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Rational curves are dense on K3 for all K3 outside of a Baire set (countable union of closed, nowhere dense). Here is the reference: https://arxiv.org/abs/1004.5167, Density of Rational Curves on K3 S …

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find wha …

Let $M$ be a holomorphically symplectic complex manifold, and $f: M \to X$ a holomorphic, birational contraction to a Stein variety $X$, contracting a subvariety $E$ to a point, and bijective outside …

You can define "meromorphic vector bundle" as locally free sheaf of modules over a sheaf of meromorphic functions. This is a highly non-trivial object, because (in contrast with rational functions) me …

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Mors …

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 …

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 guess this is always true, if you adjust the statement appropriately. Consider the Bott–Chern cohomology $H^*_{BC}(M):=\dfrac{\ker d\cap \ker d^c}{\operatorname{im} dd^c}$. Since the curvature of a …

Just blow up the singular point in a variety $X_1$ which is obtained from a smooth, irreducible manifold $X$ by identifying points $x$ and $y$. The blow-up divisor is ${\Bbb P} T_xX\coprod {\Bbb P}T_y …

Do you know other examples of non-isomorphic algebraic structures on complements to square-zero curves The easiest example is the twisted cotangent bundle to an elliptic curve. This space can be rea …

This is an update, sorry for the trouble. Let $M$ be a metric space. We say that $X \subset M$ is coarse equivalent to $Y \subset M$ if the the Hausdorff distance $d_H(X, Y)$ is finite, that is, there …

By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense. Suppose we have a Zariski de …

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))= …

Convergence is taken in Hausdorff sense, though you can define the structure of a complex variety (the Barlet space) on the set of cycles, taking every irreducible component with positive integer mult …

Calabi-Yau theorem implies that any diffeomorphism of a Calabi-Yau manifold which preserves the complex structure and the Kahler class also preserves the Calabi-Yau metric. However, the group of isom …