On general type surfaces with $$q=1$$ q = 1 and $$c_2 = 3 p_g$$ c 2 = 3 p g

Let S be a minimal surface of general type with irregularity $$q(S) = 1$$ . Well-known inequalities between characteristic numbers imply that $$\begin{aligned} 3 p_g(S) \le c_2(S) \le 10 p_g(S), \end{aligned}$$ 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 $$\#\mathcal {M}_n \rightarrow \infty $$ as $$n \rightarrow \infty $$ . 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.