Acta Mathematica

Ergodic complex structures on hyperkähler manifolds

Misha Verbitsky

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Let M be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on M up to the action of the group Diff0(M) of isotopies. The mapping class group Γ:=Diff(M)/Diff0(M) acts on Teich in a natural way. An ergodic complex structure is a complex structure with a Γ-orbit dense in Teich. Let M be a complex torus of complex dimension 2 or a hyperkähler manifold with b2>3. We prove that M is ergodic, unless M has maximal Picard rank (there are countably many such M). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic.


Partially supported by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.

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Acta Math., Volume 215, Number 1 (2015), 161-182.

Received: 8 June 2014
Revised: 18 March 2015
First available in Project Euclid: 30 January 2017

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2015 © Institut Mittag-Leffler


Verbitsky, Misha. Ergodic complex structures on hyperkähler manifolds. Acta Math. 215 (2015), no. 1, 161--182. doi:10.1007/s11511-015-0131-z.

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