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Register a new userOn topologically trivial automorphisms of compact Kahler manifolds and algebraic surfaces
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In this paper, we investigate automorphisms of compact Kahler manifolds with different levels of topological triviality. In particular, we provide several examples of smooth complex projective surfaces X whose groups of $C^infty$-isotopically trivial automorphisms, resp. cohomologically trivial automorphisms, have a number of connected components which can be arbitrarily large.
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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.
In this paper, we prove that the group $mathrm{Aut}_mathbb{Q}(X)$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds $X$ of general type which either satisfy $q(X)geq 3$ or have a Gorenstein minimal model. If $X$ is furthermore of maximal Albanese dimension, then $|mathrm{Aut}_mathbb{Q}(X)|leq 4$, and equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset $mathcal{C}subset (0,1]$ such that $mathcal{C}cup{1}$ attains the minimum.
A special Kahler-Ricci potential on a Kahler manifold is any nonconstant $C^infty$ function $tau$ such that $J( ablatau)$ is a Killing vector field and, at every point with $dtau e 0$, all nonzero tangent vectors orthogonal to $ ablatau$ and $J( ablatau)$ are eigenvectors of both $ abla dtau$ and the Ricci tensor. For instance, this is always the case if $tau$ is a nonconstant $C^infty$ function on a Kahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $tilde g=g/tau^2$, defined wherever $tau e 0$, is Einstein. (When such $tau$ exists, $(M,g)$ may be called {it almost-everywhere conformally Einstein}.) We provide a complete classification of compact Kahler manifolds with special Kahler-Ricci potentials and use it to prove a structure theorem for compact Kahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.
Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who treated the case $e=14$), we prove that the moduli space of polarized K3 surfaces of degree $2e$ contains complete curves for all $egeq 62$ and for some sporadic lower values of $e$ (starting at $14$). We also construct complete curves in the moduli spaces of polarized hyper-Kahler manifolds of $mathrm{K3}^{[n]}$-type or $mathrm{Kum}_n$-type for all $nge 1$ and polarizations of various degrees and divisibilities.
In the present paper, we show that given a compact Kahler manifold $(X,omega)$ with a Kahler metric $omega$, and a complex submanifold $Vsubset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any quasi-plurisubharmonic function $varphi$ on $V$ such that $omega|_V+sqrt{-1}partialbarpartialvarphigeq varepsilonomega|_V$ with $varepsilon>0$ can be extended to a quasi-plurisubharmonic function $Phi$ on $X$, such that $omega+sqrt{-1}partialbarpartial Phigeq varepsilonomega$ for some $varepsilon>0$. This is an improvement of results in cite{WZ20}. Examples satisfying the assumption that there exists a holomorphic retraction structure contain product manifolds, thus contains many compact Kahler manifolds which are not necessarily projective.
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