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Are there any stability theorems of analytic automorphism groups concerning the deformation of complex manifolds. For example, in the case of K3 or Calabi-Yau.

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    $\begingroup$ Could you please explain a little bit better which kind of stability theorems you are looking for? $\endgroup$
    – diverietti
    Nov 12, 2015 at 11:31
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    $\begingroup$ "Are automorphism groups of small deformed complex manifolds conjugate to the automorphism group of the given complex manifold?". If you literally mean "small deformation" rather than "generizations", then this is maximally false. Inside the local Kuranishi space of a K3 surface, the subset parameterizing Kummer K3 surfaces is an analytically dense subset. Each Kummer K3 has nontrivial automorphisms: translations by $2$-torsion of the associated Abelian surface. Yet the central K3 often has no automorphisms. $\endgroup$
    – Jason Starr
    Nov 12, 2015 at 13:42
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    $\begingroup$ "Is it known the similar phenomena in the case of Calabi-Yau threefolds?" For the standard definition of Calabi-Yau threefold, there is one big difference. Because $h^2(X,\mathcal{O}_X)$ vanishes, the Picard group is a deformation invariant. In particular, if the central fiber has Picard number $1$, then there is a (quasi-compact, finite type) affine scheme over the base of your deformation parameterizing automorphisms of the deformed varieties. So the Kummer situation above, where the "special locus" in the base is a countable union of locally closed subsets, does not occur in that case. $\endgroup$
    – Jason Starr
    Nov 12, 2015 at 15:00
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    $\begingroup$ I think there is a Calabi-Yau threefold counterexample via intersections in $\mathbb{P}^3\times \mathbb{P}^2$ of a hypersurface of type $(2,0)$, i.e., a product of a quadric surface with $\mathbb{P}^2$, and a hypersurface of type $(2,3)$, i.e., a conic bundle over $\mathbb{P}^3$ with discriminant a degree $9$ surface. If everything is stable under an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{P}^3\times \mathbb{P}^2$, and if the quadric surfaces specialize to an invariant nodal quadric, then the two small blowups of the total space of the family of quadrics are usually not equivariant. $\endgroup$
    – Jason Starr
    Nov 12, 2015 at 16:05
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    $\begingroup$ I don't know CY examples, but something similar to what Jason Starr mentions also happens in families of rational surfaces, where the Picard rank is constant. The blow-up of P^2 at 9 or more very general points has trivial automorphism group, but there are countably many types of configurations of points for which the group becomes infinite. (I think the opposite phenomenon, where the automorphism group of the special fiber drops, occurs in the Enriques case.) $\endgroup$
    – user47305
    Nov 12, 2015 at 21:39

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