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Complex hypersurfaces in direct products of Riemann surfaces

Claudio Llosa Isenrich

Abstract

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Gromov's question of which subgroups of a direct product of surface groups are Kähler for two classes: subgroups of direct products of three surface groups; and subgroups arising as kernel of a homomorphism from the product of surface groups to $\mathbb{Z}^3$. These results will be a consequence of answering the more general question of which subgroups of a direct product of surface groups are the image of a homomorphism, which is induced by a holomorphic map, for the same two classes. This provides new constraints on Kähler groups.

Complex hypersurfaces in direct products of Riemann surfaces

Abstract

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Gromov's question of which subgroups of a direct product of surface groups are Kähler for two classes: subgroups of direct products of three surface groups; and subgroups arising as kernel of a homomorphism from the product of surface groups to . These results will be a consequence of answering the more general question of which subgroups of a direct product of surface groups are the image of a homomorphism, which is induced by a holomorphic map, for the same two classes. This provides new constraints on Kähler groups.

Paper Structure

This paper contains 6 sections, 17 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Background
  3. From homomorphisms to complex hypersurfaces
  4. The three factor case
  5. Complex hypersurfaces
  6. Maps to ${\mathds Z}^3$

Key Result

Theorem 1.1

Let $G=\pi_1(X)$ be the fundamental group of a compact Kähler manifold $X$, and let $\phi: G\to \Gamma_{g_1}\times \Gamma_{g_2}\times \Gamma_{g_3}$ be a homomorphism with finitely presented full subdirect image $\overline{G}:=\phi(G)$ of infinite index. Assume that ${\rm{ker}} (p_i\circ \phi)$ is fi induced by branched holomorphic coverings $f_i: S_{\gamma_i}\to E$, such that $\overline{G}_0={\rm{

Theorems & Definitions (34)

  • Definition
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1: Llo-16-II
  • Theorem 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thmMainTheorem']}
  • ...and 24 more