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Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers?

Edit: as written by Daniel Loughran in the comments below, complete intersections with the same multidegree are diffeomorphic (apparently this result is attributed to R. Thom). And we know that he multidegree determines the Hodge numbers see the appendix of F. Hirzebruch's book "Topological methods in algebraic geometry".

Thus we need to find two diffeomorphic complete intersections with different multidegrees such that their Hodge numbers are different.

Edit 2: Oscar Randall-Williams suggested to test examples of 3-folds due to Libgober and Wood, this is a very good idea, but they have the same Hodge numbers (I made the computations via Sage macros written by Donu Arapura). I also tested examples in W. Ebeling "An example of two homeomorphic, nondiffeomorphic complete intersection surfaces." Inventiones mathematicae 99.3 (1990): 651-654 where we can encounter two homeorphic nondiffeomorphic complete intersection surfaces, and get that they have the same Hodge numbers.

Edit 3: Related to this question there is this paper: "The Hodge ring of Kähler manifolds", Compositio Math. 149 (2013), 637--657 by D. Kotschick and S. Schreieder where they determine which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants.

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  • $\begingroup$ somehow related: mathoverflow.net/questions/42744/… $\endgroup$
    – Francesco Polizzi
    Commented Jul 1, 2014 at 20:06
  • $\begingroup$ and also mathoverflow.net/questions/42709/question-about-hodge-number/… $\endgroup$
    – Francesco Polizzi
    Commented Jul 1, 2014 at 20:07
  • $\begingroup$ A result of Hartshorne says that the Hilbert scheme of closed subschemes of projective space with given Hilbert polynomial is connected. Therefore it seems that any two smooth complete intersections with the same type (i.e. number and degrees of equations) are deformation equivalent (hence diffeomorphic) and so have the same Hodge numbers. $\endgroup$
    – Daniel Loughran
    Commented Jul 1, 2014 at 21:40
  • $\begingroup$ Thank you Daniel, apparently R. Thom also proved this result thanks to purely topological arguments. $\endgroup$
    – David C
    Commented Jul 2, 2014 at 6:25
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    $\begingroup$ Thom's result follows from the simple fact that it is a certain projective space (one coordinate for each possible coefficient in the defining polynomials) minus the Zariski closed subspace corresponding to coefficients defining singular intersections, and the latter has positive complex codimension, so real codimension > 2. $\endgroup$
    – nsrt
    Commented Jul 2, 2014 at 9:17

2 Answers 2

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We got four pairs of diffeomorphic complete intersections but with Hodge numbers different. Please check the link:here

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In

Libgober, Anatoly S., Wood, John W. ``Differentiable structures on complete intersections. II." Singularities, Part 2 (Arcata, Calif., 1981), 123–133.

the authors claim that a computer search, based on their classification technique, shows that $X_3(16,10,7,7,2,2,2)$ and $X_3(14,14,5,4,4,4)$ are diffeomorphic. Unfortunately I don't know how to compute Hodge numbers, but these would seem to be good candidates.

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  • $\begingroup$ Thank you Oscar, unfortunately they have the same Hodge numbers which are: [76832001, 457542401, 457542401, 76832001] $\endgroup$
    – David C
    Commented Jul 2, 2014 at 8:34

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