A complete intersection has a multidegree, written as the tuple (properly though a multiset) of the degrees of defining hypersurfaces. For example taking quadrics in P3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position is an elliptic curve. The Hodge numbers of complex smooth complete intersections were worked out by Kunihiko Kodaira.
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