Bounded cohomology property on a smooth projective surface with Picard number two

TL;DR

This paper establishes conditions under which smooth projective surfaces with Picard number two satisfy a bounded cohomology property relating the dimensions of certain cohomology groups.

Contribution

It proves that surfaces with Picard number two meet the bounded cohomology property under specific geometric and Kodaira dimension conditions.

Findings

Surfaces with Picard number two satisfy the bounded cohomology property when 1) 01) the Kodaira dimension is at most 1 or (ii) the Kodaira dimension is 2, irregularity is zero, and the Iitaka dimension of a curve is 1.

The paper characterizes the bounded cohomology property in terms of the Mori cone and Kodaira dimension.

Provides new criteria for bounded cohomology property on algebraic surfaces.

Abstract

We say a smooth projective surface $X$ satisfies the bounded cohomology property if there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every prime divisor $C$ on $X$. Let the closed Mori cone $\mathrm{NE}(X)=\mathbb R_{\ge0}[C_1]+\mathbb R_{\ge0}[C_2]$ such that $C_1$ and $C_2$ with $C_2^2<0$ are some curves on $X$. If either (i) the Kodaira dimension $\kappa(X)\le1$ or (ii) $\kappa(X)=2$, the irregularity $q(X)=0$ and the Iitaka dimension $\kappa(X,C_1)=1$, then we prove that $X$ satisfies the bounded cohomology property.