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Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property?

Being virtual, defined by inclusion-exclusion from complete varieties, they're not individually positive. What I'd most like is to hear "The Euler characteristic is nonnegative". But that's not true for $X$ a quintic 3-fold, $D$ empty.

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Allen, could you please give a bit of motivation so it would be more clear what you are looking for? – Dmitri Mar 5 '10 at 13:26
"The following linear combination of the (p,p) Hodge numbers is always positive" would be particularly nice. One example: X a toric variety, D its boundary. Then X \ D has zero Euler characteristic, since it's a torus. – Allen Knutson Mar 5 '10 at 19:41
Ack, that's not what I meant to say. The virtual Poincare polynomial p(d) of the n-torus is (d-1)^n, computable by e.g. sticking it in (P^1)^n. There the sort of positivity statement I'm looking for is that p(d+1) has nonnegative coefficients (namely, it is d^n). – Allen Knutson Mar 8 '10 at 12:06

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