Ergodic complex structures on hyperkähler manifolds

Misha Verbitsky National Research University Higher School of Economics, Laboratory of Algebraic Geometry, 7 Vavilova Street, Moscow, Russia

Complex Variables and Complex Analysis mathscidoc:1701.08005

Acta Mathematica, 215, (1), 161-182, 2014.6
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.
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@inproceedings{misha2014ergodic,
  title={Ergodic complex structures on hyperkähler manifolds},
  author={Misha Verbitsky},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203405552661807},
  booktitle={Acta Mathematica},
  volume={215},
  number={1},
  pages={161-182},
  year={2014},
}
Misha Verbitsky. Ergodic complex structures on hyperkähler manifolds. 2014. Vol. 215. In Acta Mathematica. pp.161-182. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203405552661807.
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