Index formulae for line bundle cohomology on complex surfaces. (English) Zbl 1537.14060
Summary: 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.
© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
MSC:
| 14J30 | \(3\)-folds |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 32S25 | Complex surface and hypersurface singularities |
| 32J15 | Compact complex surfaces |
| 32J17 | Compact complex \(3\)-folds |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
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