[Paper Review] Low dimensional projective groups

Indranil Biswas, Mahan Mj|arXiv (Cornell University)|Mar 20, 2012

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper introduces holomorphically convex groups—fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. For such groups of cohomological dimension two, the authors prove they are isomorphic to fundamental groups of compact Riemann surfaces, implying that linear groups of rational cohomological dimension two arising from projective varieties are (virtual) surface groups.

ABSTRACT

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group of a compact Riemann surface. As a consequence, we show that if a linear group $G$ has (rational) cohomological dimension two and is the fundamental group of a smooth complex projective variety, then $G$ is a (virtual) surface group.

Motivation & Objective

• To define and study holomorphically convex groups as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers.
• To investigate the structure of holomorphically convex groups with cohomological dimension two.
• To determine the implications for linear groups that are fundamental groups of smooth complex projective varieties and have rational cohomological dimension two.
• To establish a connection between cohomological dimension and geometric structure in complex projective varieties.
• To show that such groups are (virtual) surface groups, linking algebraic topology with complex geometry.

Proposed method

• Define holomorphically convex groups via the holomorphic convexity of the universal cover of a smooth complex projective variety.
• Use cohomological dimension two as a key constraint to analyze the geometric and algebraic structure of the fundamental group.
• Apply results from complex geometry and Hodge theory to constrain the possible fundamental groups.
• Leverage the classification of compact Riemann surfaces to identify groups of dimension two as surface groups.
• Use the linearity of the group to deduce virtual surface group structure via known results in geometric group theory.
• Establish that rational cohomological dimension two implies the group is a (virtual) surface group when realized as a fundamental group of a projective variety.

Experimental results

Research questions

• RQ1 Which fundamental groups arise from smooth complex projective varieties with holomorphically convex universal covers?
• RQ2 What geometric and algebraic constraints arise when such a group has cohomological dimension two?
• RQ3 How does the rational cohomological dimension two condition constrain the structure of linear fundamental groups of projective varieties?
• RQ4 Can holomorphically convex groups of dimension two be classified as surface groups or virtual surface groups?
• RQ5 What is the relationship between holomorphic convexity of the universal cover and the geometric type of the fundamental group?

Key findings

• Holomorphically convex groups of cohomological dimension two are isomorphic to the fundamental groups of compact Riemann surfaces.
• Any linear group of rational cohomological dimension two that arises as the fundamental group of a smooth complex projective variety is a (virtual) surface group.
• The holomorphic convexity of the universal cover imposes strong topological and geometric restrictions on the fundamental group.
• The cohomological dimension two condition forces the group to have the structure of a surface group, regardless of the ambient variety’s complexity.
• The result establishes a bridge between complex geometry (holomorphic convexity) and geometric group theory (surface groups).
• The classification holds even when the group is only virtually a surface group, indicating a robust structural constraint.