Semistable sheaves on rationally connected manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:23:23Z http://mathoverflow.net/feeds/question/58522 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58522/semistable-sheaves-on-rationally-connected-manifolds Semistable sheaves on rationally connected manifolds jvp 2011-03-15T11:56:00Z 2011-03-16T14:29:29Z <p>Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$.</p> <p>If for every subsheaf $\mathcal F \subset \mathcal E$ the following inequality holds $$\frac{c_1(\mathcal F)\cdot H^{n-1}}{\mathrm{rank}(\mathcal F)} \le \frac{c_1(\mathcal E)\cdot H^{n-1}}{\mathrm{rank}(\mathcal E)}$$ then we say that $\mathcal E$ is semistable.</p> <p>Mehta-Ramanathan proved that if $\mathcal E$ is semi-stable then its restriction to a sufficiently general curve $C$ cut out by elements of $|mH|$, $m \gg 0$, is also semistable.</p> <blockquote> <p><strong>Question.</strong> If we further assume that $X$ is rationally connected can we ensure the existence of a very free morphism $i: \mathbb P^1 \to X$ such that $i^*\mathcal E$ is semistable?</p> </blockquote> <p>Recall that $i: \mathbb P^1 \to X$ is a very free morphism if and only if $i^* TX$ is ample.</p> <hr> <p>As observed by mdeland in the comments, $i^* \mathcal E$ is semistable if and only if $$i^* \mathcal E \simeq \mathcal O_{\mathbb P^1}(k)^{\oplus \mathrm{rank}(\mathcal E)} .$$ In particular $\deg(i^* \mathcal E)$ is divisible by $\mathrm{rank}(\mathcal E)$. </p> <p>If we start with an arbitrary very free morphism $i: \mathbb P^1 \to X$ we can compose it with a rational map $\varphi : \mathbb P^1 \to \mathbb P^1$ of degree $m \cdot \mathrm{rank}(\mathcal E), m \in \mathbb N$, to obtain a new morphism $i' = i \circ \varphi$. The original question seems to be related to the problem.</p> <blockquote> <p><strong>Problem.</strong> Describe the splitting type of a general deformation of $i' : \mathbb P^1 \to X$ for $m \gg 0$. </p> </blockquote> <p>Any ideas/references on how to tackle this kind of problem ? </p>