A characterization of irreducible Hermitian symmetric spaces of tube type by $\mathbb{C}^{*}$-actions

Rigidity of irreducible Hermitian symmetric spaces of the compact type under K"ahler deformation

We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic $G$-structure, corresponding to a reduction of the structure group of the tangent bundle. $S$ is characterized as the unique simply-connected compact complex manifold admitting such a $G$-structure which is at the same time integrable. To prove the deformation rigidity of $S$ it …

Projective Ranks of Compact Hermitian Symmetric Spaces

Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.

不可约曲线与Hermitian曲线的交点

Intersection of irreducible curves and the Hermitian curve

本文研究Hermitian曲线与不可约平面曲线在有限域上的交点个数是否达到Bézout上界，证明对许多次数范围（如 $q \le d \le q^2-q+1$ 等）答案是肯定的，并给出部分结果。

Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

On some Hermitian variations of Hodge structure of Calabi-Yau type with real multiplication

We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible factor of Calabi-Yau type of the corresponding real variation of Hodge structure is the canonical real VHS of CY type over the Hermitian symmetric domain II_6, associated to the real group SO^*(12). The main point is a rationality result for the half spin representations of a form of the group SO^*(4m) defined over a number field.

Realization of an equivariant holomorphic Hermitian line bundle as a Quillen determinant bundle

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian lin…

Normal Functions over Locally Symmetric Varieties

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.

The B-orbits on a Hermitian symmetric variety in characteristic 2

Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the corresponding parabolic subgroup over $\mathbb{C}$ has abelian unipotent radical and fix a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $Φ$ is irreducible. For $Φ$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinat…

The Hodge star operator on Schubert forms

Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.

On the heights of algebraic points on curves over number fields

We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those of S.Zhang on the same question. The case of the relative dualizing sheaf with the Arakelov metric is made especially explicit.

Embedding of a maximal curve in a Hermitian variety

Let X be a projective geometrically irreducible non-singular algebraic curve defined over a finite field F of order $q^2$. If the number of F-rational points of X satisfies the Hasse-Weil upper bound, then X is said to be F-maximal. For a point P_0\in X(F), let πbe the morphism arising from the linear series D:=|(q+1)P_0|, and let N:=dim(D). It is known that N\ge 2 and that πis independent of P_0 whenever X is F-maximal. The following theorems will be proved: Theorem 0.1: If X is F-maximal, then π:X\to π(X) is a F-isomorphism. The non-singular model π(X) has degree q+1 and lies on a Hermitian …

On the Bruhat $\mathcal{G}$-order between local systems on the B-orbits of a Hermitian symmetric variety

Following Lusztig and Vogan, we study the Bruhat $G$-order on the set $\mathcal{D}$ of rank $1$ local systems on $B$-orbits over an Hermitian symmetric variety $G/L$. The main aim is to give a combinatorial characterization similar to the one on the Bruhat order given by Gandini and Maffei. The results depend on the type of the root system $Φ(G)$ which we suppose irreducible. In particular, for $Φ(G)$ simply laced either all the local systems are trivial or only a specific subset of orbits (the orbits of maximum rank) admit a non-trivial local system. In this last case the Hasse diagram of the…

A Characterization of Varieties whose Universal Cover is the Polydisk or a Tube Domain

Catanese and Franciosi defined a semispecial tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor and by a 2-torsion line bundle. A slope zero tensor is instead a section of the nm-th symmetric power of the cotangent bundle twisted by m times the anticanonical divisor. With these definitions we have: Theorem 1. The universal cover of X is the polydisk iff 1) holds. 1) X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding hypersurface in the projectivized tangent space is reduce…

Sequences of stable bundles over a compact complex surface

When identified with sequences of irreducible Hermitian-Einstein connections, sequences of stable holomorphic bundles of fixed topological type and bounded degree on a compact complex surface equipped with a Gauduchon metric are shown to have strongly convergent subsequences after blowing-up and pulling-back sufficiently many times.

Polarized Variation of Hodge Structures of Calabi-Yau Type and Characteristic Subvarieties Over Bounded Symmetric Domains

In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in \cite{G} to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by N. Mok in \cite{M}. We verified the generating property of B. Gross for all irreducible bou…

Extremal rays in the Hermitian eigenvalue problem for arbitrary types

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of $G=\operatorname{SL}(n)$ . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups $G$, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of $G$. We give a description of the extremal rays of the eig…

Homogeneous bundles and the first eigenvalue of symmetric spaces

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.