We conjecture and prove closed‐form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any line bundle cohomology in terms of an index. These formulae follow from general theorems we prove for a wider class of surfaces. In particular, we construct a map that takes any effective line bundle to a nef line bundle while preserving the zeroth cohomology dimension. For complex surfaces, these results explain the appearance of piecewise polynomial equations for cohomology and they are a first step towards understanding similar formulae recently obtained for Calabi‐Yau three‐folds.

中文翻译：

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复杂曲面上线束同调的索引公式

我们猜想并证明了del Pezzo和Hirzebruch曲面上线束的同调维度的封闭式索引表达式。此外，对于所有紧凑的复曲面，我们提供了一种简单的算法，可以根据索引来表达任何线束同调。这些公式来自我们针对更广泛的曲面类别所证明的一般定理。特别是，我们构造了一个映射，该映射将任何有效线束都转换为nef线束，同时保留了第零个同调维。对于复杂的曲面，这些结果解释了谐函数的分段多项式方程的出现，它们是了解最近针对Calabi-Yau三折式获得的相似公式的第一步。