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Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "va …

The group of symplectomorphisms $Aut(X)$ of a K3 is the group $O(\Lambda)$ of automorphisms of its period lattice $\Lambda=H^{1,1}(M,{\Bbb Z})$. For each (-2)-cohomology class $\eta\in H^{1,1}(M,{\B …

A rational singularity is a singularity of a complex variety $X$ such that for any resolution $\pi:\; \tilde X\rightarrow X$ the higher direct images $R^i\pi_*(O_{\tilde X})$ vanish for all $i>0$. Sup …

This is known, for projective (even Moishezon) manifolds as shown by Dan Popovici in his paper http://arxiv.org/abs/1003.3605 For general Kaehler manifold, this is conjectured. Popovici has proved t …

The second and the third are pretty much equivalent. Indeed, "the period" in XIX century sense is essentially the same as the discrepancy between the branches of a multi-valued function, obtained as a …

The corresponding Kahler metric is called "Weil-Petersson metric"; it is often constructed using infinite-dimensional determinants of the corresponding Laplace operators. The standard reference is a s …

There is just no definition of the moduli for complex structures on non-compact manifolds, but by any reasonable definition, it would be (generally) very bad space, certainly infinite-dimensional. For …

It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))= …

Here is the reference: Persson, Ulf, On degenerations of algebraic surfaces, Mem. Amer. Math. Soc. 11 (1977), no. 189. Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. …

There are two definitions of Gorenstein singularities in the literature. Using Grothendieck's (or Serre's) duality, one defines the "dualizing sheaf" an object $\hat K_M$ of derived category of cohere …

I think there are proofs which are much easier. For example, you can try to compute the cohomology using the Morse theory. For that you need existence of Morse functions having finitely many critical …

Being "affine" in this case does not make much sense, because the hyperkaehler deformation is a complex manifold, without a fixed algebraic structure. Simpson produced an example of a hyperkaehler …

A generic deformation of a Hilbert scheme of K3 and a generic torus have no subvarieties, hence they are "simple" in the above sense. For a torus it's well known, for a Hilbert scheme of K3 it's in my …

There are many such examples, for instance, all complex nilmanifolds (and most complex solvmanifolds) have tangent bundle which is topologically trivial. The Hopf manifolds also have topologically tri …

I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a vect …