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I want to know about people in researching complex (maybe differential) geometry are careing about what currently ? For example ,$L^2$ estimate inspired by Lars Hormander is a very useful tool,and how does this theory be developed currently ? As myself , i like this method very much ,but i don't know which is the next important problem be solved by this method . How far will this method go ? As well , just like the holomorphic morse inequalities , when it is proved by Demailly in 1985,twenty years passed , it seems that during thest twenty years there are no important results comes out in complex geometry ? I'm a beginner in complex geometry, i think that i can't scratch the direction of complex geometry ? I don't know people are careing about what in complex geometry ?Only doing some little questions or leave this field? So this is just the purpose of this question i asked , i want to communicate with all who are interested in this field.

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Complex geometry is quite a wide area, you may want to tag your question as a "big list". Could you also explain a little bit of motivation (are you interested in starting research in complex geometry?), so that the answers can be focused. – Benoît Kloeckner Apr 5 '10 at 9:15
1. It is unlikely that MathOverflow be the right place to ask your question, since I don't see a precise one. – Benoît Kloeckner Apr 5 '10 at 16:58
2. If you expect a useful answer, you should at the very least explain who you are (a graduate student?) and what is your aim exactly. – Benoît Kloeckner Apr 5 '10 at 16:58
3. One could think that you are searching for someone telling you: "with this method, you will be able to solve this important problem." Then what you are searching for is close to a PhD advisor. – Benoît Kloeckner Apr 5 '10 at 16:59
4. Claiming that seemingly nothing important in complex geometry has been done in the last twenty years is really not something that will encourage people to answer you. It is a very active research field, with loads of results, and even citing one (like the proof by Siu that the complex projective space do not admit Levi-flat hypersurfaces in dimension $3$ or more) seems odd since there are so many others. – Benoît Kloeckner Apr 5 '10 at 17:05
up vote 12 down vote accepted

One major area of research is that of canonical metrics on Kahler manifolds. The original definition of "canonical" is due to Calabi. One considers all metrics in a fixed Kahler class and attempts to minimise the L^2-norm of the curvature tensor. The Euler-Lagrange equations say that a metric is a critical point for this functional precisely when its scalar curvature has holomorphic gradient. Such metrics are called extremal. Special cases include Kahler-Einstein metrics and constant scalar curvature metrics.

Enormous amounts of research have been done in this direction. In one direction, one tries to prove existence. A random selection of results: Aubin and Yau's work on Kahler-Einstein metrics of negative and zero scalar curvature, Tian's work on Kahler-Einstein surfaces of positive scalar curvature, more recently Donaldson's work on extremal metrics on toric surfaces.

In the opposite direction, one tries to find obstructions to existence. Futaki found one such obstruction, involving holomorphic vector fields, which was then vastly generalised by Tian and Donaldson. This gives examples of many manifolds and Kahler classes which can never admit extremal metrics.

There is a conjecture (due in various forms to Donaldson, Tian and Yau) which says that the known obstructions are the only obstructions. When they vanish, an extremal metric exists. This would be a beautiful result because the obstructions are purely algebro-geometric (at least when the Kahler class is integral) yet the conclusion is analytic - we can find a metric which solves a PDE. If you know of the Hitchin-Kobayashi correspondence for Hermitian Yang-Mills metrics on vector bundles, this conjecture can be seen as the analogue for Kahler metrics.

At the moment, the conjecture is known to hold for toric surfaces (Donaldson) and is close to being settled for Kahler-Einstein metrics. This is due to Aubin-Yau for negative scalar curvature, and Yau for zero scalar curvature - in these cases the metrics always exist. The positive scalar curvature case is far more delicate. Donaldson recently announced significant progress in this direction.

If you are looking for a link with Hormander's L^2 estimates, then look no further than projective embeddings via higher and higher powers of a positive line bundle L over your Kahler manifold X. For each power of L we consider the following problem: find a basis of holomorphic sections of L so that the image of X in CPn has zero centre of mass (to define centre of mass we think of CPn as a coadjoint orbit in the linear space su(n+1)* equiped with an inner-product via the Killing form). This can be thought of as a finite dimensional analogue of the problem of finding an extremal metric representing the first Chern class of L. As the power of L tends to infinity, these projective problems converge in some sense to the problem of finding an extremal metric. Understanding the precise nature of this convergence involves difficult questions concerning the Bergman kernel and, ultimately, Hormander's estimate. (Again, this part of the story is due to Donaldson.)

To get started you could look at:

  • Extremal Kahler metrics 1 and 2, by Calabi.
  • Canonical metrics in Kahler geomtery, by Tian.
  • Remarks on gauge theory, complex geometry and 4-manifold topology, by Donaldson.
  • Scalar curvature and stability of toric varieties, by Donaldson.
  • Scalar curvature and projective embeddings 1, by Donaldson.
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thanks very much! – HKSHLZW Dec 10 '10 at 4:44

There also problems along the lines of proving L2 extension theorems for vector valued forms (there are such theorems already like the Ohsawa-Takegoshi extension theorem however, that applies to (n,1) forms. We want more general results). Besides, there is the problem of deformation invariance of plurigenera over Kahler manifolds (Siu proved it for the projective case).

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thanks very much! – HKSHLZW Dec 10 '10 at 4:44

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