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\centerline{\large \bf Equivalence relations for algebraic surfaces.}

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\centerline{Fabrizio Catanese}
\centerline{\it Universit\"at at Bayreuth.}


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 Deformation equivalent surfaces of general type are 
diffeomorphic, and actually symplectomorphic for the standard 
symplectic form corresponding to the canonical class.
Conversely, diffeomorphism and symplectomorphism do not guarantee 
deformation equivalence. A coarser equivalence is gotten is the QED 
equivalence, generated by

1) birational equivalence

2) deformations whose fibres have canonical singularities, and

3) morphisms which are etale in codimension 1.

Kodaira dimension is an invariant for QED equivalence.
Are surfaces of general type all QED equivalent ?
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