Complex differential geometry and its applications

Abstract

<p/><p content-type="inline"/> “Complex differential geometry and its applications” originated from the author's prior work in Kähler geometry revolving around bounded symmetric domains $\Omega$, their finite-volume quotients $X_\Gamma~:=~\Omega/\Gamma$ and dual Hermitian symmetric spaces $S$ of the compact type. The author's solution of the generalized Frankel conjecture has revealed the importance of the collection of tangents to minimal rational curves in the curvature characterization of $S$. Together with Hwang, the author has developed the foundation of a differential-geometric theory of uniruled projective manifolds $(X,\mathcal~K)$ based on the variety of minimal rational tangents (VMRT), encapsulated in the VMRT structure $\pi:~\mathscr~C(X)~\to~X$ and the universal family $\rho:~\mathcal~U~\to~\mathcal~K$, enabling them to resolve classical problems in algebraic geometry such as the deformation rigidity of rational homogeneous spaces $G/P$ of Picard number 1, the Lazarsfeld problem and the characterization of uniruled projective manifolds equipped with reductive G-structures. Hwang-Mok established the Cartan-Fubini extension principle for uniruled projective manifolds $(X,\mathcal~K)$ of Picard number 1 under very mild conditions for the analytic continuation of VMRT-preserving local biholomorphisms, which was further developed by Hong-Mok resp. Mok-Zhang to a non-equidimensional version of Cartan-Fubini extension principle resp. analytic continuation of germs of complex submanifolds inheriting sufficiently non-degenerate sub-VMRT structures, resulting in particular in Schubert and Schur rigidity theorems.The geometric theory of VMRT has also motivated the author to examine algebraic subsets (defined via the Borel embedding) $Z~\subset~\Omega$, and their images $\pi_\Gamma(Z)$ under the quotient map $\pi_\Gamma:~\Omega~\to~X_\Gamma$, i.e., to examine an irreducible component $\mathcal~K$ of the Chow scheme ${\rm~Chow}(S)$, its universal family $\rho:~\mathcal~U~\to~\mathcal~K$ and evaluation map $\mu:~\mathcal~U~\to~X$, together with the quotient $\mathcal~U_\Gamma~:=~\mathcal~U_0/\Gamma$ of the restriction $\mathcal~U_0~:~=~\mathcal~U|_\Omega$. The fibered space $\mathcal~U_\Gamma$ may be regarded as the support of a meromorphic distribution $\mathscr~F$, such that the image under the natural projection of a certain $\mathscr~F$-saturated subvariety $\mathscr~Z\subset~\mathcal~U_\Gamma$ gives the Zariski closure of $q(Z)$. At the same time, the prior work of Mok-Zhong for the compactification under certain curvature assumptions of complete Kähler manifolds of finite volume yields differential-geometric proofs realizing $X_\Gamma$ and various holomorphic fiber bundles over $X_\Gamma$ as quasi-projective varieties, together with a proof of the algebraicity of natural meromorphic foliations defined on them, enabling him to prove the Ax-Lindemann Theorem for not necessary arithmetic lattices $\Gamma$ of rank 1. In the setting of Shimura varieties $X_\Gamma$, in which the arithmeticity of the lattices $\Gamma$ has enabled a flourishing of results on functional transcendence with the availability of o-minimal structures from model theory and the interpretation of $X_\Gamma$ as modular varieties of (mixed) Hodge structures, a combination of the differential-geometric and algebro-geometric perspective of the author together with the variety of techniques and results from o-minimality and Hodge theory, enabled Mok-Pila-Tsimerman to establish the long awaited Ax-Schanuel theorem on Shimura varieties, which, together with its many generalizations, has proven to be applicable to deep arithmetic problems such as the Pink-Zilber conjecture and the Shafarevich conjecture in number theory.