Rational curves and special metrics on twistor spaces

By Misha Verbitsky. 2012

52 citations recorded by the source corpus.

A Hermitian metric $\omega$ on a complex manifold is called SKT or pluriclosed if $dd^c\omega=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is K\"ahler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eeda504fadeeece80a68c1d529a4395776d60a46

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Balanced Hermitian structures on almost abelian Lie algebras (2020, background): “This has been confirmed on some special classes of compact complex manifolds: twistor spaces of compact anti-self-dual Riemannian manifolds [56], non-Kähler manifolds belonging to the Fujiki class C [15], a family of non-Kähler Calabi-Yau threefolds constructed in [31], generalized Calabi-Gray manif”
Ergodic complex structures on hyperkähler manifolds (2013, background): “tation of a complex structure. Campana observed that the space of rational curves on Tw(M) transversal to the ﬁbers is never compact (in fact it is holomorphically convex, as shown in [KaV]; see also [V3] and [DDM]). Then a limit of a sequence of rational curves in Tw(M) would contain an entire curv”
On the existence of balanced and SKT metrics on nilmanifolds (2015, background): “[13, 14, 11, 20, 18, 19, 22] and the references therein), while balanced metrics were introduced and firstly studied by Michelsohn in [17], where their existence is characterized in terms of currents.”
Special Hermitian metrics on compact solvmanifolds (2015, background): “For instance: Verbitsky proved in [49] that the twistor space of a compact, anti-self-dual Riemannian manifold has no SKT metrics unless it has Kähler metrics and Chiose proved in [9] a similar result for non-Kähler manifolds belonging to the Fujiki class.”