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A complete intersection is an algebraic variety cut out by homogenous polynomials. Geometrically, this is the intersection of hypersurfaces in complex projective space.

Below, let's confine to the non-singluar case. These are smooth manifolds and much topological information could be read out from the degrees of the polynomials used. see wikipedia

https://en.wikipedia.org/wiki/Complete_intersection#Topology

and manifold atlas

http://www.map.mpim-bonn.mpg.de/Complete_intersections

for a survey.

I am interested in the smooth structures on the homeomorphism types of complete intersection.

In particular, are there two complete intersections which are homeomorphic but not diffeomorphic? In the paper

An example of two homeomorphic, nondiffeomorphic complete intersection surfaces. Invent. Math. 99 (1990), no. 3, 651–654.

Wolfgang Ebeling gave an example in the surface case. How about higher dimensions?

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    $\begingroup$ "A complete intersection is an algebraic variety cut out by homogenous polynomials" - A complete intersection in projective space is a variety cut out by a number of homogeneous equations equal to its codimension. $\endgroup$
    – Qfwfq
    May 1, 2019 at 13:34
  • $\begingroup$ The Manifold Atlas page you link to contains information on the (topological) Sullivan conjecture, which says that complete intersections are (homeomorphic) diffeomorphic iff they have the same multi-degree, Pontryagin classes and Euler number. Crowley and Nagy announced a proof of the Sullivan conjecture in complex dimension 4 (newton.ac.uk/seminar/20181206160016301) but I don't know if the details are written up yet. $\endgroup$
    – Mark Grant
    May 8, 2019 at 10:25

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