User craig - MathOverflow

Answer by Craig for Normalized Hamiltonian holomorphic vector fields on Sasakian manifolds

2013-04-04T01:14:28Z

The equation

$$\int (\Delta^h u \cdot v - u \Delta^h v)e^h (1/2 d\eta)^n \wedge \eta$$

already gives a proof. $u$ and $v$ are eigenvalues of $\Delta^h$, so the integrand is zero.

This covers the case of $c_1(\mathcal{F})=ad\eta,\ a>0,$ but I'm pretty sure the proof works for general Sasakian manifolds. We have a Lie algebra homomorphism

$$ (\operatorname{Ham},[,]) \rightarrow (\chi,[,]) $$

taking $u_X$ to $X$ with poisson bracket $[u ,v]= \nabla^i u \nabla_i v - \nabla^i v \nabla_i u$ which has intgral zero. So there is always of splitting of the Lie algebra map when $M$ is compact.

Answer by Craig for Non-Kahler manifolds and the dd^c-lemma

2011-04-21T07:56:51Z

I think this is an example.

There is an integrable complex structure on X=S^3 xS^3. For topological reasons we have Td(X)[X]=1. Thus h^0,1 =0. It's an easy exercise that a complex manifold with h^0,1 =0 satisfies the d\bar{d} lemma.

Does every smooth hypersurface in CP^n admit a Kahler-Einstein metric

2011-04-20T13:42:24Z

Is it a solved problem whether every smooth hypersurface X in CP^n admits a Kaehler-Einstein metric? Of course the c_1<0 and c_1 =0 cases are old, and every Fermat hypersurface does. It seems that existing knowledge implies they all do.

For example they are all Chow-Mumford stable. G. Tian then proved that the K-energy is proper, so if there are no hol. vector fields there is a K-E metric.

I'm reading someone's paper that has a result that gives another proof. But I'm not sure if this paper is correct yet.

Answer by Craig for Looking for almost complex structure on a contact manifold invariant under flow of Reeb vector field !?

2010-11-14T04:48:05Z

Mohammad,

When there is an invariant almost complex structure on $\xi\subset V$, then $V$ has a metric contact structure called a ``K-contact'' structure. Specifically, the metric is $$ g=\frac{1}{2}d\eta(\cdot,J\cdot) +\eta\otimes\eta$$.

There are contact structures on $S^2 \times T^{2n-1}$ for which it is easy to see there is no metric preserved by the Reeb action. (I think the example is in a paper of S. Tolman on toric contact manifolds.)

If one assumes further that $(\xi,J)$ is an integrable CR-structure, then the above metric is Sasaki. This case is much more thoroughly studied.

Holomorphic automorphism of strictly psudo-convex domain smooth on boundary

2010-11-11T08:34:28Z

I am wondering if anything is known about this. I couldn't find anything in the literature.

In '74 C. Fefferman published a solution to the following problem. Let $\sigma:D\rightarrow D$ be an automorphism of a strictly pseudoconvex domanain $D\subset\mathbb{C}^n$. Then $\sigma$ extends to a smooth map $\sigma:\overline{D}\rightarrow\overline{D}$.

My question: Is anything known about this problem for a strictly pseudoconvex domain $D\subset M$ in a complex manifold $M$?

I have an idea of how to prove it for $K_M >0$, i.e. positive canonical bundle. Fefferman's approach used the Bergman metric. The Bergman metric is nondegenerate for more general domains only if $K_M$ is very ample, which is too strong an assumption to be very interesting.

Comment by Craig

2013-03-17T03:51:50Z

That's interesting, but sounds impossible. If the deformation preserves a positive dimensional subgroup of the automorphism group of the mukai-Umemura 3-fold, then the Futaki invariant has to vanish on the subgroup. This follows from an argument in arxiv.org/abs/hep-th/0603021 and arxiv.org/abs/math/0607586 that the Futaki invariant vanishes on a subgroup iff a volume functional is minimized on a "moment cone". The symplectic structure on the cone is not changed, thus the volume functional remains minimized.

Comment by Craig

2011-04-21T17:19:00Z

I don't think the paper that inspired me is posted on the arXiv yet.

Comment by Craig

2011-04-21T15:12:25Z

It you take C^2 \{0} you get an infinite family of topologically trivial line bundles because H^(0,1)_{\bar{\partial}} is infinite dimensional.