Academic literature on the topic 'Bounded Symmetric domain'

We characterize the structure of the domain of a pure jump-type Dirichlet form which is given by a Beurling–Deny formula. In particular, we obtain su cient conditions in terms of the jumping kernel guaranteeing that the test functions are a core for the Dirichlet form and that the form is a Silverstein extension. As an application we show that for recurrent Dirichlet forms the extended Dirichlet space can be interpreted in a natural way as a homogeneous Dirichlet space. For reflected Dirichlet spaces this leads to a simple purely analytic proof that the active reflected Dirichlet space (in the sense of Chen, Fukushima and Kuwae) coincides with the extended active reflected Dirichlet space.<br>Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Includes bibliographical references.<br>We consider two complementary problems: given a compact Kähler manifold with negative first Chern Class, when is its universal cover a Bounded Symmetric Domain? And if it is, which Bounded Symmetric Domain is it? Existing literature is discussed, with particular attention given to two recent papers of Catanese and Di Scala ([CDS12] and [CDS]) which answer both questions first for Bounded Symmetric Domains of Tube Type, and then for all Bounded Symmetric Domains without Ball Factors. Using work of Yau and others on ball quotients we extend the main result of [CDS] to all bounded Symmetric Domains, including those with ball factors, thus answering the two questions posed in full generality.

Ce travail de thèse porte sur l’étude de l’hyperbolicité complexe de compactifications de quotients de domaines symétriques bornés, et plus spécifiquement, de quotients de la boule. On souhaite ainsi comprendre la géométrie des courbes entières que ces compactifications contiennent, ainsi que le type de leurs sous-variétés. On prouve d'abord un critère métrique de positivité du fibré co-tangent d’une variété complexe, reposant notamment sur les travaux de J.-P. Demailly et de S. Boucksom. Ce critère peut s’appliquer à une large classe de variétés, dépassant le cadre précédent ; avec Y. Brunebarbe, on s'est ainsi intéressé aux variétés supportant une variation de structures de Hodge complexes. Dans le cas d’un quotient de la boule, ce critère permet de montrer qu’un revêtement ramifié d’une compactification toroïdale lisse, étale sur l’intérieur, et ramifiant à des ordres supérieurs à 7 sur le bord, ne contient que des variétés de type général en dehors de son bord. Dans ce cadre, ceci fournit une version effective d’un théorème de Y. Brunebarbe. On étudie par ailleurs d’autres situations que ces compactifications lisses : avec E.Rousseau et B. Taji, on donne des critères d'hyperbolicité complexe de ces compactifications lorsque les quotients sont singuliers. On présente aussi une version effective d’un théorème de J.-P. Demailly, concernant le caractère big des différentielles de jets sur la compactification donnée. Enfin, on montre que les méthodes métriques présentées précédemment s’étendent au cas de tous les domaines symétriques bornés ; elles fournissent alors des résultats effectifs d'hyperbolicité algébrique et transcendante pour d’autres domaines que la boule<br>This work deals with the study of complex hyperbolicity of compactifications of quotients of bounded symmetric domains, and more specifically, of quotients of the ball. We are interested in the geometry of the entire curves such a compactification contains, as well as to the type of its subvarietes. We first prove a metric criterion for the positivity of the cotangent bundle of a complex manifold, based in particular on the work of J.-P. Demailly and of S. Boucksom. This criterion can be applied to a large class of manifolds, going beyond the previous frame; with Y. Brunebarbe, we apply in particular these methods to the case of manifolds supporting a complex variation of Hodge structures.In the case of a ball quotient, the criterion shows that on a ramified cover of a toroidal compactification, étale on the inside part, and ramifying at orders higher than 7 on the boundary, there is no subvariety which is not of general type, and not included in the boundary. In this setting, this gives an effective version of a theorem of Y. Brunebarbe. We also study other situations than these smooth compactifications : with E. Rousseau and B. Taji, we give metric criterions for the complex hyperbolicity of these compactifications, if the quotients are singular. In the case of the ball, we present also an effective version of a theorem of J.-P. Demailly, concerning the bigness of the Green-Griffiths jet differentials on the given compactification. Finally, we show that the previous metric methods can be applied to the case of all bounded symmetric domains ; they give effective hyperbolicity results, algebraic and transcendental, for other bounded symmetric domains than the ball

This thesis deals with two different subjects: balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space. Correspondingly we have two main results. In the first one we prove that if a holomorphic vector bundle E over a compact Kähler manifold (M,ω) admits a ω-balanced metric then this metric is unique. In the second one, after defining the diastatic exponential of a real analytic Kähler manifold, we prove that for every point p of an Hermitian symmetric space of noncompact type there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks.

Let $\boldsymbol T=(T_1, \ldots , T_d)$ be a $d$- tuple of commuting operators on a Hilbert space $\mathcal H$. Assume that $\boldsymbol T$ is hyponormal, that is, $\big [\!\!\big [ \boldsymbol T^*, \boldsymbol T \big ]\!\! \big ]:=\big (\!\!\big ( \big [ T_j^*,T_i] \big )\!\!\big )$ acting on the $d$ - fold direct sum of the Hilbert space $\mathcal H$ is non-negative definite. The commutator $[T_j^*,T_i]$, $1\leq i,j \leq d$, of a finitely ctyclic and hyponormal $d$ - tuple is not necessarily compact and therefore the question of finding trace inequalities for such a $d$- tuple does not arise. A generalization of the Berger-Shaw theorem for a commuting tuple $\boldsymbol T$ of hyponormal operators was obtained by Douglas and Yan decades ago. We discuss several examples of this generalization in an attempt to understand if the crucial hypothesis in their theorem requiring the Krull dimension of the Hilbert module over the polynomial ring defined by the map $p\to p(\boldsymbol T)$, $p\in \mathbb C[\boldsymbol z]$, is optimal or not. Indeed, we find examples $\boldsymbol T$ to show that there is a large class of operators for which $\text{trace}\,[T_j^*,T_i]$, $1\leq j,i \leq d$, is finite but the $d$ - tuple is not finitely polynomially cyclic, which is one of the hypotheses of the Douglas-Yan theorem. We also introduce the weaker notion of ``projectively hyponormal operators" and show that the Douglas-Yan thorem remains valid even under this weaker hypothesis. We introduce the determinant operator $\text{dEt}\,(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big) $, which coincides with the generalized commutator introduced by Helton and Howe earlier. We identify a class $BS_{m, \vartheta}(\Omega)$ consisting of commuting $d$- tuples of hyponormal operators $\boldsymbol T$, $\sigma(\boldsymbol T) = \overbar{\Omega}$, satisfying a growth condition for which the dEt is a non-negative definite operator. We then obtain the trace estimate given in the Theorem below. \begin{thmAbs} Let $\boldsymbol{T}=(T_1,\ldots, T_d)$ be a commuting tuple of operators on a Hilbert space $\mathcal{H}$ such that $\boldsymbol{T}$ is in the class $BS_{m, \vartheta}(\Omega)$. Then the determinant operator $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big)$ is in trace-class and \[\text{trace}\,\big (\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big]\big)\big )\leq m\, \vartheta \,d!\prod_{i=1}^{d}\|T_i\|^2.\] \end{thmAbs} In the case of a commuting $d$ - tuple $\boldsymbol T$ of operators, where $\sigma(\boldsymbol T)$ is of the form $\overbar{\Omega}_1 \times \cdots \times \overbar{\Omega}_d$, we obtain a slightly different but a related estimate for the trace of $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big]\big)\big )$. Explicit computation of $\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big ]\big)$ in several examples and based on some numerical evidence, we make the following conjecture refining the estimate from the Theorem: \begin{conjAbs} Let $\boldsymbol{T}=(T_1,\ldots, T_d)$ be a commuting tuple of operators on a Hilbert space $\mathcal{H}$ such that $\boldsymbol{T}$ is in the class $BS_{m, \vartheta}(\Omega)$. Then the determinant operator $\text{dEt}\,\big(\big[\!\!\big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\! \big ]\big)$ is in trace-class, and \[\text{\rm trace}\,\big (\text{dEt}\,\big(\big[\!\! \big [\boldsymbol{T}^*, \boldsymbol{T}\big ]\!\!\big ]\big) \big )\leq \frac{m d!}{\pi^d} \nu(\overline{\Omega}), \] where $\nu$ is the Lebesgue measure. \end{conjAbs} Let $\Omega$ be an irreducible classical bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T$ of commuting bounded linear operators by the rule: \[k\cdot\boldsymbol{T}:=\big(k_1(T_1, \ldots, T_d), \ldots, k_d(T_1, \ldots, T_d)\big),\,\,k\in \mathbb K, \] where $k_1(\boldsymbol z), \ldots, k_d(\boldsymbol z)$ are linear polynomials. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. We realize a certain class of $\mathbb{K}$-homogeneous $d$-tuples $\boldsymbol{T}$ as a $d$ -tuple of multiplication by the coordinate functions $z_1,\ldots ,z_d$ on a reproducing kernel Hilbert space $\mathcal H_K$. (The Hilbert space $\mathcal H_K$ consisting of holomorphic functions defined on $\Omega$ and $K$ is the reproducing kernel.) Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\Omega)$. For an irreducible bounded symmetric domain $\Omega$ of rank $2$, an explicit description of the operator $\sum_{i=1}^d T_i^*T_i$ is given. Based on this formula, a conjecture giving the form of this operator in any rank $r \geq 1$ was made. This conjecture was recently verified by H. Upmeier.

碩士<br>國立清華大學<br>數學系<br>92<br>In this article, we prove that there are three positive solutions of a semilinear elliptic equation in a bounded symmetric domain Dt for large t > 0 in which one is axially symmetric and the other two are nonaxially symmetric. Moreover, we prove that these three solutions are unstable.