Projectivity criterion of Moishezon spaces and density of projective symplectic varieties

Moishezon Spaces and Projectivity Criteria

We show that a smooth Moishezon space $Y$ is non-projective if and only if it contains a rational curve such that $-[C] \in \overline{\mathrm{NE}}(Y)$. More generally, this holds if $Y$ has $\mathbb{Q}$-factorial, log terminal singularities. We derive this as a consequence of our main technical result: that we can run the relative minimal model program when the base is a normal algebraic space $Y$ of finite type over a field of characteristic $0$. As a second application, we show that every log canonical pair $(Y, Δ)$, where $Y$ is an algebraic space of finite type over a field of characterist…

纤维Fujiki/Kähler族为局部Moishezon/射影的判据

Criteria for a fiberwise Fujiki/Kahler family to be locally Moishezon/projective

利用非Kähler轨迹理论构造积分2-上同调类，通过指数序列构造相对大线丛，得到Fujiki C类空间纤维化的局部Moishezon性判据，并推广了Campana局部射影定理的双亚纯版本；结合奇异Demailly-Paun定理，得到紧Kähler流形纤维化的局部射影性判据，给出Campana定理的新证明和推广。

An Analytic Approach to the Quasi-projectivity of the Moduli Space of Higgs Bundles

The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and…

K-模空间的准映射与K-稳定的Calabi-Yau纤维模空间的准射影性

K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves

本文研究了K-稳定的Calabi-Yau纤维模空间的准射影性，通过构造具有特定log Fano条件的K-模空间，并探讨了其与准映射K-模空间的关系。