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arXiv · 1906.08769

Index Formulae for Line Bundle Cohomology on Complex Surfaces

Callum R. Brodie·Andrei Constantin·Rehan Deen·Andre Lukas
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Abstract

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.

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Callum R. Brodie, Andrei Constantin, Rehan Deen, Andre Lukas. 2019-06-20. Index Formulae for Line Bundle Cohomology on Complex Surfaces. https://doi.org/10.1002/prop.201900086

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