the geometric Bogomolov conjecture holds for nowhere degenerate abelian varieties of dimension $5$ with trivial trace. By this result together with our previous work, we see that the conjecture holds for an abelian variety such that the difference between the dimension of its maximal nowhere degenerate abelian subvariety and that of its trace equals $5$.
This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on non-archimedean analytic geometry.
Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$. We propose a Geometric Manin's Conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.
Proof of the geometric Langlands conjecture I: construction of the functor
本研究在特征零情形下构造了从自守侧到谱侧的几何朗兰兹函子，证明了猜想的多种形式等价，并探讨了Hecke本征层的结构性质，推进了几何朗兰兹猜想的证明。
We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.
We prove the geometric Bogomolov conjecture over a function field of characteristic zero.
Proof of the geometric Langlands conjecture III: compatibility with parabolic induction
本研究证明几何朗兰兹猜想中朗兰兹函子与艾森斯坦级数常数项运算的相容性，得出该函子在艾森斯坦生成子范畴上诱导等价关系的结论，推进几何朗兰兹理论研究。
Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is obtained by resolving this map. Several structural results are proven, revealing a rather orderly structure of the graph of the curves at infinity. We also exhibit and discuss a graph that may lead to a counterexample to the Jacobian Conjecture.
We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from an incompatibility of geometric invariants.
Proof of the geometric Langlands conjecture V: the multiplicity one theorem
本文证明了几何朗兰兹猜想的多重性一定理，通过分析局部系统的堆栈几何，展示了唯一对应的海克 eigensheaf。
The Tate conjecture for surfaces of geometric genus one -- embracing singularities
本文引入处理“自然模型”奇异情形的技术，推进几何亏格一曲面的 Tate 猜想证明，并据此证明特征 $p\ge11$ 的亏格一整体函数域上高度一椭圆曲线满足 BSD 猜想。
Geometric Manin's conjecture in characteristic $p$
本文提出特征 $p$ 下几何 Manin 猜想的版本，并阐述其与整体域上 Batyrev-Manin-Peyre-Tschinkel 猜想的联系。
Geometric Manin's Conjecture for Fano 3-Folds
本文通过分类光滑 Fano 三维流形上的自由有理曲线族，证明了表示固定数值曲线类的非常自由有理曲线族不可约或为空，从而在三维情形下证明了几何 Manin 猜想，并给出了模空间分量个数的显式计数。
We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus. This result extends Green's classical result on the generation of the ideal of a canonical curve by rank four quadrics to the highest linear syzygy group.
The Dolbeault geometric Langlands conjecture via limit categories
该研究引入光滑栈余切栈的极限范畴，发展其理论并探讨与 D -模范畴关系，提出多贝尔几何朗兰兹猜想精确表述，证明极限范畴半正交分解，构造赫克算子，预期其蕴含半稳定希格斯丛 BPS 范畴等价，是拓扑镜像对称猜想的范畴版本。
This is a report of the author's talk at Kinosaki Algebraic Geometry Symposium 2018. We discuss some recent progress on the geometry of thin exceptional sets in Manin's Conjecture.
Let $X$ be a smooth complete curve, and let $Bun_n$ be the moduli stack of rank $n$ vector bundles on $X$. Let $E$ be a local system on $X$. In a recent paper of E.Frenkel, K.Vilonen and the author, it was shown that the vanishing of a certian functor $Av_E^d$ acting from the category $D(Bun_n)$ to itself, implies the geometric Langlands conjecture. In this paper we establish the required vanishing result. Our proof works for sheaves with char=0 coefficients, or with torsion coefficients when the parameter $d$ is less than the characteristic.
Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE
作为几何朗兰兹猜想证明系列五篇论文的第二篇，本文在临界水平下表述并证明基本局部等价（FLE），研究Kac-Moody局部化与整体几何朗兰兹函子的相互作用，还附含发展无限型无穷相干层理论与因子范畴形式体系的附录。
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.
We construct a quartic threefold with L-rational singularities which has torsion in its middle homology group. This answers a question of Brown and Schnetz for all fields of characteristic zero.
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections.
In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler…
Parshin's method and the geometric Bombieri-Lang conjecture
本文综述 Parshin 对几何 Bombieri–Lang 猜想的证明，并用其为 Xie–Yuan 关于空特殊轨迹且可有限映射至无迹阿贝尔簇的射影簇结果提供替代证明。
Geometric Bogomolov conjecture for semiabelian varieties
本文证明了函数域上半阿贝尔簇的几何Bogomolov猜想，刻画了闭子簇包含Zariski稠密小点集的充要条件，并揭示了特殊子簇可能缺乏高度为0点的Zariski稠密集这一新现象。
We propose a 'geometric Chevalley-Warning' conjecture, that is a motivic extension of the Chevalley-Warning theorem in number theory. It is equivalent to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this paper, we show the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in $\Pbb^3$. The last section is devoted to verifying the conjecture for certain special kinds of hypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine 'Pott…
The Geometric Syzygy Conjecture in Positive Characteristic
本文证明正特征情形的几何合冲猜想：对特征 p≥2g-4 的代数闭域上亏格 g 的一般光滑曲线，所有线性合冲空间都由最小秩合冲张成。
Generic Green's Conjecture and Generic Geometric Syzygy Conjecture in Positive Characteristic
本文研究正特征中典范曲线的合冲，在 p≥(g+4)/2 时重新证明一般 Green 猜想，并在 p>g 时确立一般偶亏格曲线最后合冲空间的几何合冲猜想关键情形。
Let $S$ be a smooth projective surface with $p_g=0$, let $ι$ be a regular involution acting on $S$, and let $W$ be the resolution of singularities of the quotient surface $S/ι$. In the paper we prove that Bloch's conjecture holds for the surface $S$ if and only if it holds for the surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the same conjecture is true for the desingularized quotient $W$. In particular, Bloch's conjecture holds true for all numerical Godeaux surfaces with involutions, a "half" of Campedelli surfaces with involutions, the surface of Craighero and Ga…
Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture
本文提出利用部分高度与 Brody 引理从无穷有理点序列构造全纯曲线的新方法，并在部分高度非退化猜想下证明了特征 0 函数域上具有到阿贝尔簇有限态射的双曲射影簇的几何 Bombieri-Lang 猜想。
Let $\Cal C,\Cal C'$ be curves over a base scheme $S$ with $g(\Cal C)\ge 2$. Then the functor $T\mapsto\{$generically smooth $T$-morphisms $T\times_S\Cal C'\to T\times_S\Cal C\}$ from $((S$-schemes)) to ((sets)) is represented by a quasi-finite unramified $S$-scheme. From this one can deduce that for any two integers $g\ge 2$ and $g'$, there is an integer $M(g,g')$ such that for any two curves $C,C'$ over any field $k$ with $g(C)=g$, $g(C')=g'$, there are at most $M(g,g')$ separable $k$-morphisms $C'\to C$. It is conjectured that the arithmetic function $M(g,g')$ is bounded by a linear functio…
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.
The study of infinitesimal deformations of a variety embedded in projective space requires that of deformations of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such a scheme We interpret conjectures of Froberg and Iarrobino on the Hilbert function of a general collection of infinitesimal neighbourhoods of a collection of points in projective space. A main result gives a method for verifying the Weak Conjecture prescribed. Further, we present results on establishing the Strong Con…
We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\ell$-adic Mumford-Tate conjecture.
The geometric cone conjecture in relative dimension two
本文证明了相对维数至多二的纤维化中，klt 配对的可动有效锥在保持边界除子的相对伪自同构作用下仅有有限多个 Mori 室和 Mori 面，从而验证了几何锥猜想。
The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties
该研究针对特征 0 函数域上映射到平凡迹 Abel 簇的射影簇，借助整曲线构造并以 Lie 代数显式刻画整曲线，证明了几何 Bombieri–Lang 猜想。
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