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Is the Göttsche's formula (Eq (2.1) of this paper) expressing the Poincare polynomial (or the Euler char version) of the Hilbert scheme of points on a projective surface valid for non-compact complex surfaces? Any references would be greatly appreciated.

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Yes. This is Theorem 5.2.1 of The Douady space of a complex surface by de Cataldo and Migliorini (Math.Ann.)

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