Ergodic complex structures on hyperkähler manifolds
By Misha Verbitsky. 2013
57 citations recorded by the source corpus.
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 $${{\rm Diff}_0(M)}$$Diff0(M) of isotopies. The mapping class group $${\Gamma:={\rm Diff}(M)/{{\rm Diff}_0(M)}}$$Γ:=Diff(M)/Diff0(M) acts on Teich in a natural way. An ergodic complex structure is a complex structure with a $${\Gamma}$$Γ-orbit dense in Teich. Let M be a complex torus of complex dimension $${\ge 2}$$≥2 or a hyperkähler manifold with $${b_2 > 3}$$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.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/8e33ca029b17a1b4d7d5e168d4ff8ff2b3fc161d
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In Geometry and Topology
- Benjamin Bakker
- C. Lehn
- Ekaterina Amerik
- Misha Verbitsky
- D. Huybrechts
- Nicolas Addington
- Anna Fino
- Bruno Klingler
Citations in the corpus
- Morrison-Kawamata cone conjecture for hyperkahler manifolds (2014, background): “rally acts on Teich. The quotient of Teich by this action may be viewed as the “moduli space” for M. However, this space is too non-Hausdorff to be useful: any two open subsets of Teich/Diff intersect ([V3], [V4]). By a result of Huybrechts (see [H3]), in the hyperk¨ahler case Teich has only finitely m”
- Rational curves on hyperkahler manifolds (2014, background): “imeromorphic equivalence. Remark 2.19: The word “space” in this context is misleading. In fact, the quotient topology on Teichb /Γ is extremely non-Hausdorff, e.g. every two open sets would intersect ([V5]). The Global Torelli theorem can be stated as a result about the birational moduli space. Theor”
- The global moduli theory of symplectic varieties (2018, methodology): “2 directly using Verbitsky’s classification [Ver15] (see also [BL16]) of orbit closures of the monodromy action on the period domain via Ratner theory, which is responsible for the assumption b2(X) ≥ 5 in the theorem.”