I am currently studying complex geometry following the book by Huybrechts. I wonder if two surfaces(smooth) in $\mathbb{CP}^n$ with the same hodge number are isomorphic. If yes, I would like some hints for the proof. If not, I would appreciate a counterexample. Recall hodge numbers are the dimensions of the Dolbeault cohomology. Thank you.
2026-04-01 15:46:03.1775058363
hodge number of complex surfaces
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Here are a class of counterexamples.
One can just consider the ruled surfaces $F_n$. These surfaces are not isomorphic for different $n$. However they have the same Hodge diamonds.
One reference is Arapura's notes https://www.math.purdue.edu/~arapura/preprints/partIV.pdf see Corollary 17.3.4 (Durfee).
And https://www.math.purdue.edu/~arapura/preprints/partII.pdf see Example 11.1.2