Smooth projective surfaces with bounded cohomology property

TL;DR

This work addresses when smooth projective surfaces satisfy the bounded cohomology property ($BCP$). It proves that all Mori dream surfaces admit uniform control of $l_C$ and hence $BCP$, using the finite semiample generating set of $ ext{Nef}(X)$ and Li's results for curves with $D^2=0$; it further establishes a bound on $l_C$ for ample curves on geometrically ruled surfaces over curves of genus $g$, with $BCP$ guaranteed when $g\le1$. The results connect to the Bounded Negativity Conjecture through the $BCP$ framework and leverage cone geometry, vanishing theorems, and Riemann–Roch arguments. The paper provides detailed examples and case analyses (including ρ=2 surfaces, ruled surfaces, and Mori dream surfaces) to illustrate where $BCP$ holds and how the bounds are obtained in practice.

Abstract

In this paper, we first prove that every Mori dream surface $X$ satisfies the bounded cohomology property (BCP for short). Namely, there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$. We then prove that there is a positive constant $m(Y)$ such that $l_C:=(K_Y\cdot C)(C^2)^{-1}\le m(Y)$ for every ample curve $C$ on a geometrically ruled surface $Y$ over a curve of genus $g$, and $Y$ satisfies the BCP if $g\le1$.

Theorems & Definitions (43)

• Conjecture 1.1
• Definition 1.2
• Proposition 1.4
• Proposition 1.5
• Remark 1.6
• Theorem 1.9
• Theorem 1.10
• Definition 2.1
• Definition 2.2
• Definition 2.3