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Does anyone have an example or know any references for a complex manifold $M$ with two different holomorphic structures that give rise to different Hodge numbers?

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  • $\begingroup$ pleasantpheasant -- how do you define the Hodge numbers in the non-K\"ahler case? $\endgroup$
    – algori
    Commented Jul 13 at 19:52
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    $\begingroup$ algori - The dimension of the Dolbeault cohomology groups on M $\endgroup$
    – pleasantpheasant
    Commented Jul 13 at 19:55
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    $\begingroup$ In a complex manifold the holomorphic structure is fixed. You probably mean "a smooth manifold $M$ with two different complex structures"? $\endgroup$
    – YCor
    Commented Jul 14 at 4:55

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A well-studied example is given by the Iwasawa manifold, see 1.4.3 in Angella's thesis here https://arxiv.org/abs/1302.0524.

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