Theoretical analysis demonstrates bounded cohomology in Mori dream and ruled surfaces, indicating uniform geometric constraints on curve cohomology.
In this paper, we first prove that every Mori dream surface [Formula: see text] satisfies the bounded cohomology property (BCP for short). Namely, there exists a constant [Formula: see text] such that [Formula: see text] for every curve [Formula: see text] on [Formula: see text]. We then prove that there is a positive constant [Formula: see text] such that [Formula: see text] for every ample curve [Formula: see text] on a geometrically ruled surface [Formula: see text] over a curve of genus [Formula: see text], and [Formula: see text] satisfies the BCP if [Formula: see text].
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Hua et al. (2026) studied this question.
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