Abstract
Generalized complex geometry is a theory that unifies complex geometry and symplectic geometry into one single framework. It was introduced by Hitchin and Gualtieri around 2002.
In this thesis we address the following question: given a generalized complex manifold together with a submanifold, does the blowup of that submanifold admit again a generalized complex structure?
We give the following answers to this question: If the directions that are normal to the submanifold are purely complex, then the blowup admits a generalized complex structure for which the blowdown map is generalized holomorphic if and only if the induced Lie algebra on the conormal bundle of the submanifold is degenerate. This is a rather restrictive condition, but it is always satisfied if the submanifold is of complex codimension two.
On the other extreme, if all the normal directions are symplectic, then we show that the blowup always admits a generalized complex structure, provided the submanifold is compact.
We then also give an example of a submanifold that is not of the above two types, and show that it can not be blown up.
Finally, we also address the analogous question in generalized Kähler geometry. Given a generalized Kähler manifold together with a submanifold, we show that under certain geometric constraints on the submanifold the blowup is again generalized Kähler.
In this thesis we address the following question: given a generalized complex manifold together with a submanifold, does the blowup of that submanifold admit again a generalized complex structure?
We give the following answers to this question: If the directions that are normal to the submanifold are purely complex, then the blowup admits a generalized complex structure for which the blowdown map is generalized holomorphic if and only if the induced Lie algebra on the conormal bundle of the submanifold is degenerate. This is a rather restrictive condition, but it is always satisfied if the submanifold is of complex codimension two.
On the other extreme, if all the normal directions are symplectic, then we show that the blowup always admits a generalized complex structure, provided the submanifold is compact.
We then also give an example of a submanifold that is not of the above two types, and show that it can not be blown up.
Finally, we also address the analogous question in generalized Kähler geometry. Given a generalized Kähler manifold together with a submanifold, we show that under certain geometric constraints on the submanifold the blowup is again generalized Kähler.
Original language  English 

Awarding Institution 

Supervisors/Advisors 

Award date  30 Nov 2016 
Publisher  
Print ISBNs  9789039366745 
Publication status  Published  30 Nov 2016 
Keywords
 generalized complex geometry
 generalized Kähler geometry
 blowups