Jun-Muk Hwang : Cartan-Fubini type extension of holomorphic maps preserving webs of rational curves

Posted by Kevin Tucker , part of the Algebraic Geometry Seminar.

At

Oct. 21, 2014, 4 p.m.

In

SEO 636

Abstract

Let $X_1$ and $X_2$ with $\mathrm{dim} X_1 = \mathrm{dim} X_2$ be two projective manifolds of Picard number 1 in projective space. Assume that both $X_1$ and $X_2$ are covered by lines. Let $\varphi: U_1 \to U_2$ be a biholomorphic map between two connected Euclidean open subsets $U_1 \subset X_1$ and $U_2 \subset X_2$. Suppose that both $\varphi$ and $\varphi^{-1}$ send pieces of lines to pieces of lines. We show that $\varphi$ can be extended to a biregular morphism $\Phi: X_1 \to X_2$. This was proved by Hwang-Mok in 2001 when the indices of $X_1$ and $X_2$ are bigger than 2 and the new result is when the indices are 2. In this case, the covering family of lines form webs of rational curves. We exploit the monodromy of the webs of lines to extend the holomorphic map.