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Register a new userOn the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces
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We study holomorphic $(n+1)$-chains $E_nto E_{n-1} to >... to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by the third one. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in the work of Bradlow, Garcia-Prada and Gothen. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.
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Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bundle is a pair $(E_G,, phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $phi$ assigns to each $x, in, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,, phi)$ together with a section $theta, in, H^0(X,, text{ad}(E_G)otimes K_Xotimes{mathcal O}_X(D))$ such that $theta(x)$ is compatible with the framing $phi$ for every $x, in, D$. We construct a holomorphic symplectic structure on the moduli space $mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $mathcal{M}_{FH}(G)$ to the moduli space $mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,, theta)$ that forgets the framing, is Poisson. These results generalize cite{BLP} where $(G,, {H_x}_{xin D})$ is taken to be $(text{GL}(r,{mathbb C}),, {text{I}_{rtimes r}}_{xin D})$. We also investigate the Hitchin system for $mathcal{M}_{FH}(G)$ and its relationship with that for $mathcal{M}_{H}(G)$.
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus. We prove that the growth is bounded by a polynomial function of degree $14+varepsilon$ for any $varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be improved to degree 10. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkaehler fourfolds.
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.
Let $F$ be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor $R$ on a general K3 in $F$. We call this divisor recognizable if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification $overline{F}^R$ for a recognizable divisor is a Looijenga semitoroidal compactification. For polarized K3 surfaces $(X,L)$ of degree $2d$, we show that the sum of rational curves in the linear system $|L|$ is a recognizable divisor, giving a modular semitoroidal compactification of $F_{2d}$ for all $d$.
We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface.
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