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Zbl 0659.53032
Cordero, Luis A.; Fernández, Marisa; de León, Manuel; Saralegui, Martín
Compact symplectic four solvmanifolds without polarizations. (English)
[J] Ann. Fac. Sci. Toulouse, V. Ser., Math. 10, No.2, 193-198 (1989). ISSN 0240-2955

The existence of symplectic manifolds which do not admit polarizations has significant implications for geometric quantization theory. In fact, few examples of such manifolds are known. For instance, $S\sp 2\times S\sp 2$ has no polarizations with nonzero real index, but it admits a Kähler polarization. On the other hand, a symplectic manifold carries totally complex (resp. Kähler) polarizations (that is, with zero real index) if and only if it admits compatible complex (resp. Kähler) structures. Therefore, the manifolds $E\sp 4$ constructed by the second author, {\it M. J. Gotay} and {\it A. Gray} [Proc. Am. Math. Soc. 103, No.4, 1209-1212 (1988; Zbl 0656.53034)] (which are circle bundles over circle bundles over a torus $T\sp 2)$ with first Betti number 2 or 3 have no Kähler polarizations and, moreover, if $b\sb 1(E\sp 4)=2$, then they have no totally complex polarizations. But all of these symplectic manifolds often have real polarizations. \par Recently, {\it M. J. Gotay} [Monatsh. Math. 103, 27-30 (1987; Zbl 0608.53032)] described a class of symplectic 4-manifolds which do not admit polarizations of any type whatever. These manifolds are constructed by repeatedly blowing up $E\sp 4$ with $b\sb 1(E\sp 4)=2$. This construction has been extended by the second author and the third author [Compact symplectic four-dimensional manifolds not admitting polarizations (preprint)] by considering circle bundles over circle bundles over a Riemann surface of genus $g>1.$ \par In this paper, following Gotay's construction, a class of compact 4- dimensional symplectic manifolds $M\sb{\lambda}(k)$ is obtained by blowing up a certain manifold $M\sp 4(k)$ at $\lambda$ distinct points. Here $M\sp 4(k)$ is a compact symplectic solvmanifold constructed in [the second author and {\it A. Gray}: Compact symplectic four-dimensional solvmanifolds not admitting complex structures (preprint)]. Although $M\sp 4(k)$ has all the topological properties of a Kähler manifold it has no complex (and hence no Kähler) structures; therefore, $M\sp 4(k)$ has no totally complex (and hence no Kähler) polarizations. Moreover, we prove that $M\sb{\lambda}(k)$ has no polarizations with nonzero real index. We don't know if $M\sb{\lambda}(k)$ admits or not totally complex polarizations; if they do, this fact would be very interesting for the Kählerian geometry realm because they would provide new examples of compact Kähler manifolds.
[L.A.Cordero]

MSC 2000:: *53C15 Geometric structures on manifolds
53C55 Complex differential geometry (global)
53D50 Geometric quantization

Keywords: Kähler polarization; symplectic manifolds; solvmanifold; Kähler manifold

Citations: Zbl 0656.53034; Zbl 0608.53032

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