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Register a new userHyperbolicity of singular spaces
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Benoit Cadorel
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We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Langs conjectures in both analytic and algebraic settings. As an application, we show that Hilbert modular varieties, except for a few possible exceptions, satisfy all expected conjectures.
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We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $mathbb D$ in $mathbb C$ into the unit ball $mathbb B^n$ in $mathbb R^n$, $nge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $nge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $mathbb Dto mathbb B^n$. This implies that conformal harmonic immersions $M to mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$mathrm{e}$ metric on $M$ and the Cayley-Klein metric on the ball $mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $mathbb D$ onto affine discs in $mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $mathbb R^n$ based on minimal surfaces.
In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the condition that a connection has regular semisimple leading term to connections with non-integer slope. Finally, we construct a symplectic moduli space of meromorphic connections on the projective line that contain a regular stratum at each singular point.
In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{omega _varphi}$, where $omega _varphi= e^{-varphi}$ and $varphi$ is a subharmonic function. We consider compact Hankel operators $H_{overline {phi}}$, with anti-analytic symbols ${overline {phi}}$, and give estimates of the trace of $h(|H_{overline phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{overline {phi}}))_n$ in terms of decreasing rearrangement of $|phi |/sqrt{Delta varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{overline {phi}}))_n$ is achieved by $(s_n (H_{overline{z}}))_n$. Namely, we establish that if $s_n(H_{overline {phi}})= o (s_n(H_{overline {z}}))$, then $H_{overline {phi}} = 0$. Then, we show that if $Delta varphi (z) asymp frac{1}{(1-|z|^2)^{2+beta}}$ with $beta geq 0$, then $s_n(H_{overline {phi}}) = O(s_n(H_{overline {z}}))$ if and only if $phi $ belongs to the Hardy space $H^p$, where $p= frac{2(1+beta)}{2+beta}$. Finally, we compute the asymptotics of $s_n(H_{overline {phi}})$ whenever $ phi in H^{p }$.
Let M be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for a given finite set I of natural numbers m>1 with h(m)>0 a projective compactification M of the reduced scheme underlying M such that the ample invertible sheaf L corresponding to the determinant of the direct image of the m-th power of the relative dualizing sheaf on the moduli stack, has a natural extension L to M. A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases natural means that the pullback of L to a curve C --> M, induced by a family f:X --> C is isomorphic to the determinant of the direct image of the m-th power of the relative dualizing sheaf whenever f is birational to a semi-stable family. Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by itself. In particular we will need a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves. Following suggestions of a referee, we reorganized the article, we added several comments explaining the main line of the proof, and we changed notations a little bit.
In this note, we introduce the notion of a singular principal G-bundle, associated to a reductive algebraic group G over the complex numbers by means of a faithful representation $varrho^pcolon Glra SL(V)$. This concept is meant to provide an analogon to the notion of a torsion free sheaf as a generalization of the notion of a vector bundle. We will construct moduli spaces for these singular principal bundles which compactify the moduli spaces of stable principal bundles.
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