Movable curves and semistable sheaves (Q2797801)

Let \(X\) be a complex projective manifold of dimension~\(n\) and \(\mathcal{E}\) a torsion-free coherent sheaf of~\(\mathcal{O}_X\)-modules. Given an ample line bundle \(H\) on~\(X\), the slope of~\(\mathcal{E}\) with respect to~\(H\) is classically defined as NEWLINE\[NEWLINE \mu_H(\mathcal{E}) := \frac{H^{n-1}\cdot c_1(\det \mathcal{E})}{\mathrm{rank }\mathcal{E}}. NEWLINE\]NEWLINE Using this definition, one introduces certain \textit{stability} notions, i.e., \(\mathcal{E}\) is said to be \textit{semistable} if \(\mu_H(\mathcal{F}) \leq \mu_H(\mathcal{E})\) for all subsheaves \(0 \neq \mathcal{F} \subseteq \mathcal{E}\).NEWLINENEWLINEUnfortunately, this classical definition does not behave very well in applications in birational geometry, since, for example, birational pullbacks of ample line bundles are no longer ample.NEWLINENEWLINEThis motivates the authors of the article at hand to consider the generalization of the above definition obtained by replacing \(H^{n-1}\) by an arbitrary \textit{movable} curve class~\(\alpha\) (i.e., \(\alpha\cdot D \geq 0\) for all effective divisors~\(D\)). Furthermore, they allow \(X\) to be an arbitrary normal, \(\mathbb{Q}\)-factorial, projective variety.NEWLINENEWLINEThe article shows that a lot of the classical results for (semi-)stable sheaves remain true in the generalized setting. The authors prove a weak Mehta-Ramanathan theorem, boundedness of slopes and the existence of Harder-Narasimhan and Jordan-Hölder filtrations. Furthermore they investigate openness properties of the set of stabilizing movable classes for a given sheaf.NEWLINENEWLINEUsing these basic results, the authors derive the semistability of reflexive tensor products of sheaves which are semistable with respect to a movable class on a possibly singular space. They also prove a Bogomolov-Gieseker inequality for semistable sheaves on smooth projective surfaces.NEWLINENEWLINEIn the last part of the article, the authors give numerical criteria for flatness and projective flatness of (semi-)stable sheaves, generalizing results by Simpson. As an application, they prove that normal \(\mathbb{Q}\)-factorial projective varieties with canonical singularities and trivial first and second Chern classes always arise as quasi-étale quotients of abelian varieties.