Index Formulae for Line Bundle Cohomology on Complex Surfaces

@article{Brodie2020IndexFF,
  title={Index Formulae for Line Bundle Cohomology on Complex Surfaces},
  author={Callum R. Brodie and Andrei Constantin and Rehan Deen and Andr{\'e} Lukas},
  journal={Fortschritte der Physik},
  year={2020},
  volume={68}
}
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… 

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