Holomorphic spheres in hyperkähler twistor spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:15:48Z http://mathoverflow.net/feeds/question/49336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49336/holomorphic-spheres-in-hyperkahler-twistor-spaces Holomorphic spheres in hyperkähler twistor spaces 680 2010-12-14T01:21:21Z 2010-12-14T04:40:01Z <p>Fix a hyperkähler manifold $X$ and an identification of $S^2$ with the hyperkähler sphere of $X$. Now consider the twistor space $T := S^2\times X$ equipped with the tautological complex structure. For each $x\in X$, we have a holomorphic map $u_x:S^2\to T$ defined by $u_x(\theta):=(\theta,x)$.</p> <p><strong>Question:</strong> Is every holomorphic map $u:S^2 \to T$ which satisfies <code>${\rm pr}_1\circ u={\rm id_{S^2}}$</code> of this form?</p> http://mathoverflow.net/questions/49336/holomorphic-spheres-in-hyperkahler-twistor-spaces/49347#49347 Answer by Sasha for Holomorphic spheres in hyperkähler twistor spaces Sasha 2010-12-14T03:57:04Z 2010-12-14T04:40:01Z <p>I think preprint arXiv:1006.0440 of Jardim is Verbitsky will answer your question. In short the answer is no, since if $\dim_C X = n$ then the deformation space of sections has dimension $\dim H^0(S^2,N_{S^2/S^2\times X}) = \dim H^0(P^1,O_{P^1}(1)^n) = 2n$ which is twice the dimension of the space which you consider.</p>