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We determine several classes of smooth complex projective surfaces on whichZariski decomposition can be combined with vanishing theorems to yieldcohomology formulae for all line bundles. The obtained formulae expresscohomologies in terms of divisor class intersections, and are adapted to thedecomposition of the effective cone into Zariski chambers. In particular, weshow this occurs on generalised del Pezzo surfaces, toric surfaces, and K3surfaces. In the second part we use these surface results to derive formulaefor all line bundle cohomology on a simple class of elliptically fiberedCalabi-Yau three-folds. Computing such quantities is a crucial step in derivingthe massless spectrum in string compactifications.

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