Large families of stable bundles on abelian varieties (Q2839293)

One of the most fundamental problems about \(\mu\)-stable vector bundles on smooth complex projective varieties is their existence. \textit{M. Maruyama} [J. Math. Kyoto Univ. 18, 557--614 (1978; Zbl 0395.14006)] showed that if \(r\geq n\), \(c_{1}\in NS(X)\) and \(s\in\mathbb{Z}\), then there is a \(\mu\)-stable vector bundle of rank \(r\), first Chern class \(c_{1}\) and second Chern class \(c_{2}\) such that \(c_{2}\cdot H^{n-2}\geq s\). As a consequence of this, there is a large family of \(\mu\)-stable vector bundles on \(X\), i. e. a family \(\{E_{m}\}_{m\in\mathbb{N}}\) of \(\mu\)-stable vector bundles on \(X\) such that \(r_{m}=\mathrm{rk}(E_{m})\rightarrow\infty\) and \(\Delta_{m}=(2r_{m}c_{2}(E_{m})-(r_{m}-1)c_{1}(E_{m})^{2})\cdot H^{n-2}\rightarrow\infty\) as \(m\) goes to infinity.NEWLINENEWLINEIf \(s,t\in\mathbb{R}_{+}\), we say that a large family \(\{E_{m}\}_{m\in\mathbb{N}}\) of \(\mu\)-stable vector bundles on \(X\) is of order \((s,t)\) if \(r_{m}=O(m^{s})\) and \(\Delta_{m}=O(m^{t})\). The existence of a large family of \(\mu\)-stable vector bundles of a given order is related to the strong Bogomolov inequality: if \(l\in\mathbb{Q}_{+}\), the strong Bomogolov inequality of type \(l\) for a \(\mu\)-stable vector bundle \(E\) of rank \(r\) is the inequality \(\Delta(E)\geq r^{l}\alpha(X,H)\) for some positive constant \(\alpha(X,H)\) of the polarized variety \((X,H)\). If there is a large family of \(\mu\)-stable vector bundles of order \((s,t)\) with \(t<sl\), then the strong Bogomolov inequality of type \(l\) does not hold.NEWLINENEWLINEIn the paper under review, the author shows that on a principally polarized abelian variety \((X,\Theta)\) of dimension \(n\geq 2\) and such that \(NS(X)=\mathbb{Z}\cdot\Theta\), there are large families of \(\mu\)-stable vector bundles of order \((1,2)\) and of order \((1,3)\).NEWLINENEWLINEThis is done in the following way: first, choose \(a_{1},\dots,a_{m}\in X\) distinct points. We let \(D_{i}\in |2\Theta_{a_{i}}|\), \(Y_{i}:=D_{i}\cap\Theta_{-a_{i}}\) and \(Y=Y_{1}\cup\cdots\cup Y_{m}\). The author shows that \(Y\) is a locally complete intersection subscheme of \(X\) of codimension 2 which is subcanonical, hence Serre correspondence gives a rank 2 vector bundle \(E'\) defining an extension in \(\mathrm{Ext}^{1}(\mathcal{I}_{Y}(3\Theta),\mathcal{O}_{X})\). The vector bundle \(E'(-\Theta)\) is shown to be \(\mu\)-stable, its first Chern class is \(\Theta\) and its second Chern class is \(2(m-1)\Theta^{2}\).NEWLINENEWLINEFix now a subspace of dimension \(m-1\) of \(\mathrm{Ext}^{1}(E'(-\Theta),\mathcal{O}_{X})\), and let \(E_{m}\) be the universal extension of \(E'_{m}(-\Theta)\) by \(\mathcal{O}_{X}\) corresponding to it. Using general results about stability of universal extensions, the author shows that \(E_{m}\) is a \(\mu\)-stable vector bundle of rank \(m+1\), first Chern class \(\Theta\) and second Chern class \((4m^{2}-m-4)\Theta^{g}\). The family \(\{E_{m}\}_{m>1}\) is then large of order \((1,2)\).NEWLINENEWLINEThe family of order \((1,3)\) is obtained by considering a subspace of dimension \(m-1\) of \(\mathrm{Ext}^{1}(\mathcal{O}_{X}(\Theta),E'(-\Theta))\), and then letting \(E_{m}\) be the universal extension corresponding to it.