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Register a new userRandom dynamics on real and complex projective surfaces
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Serge Cantat
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We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact K{a}hler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.
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Let ${mathcal B}_g(r)$ be the moduli space of triples of the form $(X,, K^{1/2}_X,, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g, geq, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{mathcal B}_g(r)$--torsor ${mathcal H}_g(r)$ over ${mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundles of rank $r$ on $Y$, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that ${mathcal H}_g(r)$ has a holomorphic symplectic structure compatible with the $T^*{mathcal B}_g(r)$--torsor structure. We also describe ${mathcal H}_g(r)$ in terms of the second order matrix valued differential operators. It is shown that ${mathcal H}_g(r)$ is identified with the $T^*{mathcal B}_g(r)$--torsor given by the sheaf of holomorphic connections on the theta line bundle over ${mathcal B}_g(r)$.
This article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree >= 5 or an Inoue surface. We give examples of rigid manifolds of dimension n >= 3 and Kodaira dimensions 0, and 2 <=k <= n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n >= 4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.
We construct a compactification $M^{mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $gamma colon M^{ss} to M^{mu ss}$, where $M^{ss}$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.
We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.
We study foliations $mathcal{F}$ on Hirzebruch surfaces $S_delta$ and prove that, similarly to those on the projective plane, any $mathcal{F}$ can be represented by a bi-homogeneous polynomial affine $1$-form. In case $mathcal{F}$ has isolated singularities, we show that, for $ delta=1 $, the singular scheme of $mathcal{F}$ does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For $delta eq 1$, we prove that the singular scheme of $mathcal{F}$ does not determine the foliation. However we prove that, in most cases, two foliations $mathcal{F}$ and $mathcal{F}$ given by sections $s$ and $s$ have the same singular scheme if and only if $s=Phi(s)$, for some global endomorphism $Phi $ of the tangent bundle of $S_delta$.
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