Index formulae for line bundle cohomology on complex surfaces

Journal article

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.

Authors: Brodie, CR; Constantin, A; Deen, R; Lukas, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Fortschritte der Physik / Progress of Physics
• Volume: 68
• Issue: 2
• Article number: 1900086
• Publication date: 2020-01-26
• DOI: 10.1002/prop.201900086
• EISSN: 1521-3978
• ISSN: 0015-8208
• Language: English
• Keywords: FFR; del Pezzo surfaces; cohomology; Hirzebruch surfaces; string theory