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. 



Hodge numbers of diffeomorphic complete intersections
We got four pairs of diffeomorphic complete intersections but with Hodge numbers different. Please check the link:[here](https://arxiv.org/abs/2004.07142)


