Integral affine structures on spheres III: complete intersections

Coupled Painlevé III system with affine Weyl group symmetry of type $D_6^{(1)}$

We find and study a six-parameter family of coupled Painlevé III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types $B_5^{(1)}$ and $D_5^{(2)}$.

Toric Intersection Theory for Affine Root Counting

Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for o…

Characteristic classes of affine varieties and Plucker formulas for affine morphisms

An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of $V$. We call this limit the affine cohomology of $V$ and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make t…

Coupled Painlevé III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$

We find and study four kinds of a 4-parameter family of four-dimensional coupled Painlevé III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$. We also show that these systems are equivalent by an explicit birational and symplectic transformation, respectively.

Pursuing the double affine Grassmannian III: Convolution with affine Zastava

This is the third paper of a series (started by arXiv:0711.2083, arXiv:0908.3390) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian and affine Zastava.

Coupled Painlevé III systems with affine Weyl group symmetry of types $B_5^{(1)},D_5^{(1)}$ and $D_6^{(2)}$

We find and study four kinds of five-parameter family of six-dimensional coupled Painlevé III systems with affine Weyl group symmetry of types $D_5^{(1)},B_5^{(1)}$ and $D_6^{(2)}$. We show that each system is equivalent by an explicit birational and symplectic transformation, respectively. We also show that we characterize each system from the viewpoint of holomorphy.

Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety

This paper describes a paving by affines for regular nilpotent Hessenberg varieties in all Lie types, namely a kind of cell decomposition that can be used to compute homology despite its weak closure conditions. Precup recently proved a stronger result; we include ours because we use different methods. We then use this paving to prove that the homology of the Peterson variety injects into the homology of the full flag variety. The proof uses intersection theory and expands the class of the Peterson variety in the homology of the flag variety in terms of the basis of Schubert classes. We explic…

非常仿射完全交的拉格朗日骨架

Lagrangian skeleta of very affine complete intersections

研究$(\mathbb{C}^*)^n$内完全交$Z^\circ$的拉格朗日骨架，通过计算其骨架并描述分解，应用结果在大体积极限下为巴蒂列夫 - 鲍里索夫对建立同调镜像对称，还证明等价性与环面HMS兼容及相关函子性质。

仿射曲面的斜驶自同构的轨道的交

Intersection of orbits of loxodromic automorphisms of affine surfaces

该研究证明域上仿射曲面的两个斜驶自同构若轨道有无穷多交点则必有公共迭代，借助前期仿射曲面自同态动力学工作与算术动力学论证，还得到乘积空间中曲面的动力学Mordell-Lang型结果。

Birational Mori fiber structures of $\mathbb{Q}$-Fano $3$-fold weighted complete intersections, III

This is a continuation of a series of papers studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consists of birationally rigid varieties and $14$ families consists of birationally birigid varieties. The aim of this paper is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.

关于单项式曲线的完全交射影闭包

On complete intersection projective closures of monomial curves

研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。

The semi-infinite intersection cohomology sheaf

We introduce the semi-infinite category of sheaves on the affine Grassmannian, and construct a particular object in it, which we call the the semi-infinite intersection cohomology sheaf. We relate it to several other entities naturally appearing in the geometric Langlands theory.

Transformée de Mellin des intégrales- fibres associées à l'intersection complète non-dégénérée

Mellin transform of fibre integral is calculated for certain classes of non-degenerate affine complete intersections. The lattice structure of the poles of the Mellin transform is clarified by means of the mixed Hodge structure of the affine variety in question. One establishes the relation of the fibre integral with the hypergeometric function of Horn and that of Gel'fand-Kapranov-Zelevinski. As an application we prove an hypothesis proposed by Berglund, Candelas et altri on the mirror symmetry.

Intersection Pairings for Higher Laminations

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the m…

Affine representability results in A^1-homotopy theory III: finite fields and complements

We give a streamlined proof of ${\mathbb A}^1$-representability for $G$-torsors under "isotropic" reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in ${\mathbb A}^1$-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.

仿射球面质量与热带双扇的高斯运动学公式

Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans

本文在秩四格中引入仿射球面质量并证明其与稳定自交和最小次数的双边不等式，结合高斯-运动学公式与仿射规范化原理，给出四维横截性估计及协变性，并探讨最优常数问题。

On the structure of homogeneous symplectic varieties of complete intersection

If X is a symplectic variety emedded in an affine space as a complete intersection of homogeneous polynomials, then X coincides with the nilpotent variety of a semisimple Lie algebra.

Cylinders over affine surfaces

For an affine variety $S$ we consider the ring $AK(S),$ which is the intersection of the rings of constants of all locally-nilpotent derivations of the ring $\Cal {O}(S).$ We show that $AK(S\times\Bbb {C}^n)=AK(S)$ for a smooth affine surface $S$ with $H^2(S,\Bbb {Z})=\{0\}.$

到带奇点的积分仿射流形的热带收缩

Tropical contractions to integral affine manifolds with singularities

本文在 Gross--Siebert 纲领中研究 Batyrev--Borisov 型 Calabi--Yau 完全交的环面退化，证明其对偶交复形包含于热带化所得热带簇，并构造保持热带上同调且将特征波映到辐射阻碍的积分仿射收缩映射。

On toric varieties which are almost set-theoretic complete intersections

We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.

On binomial set-theoretic complete intersections in characteristic p

Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$.

Minimum distance functions of complete intersections

We study the footprint function, with respect to a monomial order, of complete intersection graded ideals in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we give a formula for the footprint function and a sharp lower bound for the corresponding minimum distance function. This allows us to recover a formula for the minimum distance of an affine cartesian code and the fact that in this case the minimum distance and the footprint functions coincide. Then we present an extension of a result of Alon and Füredi, a…

Equivariant geometry of odd-dimensional complete intersections of two quadrics

Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants -- both classical cohomological invariants and recent symbol constructions -- control rationality in some cases.

反向 adjunction 公式对于商奇点 III：半不变情况

Inversion of adjunction for quotient singularities III: semi-invariant case

本文证明了有限线性群商的完全交集变种中半不变方程的反向 adjunction 公式，并证明了这些变种的最小 log 可辨性半连续性，扩展了前文关于不变方程的结果。

Toric Degenerations and Batyrev-Borisov Duality

This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersecti…

The density of integral points on complete intersections

In this paper, an upper bound for the number of integral points of bounded height on an affine complete intersection defined over $\mathbb{Z}$ is proven. The proof uses an extension to complete intersections of the method used for hypersurfaces by Heath-Brown, the so called q-analogue of van der Corput's AB process.

通过热带收缩的非阿基米德SYZ纤维化

Non-archimedean SYZ fibrations via tropical contractions

本文通过热带收缩映射，证明了Batyrev-Borisov完全交的环退化中，对偶相交复形与Berkovich解析化的基本骨架同构，并构造了仿射环面纤维化。

Complete intersection monomial curves and the Cohen-Macaulayness of their tangent cones

Let $C({\bf n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C({\bf n}+w{\bf v})$.

Intersection Homology of Toric Varieties and a Conjecture of Kalai

We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.

On superheight conditions for the affineness of open subsets

In this paper we consider the open complement U of a hypersurface Y=V(a) in an affine scheme X. We study the relations between the affineness of U, the intersection of Y with closed subschemes, the property that every closed surface in U is affine, the property that every analytic closed surface is Stein and the superheight of the defining ideal a.

Localization of $\frak{u}$-modules. I. Intersection cohomology of real arrangements

This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a root of unity. In the present paper we study perverse sheaves over a complex affine space which are smooth along the stratification determined by a finite arrangement of complex affine hyperplanes defined by real equations. In particular, we construct explicitely (in terms of combinatorial data) complexes computing cohomology of Goresky-MacPherson extensions o…

关于奇变分的奇异品种III

On infinitesimal deformations of singular varieties III

本文研究了通过粘合三条射影线形成的奇异曲线的仿射锥，并证明其线丛非常量，从而嵌入到$\mathbb{P}^9$中。

光滑完全交射影簇上锥体的最小指数

The minimal exponent of cones over smooth complete intersection projective varieties

本文计算了光滑射影超曲面横截完全交上仿射锥的最小指数，在加权齐次情形下证明上界，并利用强因子分解消解推导下界。

多项式在某些仿射超曲面非负部分上的非负性

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

研究多项式在特定仿射超曲面非负部分上的非负性，证明严格正多项式可由正系数多项式表示，推广Pólya定理。

Popov-Vinberg 簇的交点上同调

Intersection cohomology of Popov-Vinberg varieties

针对单连通分裂半单代数群 G 的 Popov-Vinberg 簇，提出递归计算其交点上同调的方法，并重点分析 G=SL_n 的情形。

Intersections of two Grassmannians in $\mathbf{P}^9$

We study the intersection of two copies of $\mathrm{Gr}(2,5)$ embedded in $\mathbf{P}^9$, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi-Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi-Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by…

两个二次曲面交集的强逼近

Strong approximation for the intersection of two quadrics

本文研究两个仿射二次曲面交集的强逼近问题，并将其应用于证明秩四数域上非分裂纤维由二次扩张分裂情形下的弱逼近纤维化方法。

Intersection complexes and unramified $L$-factors

Let X be an affine spherical variety, possibly singular, and $L^+X$ its arc space. The intersection complex of $L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L-functions. Such relationships were previously established in Braverman-Finkelberg-Gaitsgory-Mirkovic for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier-Ngo-Sakellaridis for toric varieties and L-monoids. In this paper, we compute this intersection complex for the large class of those spherical …

Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

Let $k$ be a commutative ring and $S=k[x_0, \ldots, x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if $J$ defines a complete intersection.

Open $r$-spin theory III: a prediction for higher genus

In our previous two papers, we constructed an $r$-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's $r$-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open $r$-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand-Dickey hierarchy, and we conjectured that the whole solution controls the open $r$-spin intersection numbers in all genera, which do not yet have a geom…

A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincaré-Lefschetz by simplicial methods for constructible sheaves might be of i…

Hilbert schemes, integrable hierarchies, and Gromov-Witten theory

Various equivariant intersection numbers on Hilbert schemes of points on the affine plane are computed, some of which are organized into tau-functions of 2-Toda hierarchies. A correspondence between the equivariant intersection on Hilbert schemes and stationary Gromov-Witten theory is established.