Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

Understand

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles.

• The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers.
• In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces.
• In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi-Yau three-folds.