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If two Fano projective manifolds are homeomorphic are they diffeomorphic?

There are examples with one manifold being Fano and the other of general type (Barlow surfaces).

  Moreover, the number of distinct smooth structures among sets of homeomorphic surfaces of general type can be arbitrarily large. There is also infinitely many non-diffeomorphic homeomorphic elliptic surfaces.

However, there is only finitely many deformation classes of Fano projective manifolds of any given dimension.

If two Fano projective manifolds are homeomorphic are they diffeomorphic?

There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of distinct smooth structures among sets of homeomorphic surfaces of general type can be arbitrarily large. There is also infinitely many non-diffeomorphic homeomorphic elliptic surfaces.

However, there is only finitely many deformation classes of Fano projective manifolds of any given dimension.

This paper may be relevant.

If two Fano projective manifolds are homeomorphic are they diffeomorphic?

There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of distinct smooth structures among sets of homeomorphic surfaces of general type can be arbitrarily large. There is, however, only finitely many deformation classes of Fano projective manifolds of any given dimension.

If two Fano projective manifolds are homeomorphic are they diffeomorphic?

There are examples with one manifold being Fano and the other of general type (Barlow surfaces). 

Moreover, the number of distinct smooth structures among sets of homeomorphic surfaces of general type can be arbitrarily large. There is also infinitely many non-diffeomorphic homeomorphic elliptic surfaces.

However, there is only finitely many deformation classes of Fano projective manifolds of any given dimension.

If two Fano projective manifolds are homeomorphic are they diffeomorphic? There are examples with one manifold being Fano and the other of general type.

If two Fano projective manifolds are homeomorphic are they diffeomorphic? 

There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of distinct smooth structures among sets of homeomorphic surfaces of general type can be arbitrarily large. There is, however, only finitely many deformation classes of Fano projective manifolds of any given dimension.