Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Good morning,

I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.

My question: Are there other holomorphic objects associated with a compact complex manifold? I'm interested in the objects whose shape is well understood. E.g, an Albanese torus is just a torus, and we know its cohomology, its kahler form etc.

Any help is appreciated. Thanks in advance,

Duc Anh

share|improve this question
    
Intermediate Jacobians? –  S. Carnahan Mar 18 '13 at 7:16
    
Thank you. I hear about it for the first time. –  Đức Anh Mar 18 '13 at 7:22
    
en.wikipedia.org/wiki/Intermediate_Jacobian, but this is for the Kähler case. Aren't you mostly interested in the more general case of compact complex manifolds? –  diverietti Mar 18 '13 at 7:49
    
Thank you. I'm interested in the both cases : compact general complex manifolds, and Kahler manifolds. I need some objects to orient my studies. If you know anything, please give me some informations. –  Đức Anh Mar 18 '13 at 9:47

1 Answer 1

up vote 1 down vote accepted

Besides the intermediate Jacobians for Kaehler manifolds, any compact complex manifold has a rational morphism to an algebraic variety, called the algebraic reduction, so that the morphism is an isomorphism of the fields of rational functions: K. Ueno, Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics, Vol. 439. Springer-Verlag, Berlin-New York, 1975. Careful: the algebraic reduction is only defined up to birational isomorphism. If the canonical bundle is effective, then one can also define the canonical map, from the compact complex manifold to projective space. There are also moduli spaces of coherent sheaves. None of these really have their shape understood though. The biholomorphism group of a compact complex manifold is a finite dimensional complex Lie group, so its shape is essentially understood.

share|improve this answer
    
thank you very much for the informations. –  Đức Anh Mar 18 '13 at 18:50

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.