# On general type surfaces with $q=1$ and $c_2 = 3 p_g$

## TL;DR

This paper constructs and analyzes minimal general type surfaces with irregularity one, geometric genus n, and Euler characteristic 3n, showing the moduli space is finite but grows infinitely large as n increases, and relates to complex hyperbolic 2-manifolds.

## Contribution

It demonstrates the existence of surfaces with specific invariants and characterizes the growth of their moduli space, linking algebraic geometry with complex hyperbolic geometry.

## Key findings

- Existence of surfaces with q=1, p_g=n, c_2=3n for all n ≥ 1.
- Moduli space of such surfaces is finite for each n.
- Number of such surfaces tends to infinity as n increases.

## Abstract

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the topological Euler characteristic. Surfaces achieving equality for the upper bound are classified, starting with work of Debarre. We study equality in the lower bound, showing that for each $n \ge 1$ there exists a surface with $q = 1$, $p_g = n$, and $c_2 = 3n$. The moduli space $\mathcal{M}_n$ of such surfaces is a finite set of points, and we prove that $\#\mathcal{M}_n \to \infty$ as $n \to \infty$. Equivalently, this paper studies the number of closed complex hyperbolic $2$-manifolds of first betti number $2$ as a function of volume; in particular, such a manifold exists for every possible volume.