Mathematics – Differential Geometry
Scientific paper
2008-06-04
Mathematics
Differential Geometry
20 pages. To be published in J. Lond. Math. Soc
Scientific paper
A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-K\"ahler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the twist construction, we construct simply-connected astheno-K\"ahler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-K\"ahler (non strong KT) $2$-step nilmanifolds of complex dimension $4$ and we study deformations of strong KT structures on nilmanifolds of complex dimension $3$. Finally, we study the relation between astheno-K\"ahler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-K\"ahler metrics on $\T^2$-bundles over (non-K\"ahler) homogeneous complex surfaces.
Fino Anna
Tomassini Adriano
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