Hodge Numbers of Complete Intersections

Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p ≤ dim C X we have a sheaf Ω p (E) whose sections are holomorphic (p, 0)-forms with coefficients in E. We set and we define the holomorphic Euler characteristics χ p (X, E) := q≥0 (−1) q h p,q (X, E). It is convenient to introduce the generating function of these numbers χ y (X, E) := p≥0 y p χ p (X, E). Observe that Ω p (E) ∼ = Ω 0 (Λ p T * X 1,0) so that h p,q (X, E) = h 0,q (X, Λ p T * X 1,0 ⊗ E) and χ p (X, E) = χ 0 (X, Λ p T * X 1,0 ⊗ E). If E is the trivial holomorphic line bundle C then we write χ y (X) instead of χ y (X, C). Observe that χ y (X) | y=−1 = p,q (−1) p+q h p,q (X) = χ(X), χ n−p (X) = (−1) n χ p (X). Hence for n = 1 we have χ(X) = 2χ 0 (X), while for n = 2 we have χ(X) = 2χ 0 (X) − χ 1 (X). Example 1.1. X = P N then h p,q (P N) = 1 if 0 ≤ p = q ≤ N 0 if p = q. Hence χ p (P N) = (−1) p , ; χ −y (P N) = N p=0 y p = y N +1 − 1 y − 1 .