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Register a new user$L^2$-Extension of Adjoint bundles and Kollars Conjecture
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We give a new proof of Kollars conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^2$-method and we give a concrete construction and proofs of the conjecture. The $L^2$ point of view allows us to generalize Kollars conjecture to the context of non-abelian Hodge theory.
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For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, phi)$ over it, we establish a metric extension result for sections of $L^{otimes n}$ from a sub-variety $Y$ to $X$. We form normed section algebras from $(L, phi)$ and study their Berkovich spectra. To compare the supremum algebra norm and the quotient algebra norm on the restricted section algebra $V(L_{X|Y})$, two different methods are used: one exploits the holomorphic convexity of the spectrum, following an argument of Grauert; another relies on finiteness properties of affinoid algebra norms.
A conic bundle is a contraction $Xto Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $Xto Z$ is a conic bundle such that $X$ has canonical singularities and $Z$ is $mathbb{Q}$-Gorenstein, then $Z$ is always $frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $frac{1}{2}$ and $2$ are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension $1$ with canonical singularities.
The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.
We treat Kollars injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kollar type cohomology injectivity theorems. Our main theorem is formulated for a compact Kahler
manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kahler metric. We need neither covering tricks, desingularizations, nor Lerays spectral sequence.
In this article we study the Gieseker-Maruyama moduli spaces $mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=ein{-1,0}, c_2=nge1$ on the projective space $mathbb{P}^3$. We construct two new infinite series $Sigma_0$ and $Sigma_1$ of irreducible components of the spaces $mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $Sigma_0$ contains components for all big enough values of $n$ (more precisely, at least for $nge146$). $Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $mathcal{B}(0,n)$ satisfying this property.
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