Existence of closed manifolds with more than 3 linearly independent complex structures? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:01:07Z http://mathoverflow.net/feeds/question/57900 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57900/existence-of-closed-manifolds-with-more-than-3-linearly-independent-complex-struc Existence of closed manifolds with more than 3 linearly independent complex structures? Doris Hein 2011-03-09T01:46:56Z 2011-03-09T13:44:48Z <p>A Riemannian manifold is <em>hyperkähler</em>, if there are three complex structures \$I,J,K\$, which are all compatible with the Riemannian metric (i.e., \$(v,Iw)\$ defines a symplectic form and similarly for \$J\$ and \$K\$). Furthermore, we also need the complex structures to satisfy the quaternionic relations \$I^2=J^2=K^2=-1\$ and \$IJ+JI=0=IK+KI=JK+KJ\$.</p> <p>I was able to find only very few examples of closed (compact without boundary) hyperkähler manifolds, basically complex tori and K3-surfaces. In higher dimensions than complex dimension 2, there are also generalized Kummer varieties of tori. Based on K3-surfaces, there are Hilbert schemes and resolutions of singularities in some moduli spaces.</p> <hr> <p>I am searching for an example of a slightly more general setting: Let \$X\$ be a closed Riemannian manifold and \$J_1,..., J_r\$ be complex structures, all compatible with the Riemannian metric. Furthermore, assume that the \$J_l\$ to satisfy the relations \$J_l^2=-1\$ and \$J_lJ_k+J_kJ_l=0\$ for \$k\neq l\$.</p> <p>If we take \$X\$ to be a vector space of dimension \$2^{4a+b}c\$ with \$c\$ odd, I know the maximal number for \$r\$ is \$8a+2^b-1\$, i.e. the maximal number is always odd. This carries over to quotients of vector spaces, i.e. for tori and quotients of tori. These examples are flat.</p> <p>This yields the question: Is there a closed, non-flat manifold admitting more than three such complex structures?</p> <p>If \$X\$ admits more than three complex structures, it also admits three such structures. Therefore every such manifold has to be a hyperkähler manifold, i.e. one of the above mentioned examples or a product of them. Are there different examples and does one of them admit more complex structures than the 3 for hyperkähler?</p> http://mathoverflow.net/questions/57900/existence-of-closed-manifolds-with-more-than-3-linearly-independent-complex-struc/57932#57932 Answer by Misha Verbitsky for Existence of closed manifolds with more than 3 linearly independent complex structures? Misha Verbitsky 2011-03-09T08:27:50Z 2011-03-09T08:27:50Z <p>A manifold admitting a triple of complex structures satisfying quaternionic relations also admits a torsion-free connection (called "Obata connection") preserving the quaternionic structure. Such a connection is unique. For a hyperkaehler manifold, the Obata connection coinsides with the Levi-Civita. Therefore, the manifolds with this structure have Levi-Civita connection which preserves the complex structures \$J_i\$. From Berger's classification of holonomy it follows that they are products of hyperkaehler manifolds, and actually flat if \$i>3\$ (because \$J_4\$ must exchange the tangent bundles to the factors of the product).</p>