Hyper-Kähler quotients of solvable Lie groups

M. L. BARBERIS, I. DOTTI, A. FINO

Madrid

Congreso; International Congress of Mathematicians; 2006

International Mathematical Union,

Given a hyper-K\"ahler metric $g$ on a manifold $M$ and an action of a Lie group $K$ on $M$ preserving the hyper-K\"ahler structure, under certain topological assumptions there exists a $K$-equivariant moment map$$ \mu : M \to \frak k ^* \otimes \text{Im}\, \Bbb H$$where  $\Bbb H$ denotes the quaternions.  The quotient construction,due to N. Hitchin et al., gives a procedure to obtain anew hyper-K\"ahler structure as follows: let  $\xi =(\xi _1 , \xi _2 , \xi_3) \in k ^* \otimes \text{Im}\, \Bbb H$ be a regular value of  $\mu$ such that $\xi _i$ lies in the center of $\frak k ^*, \;  i=1,2,3$. If  the action of $K$ on $\mu ^{-1}(\xi)$ is free  and proper, then the quotient  $\mu ^{-1}(\xi) / K$ is hyper-K\"ahler. Although this construction is a  powerful method for showing existence of hyper-K\"ahler metrics,  finding the metric explicitely and studying completeness may be  difficult.  In this work we consider the case $M=G$, a Lie group with a left invariant hyper-K\"ahlermetric $g$ which is necessarily flat. Any such Lie group is two stepsolvable: its universal cover  is a semidirect product  $\Bbb H ^p\ltimes \Bbb H ^q$ where the action of the first  factor on thesecond is determined by fixing a torus in Sp$(q)$. We apply thequotient construction with $K=\Bbb R ^l \hookrightarrow \Bbb H ^p$acting on $G$ by left  translations, $1\leq l \leq 4p, \; l\leq q$.We can show that in this case $\mu ^{-1}(0) / K$ is closed in thehomogeneous space $K \backslash G$, thus the hyper-K\"ahler metricon the quotient is complete. When $l\leq p$ we show that $\mu^{-1}(\xi) / K$ is diffeomorphic to $4(p+q-l)$ dimensional Euclideanspace and we obtain the explicit description of the hyper-K\"ahlermetric on the quotient in local coordinates. We relate some of ourresults with those obtained by Gibbons-Rychenkova-Goto.