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Register a new userClosed symmetric 3-differentials on complex surfaces
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We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give t
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For a noncompact complex hyperbolic space form of finite volume $X=mathbb{B}^n/Gamma$, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $overline{X}$ of $X$ similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $overline{X}-X$ called `canonical radius of a cusp of $Gamma$. The main result in the article is that there is a constant $r^*=r^*(n)$ depending only on the dimension, so that if the canonical radii of all cusps of $Gamma$ are larger than $r^*$, then there exist symmetric differentials of $overline{X}$ vanishing at infinity. As a corollary, we show that the cotangent bundle $T_{overline{X}}$ is ample modulo the infinity if moreover the injectivity radius in the interior of $overline{X}$ is larger than some constant $d^*=d^*(n)$ which depends only on the dimension.
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.
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 ${mathcal B}_g(r)$.
The paper provides a description of the sheaves of Kahler differentials of the arc space and jet schemes of an arbitrary scheme where these sheaves are computed directly from the sheaf of differentials of the given scheme. Several applications on the structure of arc spaces are presented.
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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