Acta Mathematica

Ergodic complex structures on hyperkähler manifolds

Misha Verbitsky

Abstract

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 ${{\rm Diff}_0(M)}$ of isotopies. The mapping class group ${\Gamma:={\rm Diff}(M)/{{\rm Diff}_0(M)}}$ acts on Teich in a natural way. An ergodic complex structure is a complex structure with a ${\Gamma}$-orbit dense in Teich. Let M be a complex torus of complex dimension ${\ge 2}$ or a hyperkähler manifold with ${b_2 > 3}$. 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.

Note

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

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802445

Digital Object Identifier
doi:10.1007/s11511-015-0131-z

Mathematical Reviews number (MathSciNet)
MR3413979

Zentralblatt MATH identifier
1332.53092

Rights
2015 © Institut Mittag-Leffler

Citation

Verbitsky, Misha. Ergodic complex structures on hyperkähler manifolds. Acta Math. 215 (2015), no. 1, 161--182. doi:10.1007/s11511-015-0131-z. https://projecteuclid.org/euclid.acta/1485802445

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