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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 …

Explicit description of a Kahler cone for all hyperkahler manifolds is here: https://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds, Ekaterina Amerik, Misha Verbitsky)

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 …

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 …

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 …

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 …

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 …

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))= …