Seminar/exam AC and GC geometries course

[Rafael Torres]

In fulfillment of the requirements for the course “Almost-complex and 
generalized complex geometries”, Giuseppe Bargagnati (Pisa) will give 
the following seminar on May 28th, 2021 at 11 am.


Title: Uniqueness of CP^n?

Abstract: 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.

See attachment for the bibliography




Sujet : Uniqueness of CP^n
Heure : 28 mai 2021 11:00 AM Rome

Participer à la réunion Zoom
https://sissa-it.zoom.us/j/83643386694?pwd=K29QeUtoVk0wZkY1Rk9rMU9LV0dHUT09

ID de réunion : 836 4338 6694
Code secret : 655409