Hussein Mourtada
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on the minimal resolution of $S.$
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Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.
For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give rise to projective systems, to which we associate a weighted graph. We prove that the data of this graph is equivalent to the data of the analytical type of $S.$ Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for $m$ large enough, the set of components with index of speciality $1,$ i…
Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is $k$-jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope, in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power $k+n-2$ of an ample line bund…
In this notes we study $k$-jet ample line bundles $L$ on a non singular toric variety $X$, i.e. line bundles with global sections having arbitrarily prescribed $k$-jets at a finite number of points. We introduce the notion of an associated $k$-convex $\D$-support function, $ψ_L$, requiring that the polyhedra $P_L$ has edges of length at least $k$. This translates to the property that the intersection of $L$ with the invariant curves, associated to every edge, is $\geq k$. We also state an equivalent criterion in terms of a bound of the Seshadri constant $\e(L,x)$. More precisely we prove the e…
We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.
The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson conjecture for polarized toric manifolds. Also, we show a combinatorial sufficient condition for uniform relative K-polystability.
In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.
We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.
We prove the following result of Bondal's: that there is a fully faithful embedding $κ$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $κ$ is an open problem.
For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical.
A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…
Logarithmic vector fields and foliations on toric varieties
该研究提出单纯环面簇除子上对数向量场层的环面版本，推广Saito自由性判定准则，明确其与普通对数向量场层及环面簇全纯叶状结构的关联，丰富了环面几何与对数向量场理论的研究。
We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter $β\in N \otimes {\mathbb C}.$ As an application, we construct a topological mirror symmetry map that produces a complete system of $Γ$--series solutions to the better behaved version of the GKZ hypergeometric system for $β\in N \otimes {\mathbb C}.$
In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.
Equivariant vector bundles on toric schemes over semirings
该研究定义了半环上概形的等变向量丛，证明幂等半域上环面概形的环面向量丛可等变分裂为线丛直和，还研究了等变皮卡群并给出对应Klyachko分类定理版本。
In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne using different techniques.
A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define a tropical version of a 1-tacnodal curve, that is, a curve having exactly one singular point whose topological type is $A_3$, and by applying the tropicalization method, we classify tropical curves which correspond to 1-tacnodal curves.
We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.
Stratification of Projection Maps From Toric Varieties
该研究证明了适用于复/实环面簇投影映射的Thom同痕引理组合版本，提出将投影Whitney分层与对应环面簇多面体面关联的构造性方法，且算法在映射分层计算上优于现有通用方法。
Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
本文证明环面Calabi-Yau 3-fold上开Gromov-Witten理论的两种open WDVV方程，分别构造半单Frobenius流形与平坦$F$-流形。
Toric extensions of Pólya's theorem
本文利用正环面几何技术，将Pólya定理推广到稀疏多项式，并给出逆命题，同时探讨了该方法在粒子物理Feynman积分收敛性研究中的应用。
Quantum cohomology and toric minimal model programs
该研究构造了紧环面轨形量子上同调的典范同构，推广了已有环面流形与加权射影空间相关结果，借助量子Kirwan满射性和环面极小模型纲领证明结论，并揭示量子上同调分解与奇点及哈密顿不可位移环面的对应关系。
We explicitly describe infintesimal deformations of cyclic quotient singularities that satisfy one of the deformation conditions introduced by Wahl, Kollár-Shepherd-Barron and Viehweg. The conclusion is that in many cases these three notions are different from each other. In particular, we see that while the KSB and the Viehweg versions of the moduli space of surfaces of general type have the same underlying reduced subscheme, their infinitesimal structures are different.
Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on $\textbf{A}^2.$ For this purpose, we show how one can recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation $v$ centered at the origin of $\textbf{A}^2,$ we construct an algebraic embedding $\textbf{A}^2\hookrightarrow \textbf{A}^N,N\geq 2$ such that $v$ is the trace of a monomial valuation on $\textbf{A}^N.$ We explain how results in this direction …
Studying toric varieties from a scheme-theoretical point of view leads to toric schemes, i.e. "toric varieties over arbitrary base rings". It is shown how the base ring affects the geometry of a toric scheme. Moreover, generalisations of results by Cox and Mustata allow to describe quasicoherent sheaves on toric schemes in terms of graded modules. Finally, a toric version of the Serre-Grothendieck correspondence relates cohomology of quasicoherent sheaves on toric schemes to local cohomology of graded modules.
A Poincaré-Lefschetz Theorem for Cellular Cosheaves and an Application to the Tropical Homology of Orbifold Toric Varieties
本文证明特定 CW 细分上胞腔余层的 Poincaré-Lefschetz 型定理，并据此将热带 Lefschetz 超平面截面定理推广至奇异热带环面簇与超曲面。
Given a lattice polytope Q in R^n, we define an affine scheme M(Q) that reflects the possibilities of splitting Q into a Minkowski sum. On the other hand, Q induces a toric Gorenstein singularity Y, and we construct a flat family over M(Q) with Y as special fiber. In case Y has an isolated singularity only, this family is versal. (This revised version contains the proof now.)
We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Ampère equation on certain polytopes. This does not rely on discrete symmetry.
In this paper we present a version of the general polynomial involutive algorithm for computing Janet bases specialized to toric ideals. The relevant data structures are Janet trees which provide a very fast search for a Janet divisor. We broach also efficiency issues in view of application of the algorithm presented to computation of toric ideals.
Equivariant non-archimedean Arakelov theory of toric varieties
本文为环面簇建立了等变非阿基米德Arakelov理论，将等变微分形式与流关联到多面体复形上的分段多项式函数，并给出Green流与算术Chow群的组合刻画。
Ardila and Block used tropical results of Brugalle and Mikhalkin to count nodal curves on a certain family of toric surfaces. Building on a linearity result of the first author, we revisit their work in the context of the Goettsche-Yau-Zaslow formula for counting nodal curves on arbitrary smooth surfaces, addressing several questions they raised by proving stronger versions of their main theorems. In the process, we give new combinatorial formulas for the coefficients arising in the Goettsche-Yau-Zaslow formulas, and give correction terms arising from rational double points in the relevant fam…
Hyperbola method on toric varieties
本文将 Blomer–Brüdern 双曲线法推广到完备光滑分裂环簇，用以计数有界对数反典范高度的 Campana 点，并借助线性规划强对偶性验证其与猜想渐近式相容。
Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials
本文通过代数方法证明了Demailly定理，展示了一般型复射影流形上存在大量Green-Griffiths jet微分，并引入新的代数Morse不等式，适用于正特征情况，证明了光滑射影流形上存在Hasse-Schmidt jet微分。
A weak version of the McKay correspondence for cyclic quotient singularities
本文将McKay对应弱版本从SL(n,C)有限子群推广至GL(n,C)有限循环群，利用连分数所得环面解消构建对应，证明循环商奇点的相关欧拉数与群共轭类数的关系。
Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds
本文利用Witt层上同调的虚拟局部化方法，证明了取值于实Witt环的(P¹)³二次Donaldson-Thomas不变量生成级数为MacMahon函数M(q²)⁻⁸，证实修正的Viergever猜想，并证明其局部化版本对(P¹)³某些迭代胀开及相关光滑proper托里拆利簇成立。
We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parametrizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build insta…
The Nekrasov conjecture predicts a relation between the partition function for N=2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R^4, the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.
In this article, we introduce the generalized complexity of a generalized Calabi--Yau pair $(X,B,\textbf{M})$. This invariant compares the dimension of $X$ and Picard rank of $X$ with the sum of the coefficients of $B$ and $\textbf{M}$. It generalizes the complexity introduced by Shokurov. We show that a generalized log Calabi-Yau pair $(X,B,\textbf{M})$ of dimension $2$ with generalized complexity $0$ satisfies that $X$ is toric. This generalizes a result due to Brown, McKernan, Svaldi, and Zhong in the case of surfaces. Furthermore, we show that a generalized klt log Calabi-Yau surface $(X,\…
Let X be the toric scheme over a ring R associated with a fan Sigma. It is shown that there are a group B, a B-graded R-algebra S and a graded ideal I of S such that there is an essentially surjective, exact functor ~ from the category of B-graded S-modules to the category of quasicoherent O_X-modules that vanishes on I-torsion modules and that induces for every B-graded S-module F a surjection Xi_F from the set of I-saturated graded sub-S-modules of F onto the set of quasicoherent sub-O_X-modules of ~F. If Sigma is simplicial, the above data can be chosen such that ~ vanishes precisely on I-t…
A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2
本文针对皮卡秩为2的光滑射影环面簇，用线丛直和构成的线性复形分解对角层，并据此证明 Berkesch-Erman-Smith 猜想的新情形、建立 Horrocks 型分裂判别准则，以及给出 Orlov 关于 Rouquier 维数猜想的新证明。
Bulk-deformed enumerative mirror symmetry for toric Fano surfaces
本文证明环面Fano曲面在大量子上同调层面的亏格0镜像对称，通过热带方法建立Jacobian环与大量子上同调的同构，给出配边论证的纯代数类比。
The geometric Merkurjev-Panin Conjecture for the Cox category
本文证明了Cox范畴中几何Merkurjev-Panin猜想的强版本成立，揭示了该范畴在研究代数簇同调代数中的重要性。
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of canonical toric singularities. We do this with an eye towards understanding the global structure and discriminant loci of such fibrations. The paper ends with some speculation, both about local mirror symmetry, and the connections of this work with the work of W.D. Ruan and D. Joy…
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