User craig - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:13:55Z http://mathoverflow.net/feeds/user/10749 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119590/normalized-hamiltonian-holomorphic-vector-fields-on-sasakian-manifolds/126465#126465 Answer by Craig for Normalized Hamiltonian holomorphic vector fields on Sasakian manifolds Craig 2013-04-04T01:14:28Z 2013-04-04T01:14:28Z <p>The equation </p> <p>$$\int (\Delta^h u \cdot v - u \Delta^h v)e^h (1/2 d\eta)^n \wedge \eta$$</p> <p>already gives a proof. $u$ and $v$ are eigenvalues of $\Delta^h$, so the integrand is zero.</p> <p>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</p> <p>$$(\operatorname{Ham},[,]) \rightarrow (\chi,[,])$$</p> <p>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.</p> http://mathoverflow.net/questions/62492/non-kahler-manifolds-and-the-ddc-lemma/62507#62507 Answer by Craig for Non-Kahler manifolds and the dd^c-lemma Craig 2011-04-21T07:56:51Z 2011-04-21T07:56:51Z <p>I think this is an example.</p> <p>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.</p> http://mathoverflow.net/questions/62409/does-every-smooth-hypersurface-in-cpn-admit-a-kahler-einstein-metric Does every smooth hypersurface in CP^n admit a Kahler-Einstein metric Craig 2011-04-20T13:42:24Z 2011-04-20T13:42:24Z <p>Is it a solved problem whether every smooth hypersurface X in CP^n admits a Kaehler-Einstein metric? Of course the c_1&lt;0 and c_1 =0 cases are old, and every Fermat hypersurface does. It seems that existing knowledge implies they all do.</p> <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/21258/looking-for-almost-complex-structure-on-a-contact-manifold-invariant-under-flow-o/46017#46017 Answer by Craig for Looking for almost complex structure on a contact manifold invariant under flow of Reeb vector field !? Craig 2010-11-14T04:48:05Z 2010-11-14T04:48:05Z <p>Mohammad,</p> <p>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 $$g=\frac{1}{2}d\eta(\cdot,J\cdot) +\eta\otimes\eta$$.</p> <p>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.) </p> <p>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. </p> http://mathoverflow.net/questions/45666/holomorphic-automorphism-of-strictly-psudo-convex-domain-smooth-on-boundary Holomorphic automorphism of strictly psudo-convex domain smooth on boundary Craig 2010-11-11T08:34:28Z 2010-11-11T08:34:28Z <p>I am wondering if anything is known about this. I couldn't find anything in the literature.</p> <p>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 $\sigma:\overline{D}\rightarrow\overline{D}$.</p> <p>My question: Is anything known about this problem for a strictly pseudoconvex domain $D\subset M$ in a complex manifold $M$?</p> <p>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. </p> http://mathoverflow.net/questions/68316/extremal-fano-with-non-constant-scalar-curvature-vs-kaehler-einstein-fano-manifol/68408#68408 Comment by Craig Craig 2013-03-17T03:51:50Z 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 <a href="http://arxiv.org/abs/hep-th/0603021" rel="nofollow">arxiv.org/abs/hep-th/0603021</a> and <a href="http://arxiv.org/abs/math/0607586" rel="nofollow">arxiv.org/abs/math/0607586</a> that the Futaki invariant vanishes on a subgroup iff a volume functional is minimized on a &quot;moment cone&quot;. The symplectic structure on the cone is not changed, thus the volume functional remains minimized. http://mathoverflow.net/questions/62409/does-every-smooth-hypersurface-in-cpn-admit-a-kahler-einstein-metric Comment by Craig Craig 2011-04-21T17:19:00Z 2011-04-21T17:19:00Z I don't think the paper that inspired me is posted on the arXiv yet. http://mathoverflow.net/questions/62516/holomorphic-line-bundles-on-a-punctured-disc/62521#62521 Comment by Craig Craig 2011-04-21T15:12:25Z 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.