Mathematics

It  follows  from  the  very  definition  that  complex  structures  are  more  rigid than topological (or smooth) structures;  in other words, if two complex manifolds M and N are homeomorphic, it is not true in general that they are also biholomorphic.  For example, while there is only one topological closed connected surface of genus 1, it is very well known that not all the complex structures on the 2-torus are biholomorphic.  However, if one of the manifolds is the complex projective space CP^n, we will show that, if M is Kaehler and homeomorphic to CP^n, then it is also biholomorphic to it.  This result was proven by Hirzebruch and Kodaira in 1957.  A natural question is whether we can drop the Kaehler hypothesis in this theorem; we will see that this (open) problem in complex dimension 3 is strongly related to the existence of complex structures on S^6. In complex  dimension  2  there  is  an  even  stronger  result,  that  was  proven  by Yau in 1977:  if M is a compact complex surface homotopy equivalent to CP^2, then it is also biholomorphic to it.  This statement was previously known also as ”the Severi Conjecture”. In the talk, we will present these results, and we will try to give an outline of the proofs, that will involve characteristic classes and some facts about Kaehler manifolds.  Our main reference will be an expository article of Tosatti.