Abstract
In this note, we prove combinatorial formulas for the Hodge number h2,1 of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold Y4. We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than h1,1(Y4) non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric complete intersection Calabi-Yaus are the subjects of the sequel.
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