Remarks on the positivity of the cotangent bundle of a K3 surface

Abstract

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell. Comment: Published version

Examples of Fano manifolds with non‐pseudoeffective tangent bundle

“…Statement (c) is a consequence of Lemma 3.1. For (d), this is already proved in [16, Section 4.2.2] for $$S$$ being very general. On the other hand, note that $$\zeta +\frac{3}{2}\pi ^*H$$ is nef only if $$\zeta +(\frac{3}{2}+\varepsilon )\pi ^*H$$ is ample for any real numbers $$\varepsilon >0$$.…”

“…Proof Statement (a) is proved in [44, Proposition 2.3], [46, Proposition 3.14] and (b) follows from [16, Corollary 4.2]. Statement (c) is a consequence of Lemma 3.1.…”

“…Recall that by the Noether–Lefschetz theorem, $$S$$ has Picard number one if it is very general. If $$S$$ has Picard number one, Gounelas and Ottem have investigated the positivity of $$\Omega _S$$ in [16, Section 4.2]. Denote by $$U\subset \mathbb {P}(T_S)$$ the incidence variety associated to the surface of bitangents of $$S$$ (cf.…”

Contractions of hyper-Kähler fourfolds and the Brauer group

“…These K3 surfaces and their Hilbert squares are related to nodal cubic fourfolds and their Fano varieties cf. [Ha,4.2,Lemma 6.3.1], [DM,Example 6.4], [GO,4.3].…”

“…The K3 surface S ⊂ P 4 is the complete intersection of a smooth quadric and a cubic hypersurface. Taking this quadric to be a hyperplane section of the Grassmannian Gr(2, 4) ⊂ P 5 , in [GO,Lemma 4.5] it is shown that the Mukai bundle V can be chosen to be the restriction of the dual of the universal bundle on Gr(2, 4) to S.…”