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

Abstract 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. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.