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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 …

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 …

Let $S_1$, $S_2$ be homologous embedded 2-spheres in a compact smooth 4-manifold. Under which additional conditions are they smoothly isotopic? I am interested in the state of the art picture when $S_ …

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 …

Let me prove that $(L,D)\geq 0$. Consider the set of all (1,1)-classes which satisfy $\eta^2\geq 0$. It is a union of two components $P_+$ and $P_-$, intersecting in 0. If $\eta, \eta'$ are in the …

Thd equation $S(\sigma)=S_X$ is certainly not universal. Take a non-symplectic map such that $S(\sigma)=S_X$, then modify $\sigma$ replacing it by $\sigma \tau$, where $\tau$ is a non-trivial symplect …

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 …