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9-14天元基金几何与随机分析及其应用交叉讲座之133【Ruiran Sun】

Title: Higgs Bundles and  Hyperbolicity

Speaker: Ruiran Sun, University of Mainz

Time: September 14 (Friday), 14:00-15:00

Room: 1208

Abstract: 
The theory of variations of Hodge structures has shown to be powerful in the study of higher dimensional Shafarevich program, with the assumption of the injectivity of the Torelli map.  Viehweg-Zuo constructed a class of Higgs bundles on the moduli spaces of polarized varieties by combining the Kodaira-Spencer deformation theory and the Hodge theory, which provides sort of substitution for the local injectivity of the Torelli maps. More explicitly, for a family $f: X\to Y$ parametrizing $n$-folds $F$ with semi ample $\omega_F$  and with the degeneration over a closed subvariety  $D\subset Y$, Viehweg-Zuo  introduced a  Higgs bundle  $(G,\tau)$ over  $Y$ with singularities over $D$  by extending the  Kodaira-Spencer deformation $\tau$ on the higher order cohomologies of the  tangent bundle along $F$. The central feature of this Higgs bundle is that  there exists a natural comparison map  $\rho: (G,\tau)\to(E,\theta)$, where $(E,\theta)$  is the Hodge bundle associated to the cyclic covering $ g: Z_s\to X\to Y$ defined by a section in the linear system of the relative pluri-canonical line bundle on $X$ twisting with an anti-ample line bundle on $Y.$ The Hodge metric on $\text{ker}(\theta)$ induces  a non-zero (possibly degenerated) negatively curved  Finsler metric  on $Y\setminus D=:U$  via the  iterated Kodaira-Spencer map,  if the second graded piece  $\rho^{n-1,1}$  of the comparison map $\rho$ is injective on $T_U$. This Finsler metric plays a crucial  role in the study of hyperbolicity of $U$.  Indeed, $\rho^{n-1,1}$ is known to be injective for two exteme cases: $\kappa(F)=n$ and $ \kappa(F)=0$.  For the general case $0\leq \kappa(F)\leq n$ Viehweg-Zuo showed that it is generically injective along any algebraic curve in $U$. Very recently X.Lu, R.R. Sun and K. Zuo showed $\rho^{n-1,1}_{T_U}$ is injective for $\kappa(F)=1$ by investigating the Iitaka fibration. We also get some results on the bigness of the topological fundamental group of the moduli space of elliptic surfaces.

  科大主页 | 国家数学与交叉科学中心(合肥) | 中科院吴文俊数学重点实验室 |
中科院数学与系统科学研究院 | 北京国际数学研究中心 | 安徽省数学会