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On the existence of balanced metrics on six-manifolds of cohomogeneity one. (English) Zbl 1500.53079

The Fino-Vezzoni conjecture predicts that a compact Hermitian manifold with a balanced metric (i.e., \(d\omega^{n-1}=0\)) and a pluriclosed metric (i.e., \(\partial \bar{\partial} \omega =0\)) also supports a Kähler metric (i.e., \(d\omega=0\)). There have been several cases where the conjecture has been verified: If the metric is simultaneously balanced and pluriclosed it must be Kähler. Chiose verified the conjecture for complex manifolds in the Fujiki class \(\mathcal{C}\), and Verbitsky verified the conjecture for twistor spaces over anti-self-dual fourfolds, just to name a few results. An interesting case is given by the \(k\)-fold connected sum of \(\mathbb{S}^3 \times \mathbb{S}^3\), which we write as \(X_k : = \sharp_k (\mathbb{S}^3 \times \mathbb{S}^3)\). Fu-Li-Yau produced balanced metrics on \(X_k\) (\(k \geq 2\)) via conifold transitions. Michelsohn showed that \(X_1 = \mathbb{S}^3 \times \mathbb{S}^3\) with the Calabi-Eckmann complex structure does not admit a balanced metric; it is known that \(X_1\) does admit a pluriclosed metric, however. The present article addresses the existence of balanced metrics on complex threefolds using symmetries. The main results are the following:
Theorem A. Let \(M\) be a \(6\)-dimensional simply connected cohomogeneity-one manifold under the almost effective action of a connected Lie group \(G\), and let \(K\) be the principal isotropy group. Then the principal part \(M^{\text{princ}}\) admits a \(G\)-invariant balanced non-Kähler \(\text{SU}(3)\)-structure if and only if \(M\) is compact and \((\mathfrak{g}, \mathfrak{t}) = (\mathfrak{su}(2) \oplus 2\mathbb{R}, \{ 0 \})\).
The second main theorem shows that none of these local solutions can be extended to global solutions:
Theorem B. Let \(M\) be a 6-dimensional simply connected cohomogeneity-one manifold under the almost effective action of a connected Lie group \(G\). Then \(M\) admits no \(G\)-invariant balanced non-Kähler \(\text{SU}(3)\)-structures.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
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