# Boundary behavior of Kähler cone with curvature restriction

Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.

A fundamental result is due to Demailly and Paun: they proved that every nef and big class $\alpha$ contains a Kaehler current $T$. Moreover, $T$ is smooth outside some subvariety. Note that here 'big' means its mass on the manifold is positive, and this condition is crucial in their proof.

In intuition, cuvature restriction will effect the boundary behavior.For example,if $(M,\omega)$ has positive sectional curvature (Frenkel's conjecture solved by Yau and Siu),then the cone is just $R_{+}$.So in that case,the boundary is zero. Recently, Daming Wu, Fangyang Zheng and S.T. Yau proved a result which says that every nef class (without big restriction) contains a smooth non-negtive (1,1) form under a strong condition of curvature. They proved their theorem by implicit function theorem.

In order to understand Kähler cone's boundary, we can use complex Monge-Ampère equation as a tool. Indeed,it's a family of equations with the Kähler metric changing.

I interpret this method as choosing a better approximating sequence of metrics. As I know when we deal with a single complex Monge-Ampère equation, the curvature condition does not play a very important role, e.g., Yau's 1978 paper on Calabi's conjecture.

I do believe that cuvature restriction will effect Kähler cone's boundary, then my problem is how cuvature restriction comes into that family of complex Monge-Ampère equations.

I hope some expert can give me some insight.

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I've edited the question to improve readability: mostly spacing, punctuation and diacratics. I have not altered the meaning, I hope. You can, of course, always roll back to the previous version. – José Figueroa-O'Farrill Dec 29 '10 at 11:37
Dear 肖建, Can you add an alternative spelling of your name in English or Latin? Thank you, Вадим. – Wadim Zudilin Dec 30 '10 at 13:51