Blow-ups and resolutions of strong Kähler with torsion metrics. (English) Zbl 1168.53014
A Hermitian metric \(g\) on a complex manifold \((M,J)\) is said to be strong Kähler with torsion (strong \(KT)\) if the fundamental 2-form \(F\) satisfies \(\partial\overline\partial F=0\). Since this condition is weaker than the Kähler one, any Kähler metric is strong \(KT\). The authors investigate which properties that hold for Kähler manifolds are still valid in the context of strong \(KT\) geometry. Firstly, they show that the behaviour of strong \(KT\) structures under small deformations of the complex structure depends on the complex dimension of the given manifold. In fact, a result of Gauduchon helps in proving that, if \((M,J)\) is a compact complex surface, then the strong \(KT\) condition is stable under small deformations of \(J\). On the other hand, the Iwasawa manifold \(I(3)\) admits a 2-parameter family \(\{J_{t,s}\}\) of complex structures, a strong \(KT\) metric compatible with \(J_{1,1}\) and for any \(t\neq s\neq 1\) there are no strong \(KT\) metrics compatible with \(J_{t,s}\).
More generally, on a nilmanifold \((M,J)\), \(\dim_CM=3\), the condition strong \(KT\) is not stable under small deformations of \(J\). On the analogy of the Kähler case, the authors prove that the blow-up of a strong \(KT\) manifold at a point has a strong \(KT\) metric. An analogous result holds considering the blow-up \(\widetilde M_Y\) of \(M\), \(Y\) being a compact complex submanifold of \(M\). Moreover, given a point \(p\) in a complex manifold \(M\), \(\dim_CM\geq 2\), if \(M\setminus\{p\}\) admits a strong \(KT\) metric, then \(M\) admits a strong \(KT\) metric, also. In order to state the existence of non compact simply-connected strong \(KT\) manifolds, one can consider resolutions of complex orbifolds. Firstly, the authors prove the existence of a strong \(KT\) resolution of any complex orbifold endowed with a Hermitian strong \(KT\) metric. In particular, starting by the standard torus \(T^6\), a strong \(KT\) non Kähler orbifold of \(T^6\) by a suitable action of a finite group is constructed and a simply-connected strong \(KT\) resolution for this orbifold is then obtained.
Finally, in complex dimension four, one obtains strong \(KT\) resolutions on the complex orbifold constructed considering an action of a suitable finite group on a product of two Kodaira-Thurston surfaces and on a product of a Kodaira-Thurston surface with the torus \(T^4\).
More generally, on a nilmanifold \((M,J)\), \(\dim_CM=3\), the condition strong \(KT\) is not stable under small deformations of \(J\). On the analogy of the Kähler case, the authors prove that the blow-up of a strong \(KT\) manifold at a point has a strong \(KT\) metric. An analogous result holds considering the blow-up \(\widetilde M_Y\) of \(M\), \(Y\) being a compact complex submanifold of \(M\). Moreover, given a point \(p\) in a complex manifold \(M\), \(\dim_CM\geq 2\), if \(M\setminus\{p\}\) admits a strong \(KT\) metric, then \(M\) admits a strong \(KT\) metric, also. In order to state the existence of non compact simply-connected strong \(KT\) manifolds, one can consider resolutions of complex orbifolds. Firstly, the authors prove the existence of a strong \(KT\) resolution of any complex orbifold endowed with a Hermitian strong \(KT\) metric. In particular, starting by the standard torus \(T^6\), a strong \(KT\) non Kähler orbifold of \(T^6\) by a suitable action of a finite group is constructed and a simply-connected strong \(KT\) resolution for this orbifold is then obtained.
Finally, in complex dimension four, one obtains strong \(KT\) resolutions on the complex orbifold constructed considering an action of a suitable finite group on a product of two Kodaira-Thurston surfaces and on a product of a Kodaira-Thurston surface with the torus \(T^4\).
Reviewer: Maria Falcitelli (Bari)
MSC:
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 22E25 | Nilpotent and solvable Lie groups |
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