Andrea Causin, Gian Pietro Pirola
We give some upper bounds on the dimension of the kernel of the cup product map $H^{1}(X,\mathbb{C})\otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})$, where $X$ is a compact Kähler variety without Albanese fibrations.
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We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-Donaldson geometry of toric varieties. Namely we associate convex functions with hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge-Amp{è}re volume form and scalar curvature. We then provide an expression…
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
Hermitian Distance Degree of Unitary-Invariant Matrix Varieties
本文研究了酉不变矩阵流形的埃尔米特距离次数，证明其等于关联对称奇异值流形的欧几里得距离次数，并提出埃尔米特切片定理，作为Eckart-Young定理的几何埃尔米特版本。
We study the functional codes of order $h$ defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^n(\mathbb{F}_q)$ a non-degenerate Hermitian variety. We give a condition of divisibility of the weights of the codewords. For $\mathcal{X}$ a non-degenerate Hermitian surface, we list the first five weights and the corresponding codewords and give a positive answer on a conjecture formulated on this question. The paper ends with a conjecture on the minimum distance and the distribution of the codewords of the first $2h+1$ weights of the functional codes for the functional codes of order $h$ on…
Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.
In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{α,β}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters $α,β$ from the underlying field $\mathrm{GF}(q^2)$ have been constructed. In the present paper we determine the projective equivalence classes of such varieties for $r=3$ and $q$ odd.
We construct relative PEL type embeddings in mixed characteristic (0,2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type.
We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong. Key tools include: (i) a new characterization of Schubert varieties that generalizes the well known description of the smooth Schubert varieties by connected sub-diagrams of a Dynkin diagram; and (ii) an algebraic Laplacian (a la Kostant), which is used to anal…
An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications
本文给出 Hermitian 簇上 Hodge 型标准猜想的显式初等证明，利用对偶极图控制相交形式，并证明 $p$-adic 初等因子整除 $q^m$，从而得到代数环的数值等价性及猜想对有限覆盖簇的推广。
We classify the irreducible Hermitian real variations of Hodge structure admitting an infinitesimal normal function, and draw conclusions for cycle-class maps on families of abelian varieties with a given Mumford-Tate group.
Slope Stable Sheaves and Hermitian-Einstein Metrics on Normal Varieties with Big Cohomology Classes
本文引入大上同调类的斜率稳定性与埃尔米特-爱因斯坦度量概念，在特定紧正规空间上建立小林-希钦对应，证明相关Bogomolov-Gieseker不等式，双亚纯不变性与度量存在性为关键。
Let $(G,X)$ be a Shimura variety of PEL type such that $G_{{\bf Q}_2}$ is a split ${\bf GSO}_{2n}$ group with $n\ge 2$. We prove the existence of the integral canonical models of ${\rm Sh}(G,X)/H_2$ in unramified mixed characteristic $(0,2)$, where $H_2$ is a hyperspecial subgroup of $G({\bf Q}_2)$.
Jun-Muk Hwang and Ngaiming Mok have proved the rigidity of irreducible Hermitian symmetric spaces of compact type under Kaehler degeneration. I adapt their argument to the algebraic setting in positive characteristic, where cominuscule homogeneous varieties serve as an analogue of Hermitian symmetric spaces. The main result gives an explicit (computable in terms of Schubert calculus) lower bound on the characteristic of the base field, guaranteeing that a smooth projective family with cominuscule homogeneous generic fibre is isotrivial. The bound depends only on the type of the generic fibre, …
Admissible Hermitian-Yang-Mills connections over normal varieties
本文在正规簇上证明完整版 Donaldson-Uhlenbeck-Yau 定理，建立 Hitchin-Kobayashi 对应，并由此得到 Bogomolov-Gieseker 不等式及其等号刻画，还给出平凡第一 Chern 类正规 Kaehler 簇为环面有限商的新判据。
We show that the $\mathbb{Q}$-descents of the canonical $\mathbb{R}$-variation of Hodge structure of Calabi-Yau type over a tube domain of type $A$ can be realized as sub-variations of Hodge structure of certain $\mathbb{Q}$-variations of Hodge structure which are naturally associated to abelian varieties of (generalized) Weil type.
We study the functional codes of second order defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^4(\mathbb{F}_q)$ a quadric of rank($\mathcal{X}$)=3,4,5 or a non-degenerate hermitian variety. We give some bounds for %$# \mathcal{X}_{Z(\mathcal{Q})}(\mathbb{F}_{q})$, the number of points of quadratic sections of $\mathcal{X}$, which are the best possible and show that codes defined on non-degenerate quadrics are better than those defined on degenerate quadrics. We also show the geometric structure of the minimum weight codewords and estimate the second weight of these codes. For $\mathc…
The paper reviews properties of the Weyl functional calculus for several operators and its relation to the generalised numerical range of $n$ hermitian matrices. The support and singular support of the Weyl functional calculus for $n$ hermitian matrices are determined by Kippenhahn varieties in algebraic geometry.
We introduce the volume function for hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef hermitian invertible sheaf. We also give another several applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.
The cohomological Kudla conjecture for unitary Shimura varieties
通过构造紧化酉Shimura簇上特殊环的上同调类的Kudla-Millson生成级数的自然延拓，证明了它们是全纯Hermitian模形式，从而证明了Kudla和Bruinier-Rosu-Zemel猜想在所有余维数（至中维）的上同调版本，并发展了Hermitian拟模形式理论，证明了特殊环的Zariski闭包的生成级数是Hermitian拟模形式。
In this paper, we study maximal sets of skew lines on Hermitian surfaces. We give a new algorithm to compute these sets and give some computational results for Hermitian surfaces of degrees 3,4, and 5. In more generality, this algorithm solves a new variant of the clique listing problem, which may be more approachable than the classical problem. Finally, we explicitly construct a large skew set of lines on Hermitian varieties of any degree and use it to give a lower bound on the largest size of maximal skew sets and a lower bound on the possible number of maximal skew sets.
In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert--Samuel formula and the arithmetic Hilbert--Samuel formula.
We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.
On pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces
本文研究紧致型不可约 Hermitian 对称空间，给出对称张量场的不可约分解、秩的上同调刻画，并引入伪有效阈值以精确度量切丛的丰沛性且计算之。
Deformed Hermitian Yang-Mills equation on rational homogeneous varieties
本文证明了有理齐次簇上形变埃尔米特 Yang-Mills 方程在任意不变 Kähler 度量下解的存在性，利用 Lie 理论和 Cartan 矩阵刻画齐次解，并给出全纯向量丛斜率半稳定性的新判据及其在 Fano 三维簇上的应用。
Let $X$ be a normal compact Kähler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the existence of admissible Hermitian-Yang-Mills metrics on it. The existence will hold if one can prove a uniform Sobolev inequality on singular spaces.
We introduce a new arithmetic invariant for hermitian line bundles on an arithmetic variety. We use this invariant to measure the variation of the volume function with respect to the metric. The main result of this paper is a generalized Hodge index theorem on arithmetic toric varieties.
Connected Shimura varieties are the quotients of hermitian symmetric domains by discrete groups defined by congruence conditions. We examine their relation with moduli varieties. (Handbook of Moduli).
Admissible HYM metrics on klt KE varieties and the MY equality for big anticanonical K-stable varieties
本文研究了Kähler-Einstein变种上反射sheaf的稳定性、K稳定变种的Miyaoka-Yau不等式等式以及holomorphic向量包的semistability性质。
Local cohomology with Schubert support on compact Hermitian symmetric spaces
本文为紧致埃尔米特对称空间上舒伯特簇支撑的局部上同调模提供了统一的根系构造，给出权滤过公式，并应用于对称行列式与Pfaffian簇，同时计算了霍奇同调与上同调亏格。
Functional codes arising from rank $n$ Hermitian varieties and hypersurfaces in low dimensions
研究秩 $n$ 退化 Hermitian 簇 $P\\\mathcal{U}_{n-1}$ 在 $\\mathbb{P}^n(\\mathbb{F}_{q^2})$ 上的函数码 $C_d(\\mathcal{X})$，当 $d\\\leq q$ 时，通过建立交点数的上界，确定了 $n=2,3,4$ 时码的参数并刻画了达到最小距离的超曲面。
Abundance for varieties with many differential forms
假设低维极小模型纲领成立，证明具有温和奇点且存在大量带 pluricanonical 丛系数自反微分形式的代数簇$X$满足丰度猜想，推出埃尔米特半正典范除子几乎半丰富等结论，且$K_X$数值维数为1时结果无条件成立，还处理 Calabi-Yau 簇上 nef 线丛半丰富性问题。
We use the method of Bruinier--Raum to show that symmetric formal Fourier--Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla's conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.
We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular p-adic integral models for these varieties over odd primes p at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.
In this note, we study Shimura varieties for the groups $\mathrm{GU}(V)$, where $V$ is a Hermitian space relative to a CM extension $E/E^+$. We give a description of the supersingular locus of the fiber at a prime $ν$ over $p$ of such a Shimura variety, under the assumptions that $\mathrm{dim}_E(V) \leq 4$ and that the prime $p$ splits completely in $E^+$ and is unramified in $E$.
Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle.
By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic projective varieties.
Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods
本文引入CM点切空间上的双$\overline{\mathbb{Q}}$-结构，证明其可分解为一维子空间直和，并提出解析子空间猜想，以推导CM abelian簇全纯周期的二次关系。
We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively.
Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties
本文证明了任意紧Hermiteian簇上Gauduchon度量的存在性，借此在非Kähler紧正规簇上定义无挠凝聚层的斜率稳定性，并证明了斜率稳定反射层奇异Hermite-Einstein度量的存在性与唯一性。
Semi-stable models for some unitary Shimura varieties over ramified primes
本文针对签名 $(2,n-2)$ 的酉群相关的 Shimura 簇，通过显式分解局部模型，在虚二次域分歧的奇素数 $p$ 上构建了正则 $p$-adic 整模型。
We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.
The Kodaira dimension of Shimura varieties has been studied by many people. Kondo and Gritsenko-Hulek-Sankaran studied the singularities of orthogonal Shimura varieties related to the moduli spaces of polarized K3 surfaces. They proved that they have canonical singularities and are of general type if the polarization degree is sufficiently large. In this paper, we work on similar problems for unitary Shimura varieties. We show that they have canonical singularities if $n > 4$. As an application, we show that certain unitary Shimura varieties associated with Hermitian forms over the rings of in…
The Hermitian, Suzuki and Ree curves form three special families of curves with unique properties. They arise as the Deligne-Lusztig varieties of dimension one and their automorphism groups are the algebraic groups of type 2A2, 2B2 and 2G2, respectively. For the Hermitian and Suzuki curves very ample divisors are known that yield smooth projective embeddings of the curves. In this paper we establish a very ample divisor for the Ree curves. Moreover, for all three families of curves we find a symmetric set of equations for a smooth projective model, in dimensions 2, 4 and 13, respectively. Usin…
We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbb D$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a Hermitian symmetric domain $\mathcal D$, embedded via a totally geodesic embedding in $\mathbb D$. Next we discuss the case when $Z$ is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains $\mathcal D$ and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of sy…
Stability conditions for polarised varieties
本文为极化簇引入了Z稳定性概念及其解析对应物Z临界Kähler度量，证明了在大体积情形下渐近Z稳定且解析K半稳定的极化簇允许Z临界Kähler度量，并揭示了其与局部壁穿越的联系。
The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and …
Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is …
The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of …
Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified representation $\T$, a hermitian tensor bundle $\E_T$. We can prove then that there exists a lower bound for the heights of points $x\in¶(\E_T)$ with $SL_N(K)$--semistable generic fibre in terms of the degree of $\E$ and some universal constants depending only on the compactified representation. We give then three applications: a universal lower bound for general flag v…
Slope stability of tangent bundles of smooth toric Fano varieties
本文对三维至六维的8630个光滑环面Fano簇的切丛反典范斜率稳定性分类，引入根扭环面dP-纤维丛构造，明确稳定与凯勒-爱因斯坦性质的关系，完成相关分类与证明。
On the singularities and the Kodaira dimension of unitary Shimura varieties
本文研究酉 Shimura 簇的奇点与 Kodaira 维数，证明在特定条件下存在至多典范奇点的环面紧化，并利用 Arthur 公式证明有限多对参数给出非一般型簇，同时改进并证明奇点界的最优性。
A flatness criterion for pseudo-effective sheaves on compact Kähler spaces
本文将Höring–Peternell的平坦性准则从射影簇推广至凯勒空间，证明klt紧凯勒空间上第一陈类为零的伪有效层经有限拟平覆盖后自反拉回局部自由且平坦，核心依赖两项新结果。
Matrix Points on Varieties
研究交换n×n矩阵模空间C_n(X)的上同调，引入半单对应S_n(X)并证明在温和条件下σ: S_n(X)→C_n(X)诱导ℓ-adic上同调同构，给出Betti数的显式组合公式和Macdonald型生成级数。
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