The authors introduce a new bound in the restriction theorem of vector bundles to curves embedded to a smooth algebraic surface using the Bridgeland stability conditions.
Suppose that \((X,H)\) is a smooth polarized surface, C is an integral curve on \(X\), and \(E\) is a \(\mu_{H}-\) stable vector bundle of rank \(r \geq 2\) on \(X\). Then \(E|_{C}\) is stable if \[ \frac{C^2}{2H \cdot C } \leq r(r-1)\Delta(E) + \frac{1}{2r(r-1)H^2} \] They do so first by finding a formula for the Chern character of the immersion of a sheaf \(i_{*}F\) on a curve \(C\) embedded to a surface \(X\). Their strategy is as follows: first they use the short exact sequence \(0 \to \mathcal{O}_{X}(-C) \to \mathcal{O}_{X} \to i_{*}\mathcal{O}_{C} \to 0\) and get the desired formula for the Chern character by a simple calculation. Then writing the same SES for an arbitrary divisor \(D\), substracting and adding smooth points \(p \in C\) they get the result for any line bundle. Then they prove the formula for a sheaf \(F\) of a rank one and further proceed by induction to the arbitraty rank. While working with the Bridgeland stability conditions they define a stability condition for \(E, F \in \mathcal{D}^{b}(X), s, t \in \mathbb{R}\) as \(Z_{s,t}(E)=-{\mathrm{ch}_2}^{D+sH}(E) + \frac{t^2H^2}{2}{\mathrm{ch}_{0}}^{D+sH}(E)+iH \cdot {\mathrm{ch}_{1}}^{D+sH}(E)\) and slope, center and a radius of a wall \(W(E,F)\) as \[ \mu_{\sigma}(E)=\frac{(\mu_{H,D}-s)^{2}-t^2-2\Delta_{H,D}(E)}{\mu_{H,F}(E)-s}. \] \[ s_{0}=\frac{1}{2}(\mu_{H,D}(E)+\mu_{H,D}(F))-\frac{\Delta_{H,D}(E)-\Delta_{H,D}(F) }{\mu_{H,D}(E)-\mu_{H,D}(F)} \] \[ \rho_{0}^{2}=(\mu_{H,D}(E)-s)^{2}- 2\Delta_{H,D}(E) \] From the distinguished triangle \(E \to i_{*}E|_{C} \to E(-C)[1] \to E[1]\) in \(\mathcal{D}^{b}(X)\) it clearly follows that if one can find a stability condition \(\sigma=(Z_{t,s}, \mathcal{A}_{s})\) s.t. \(E\) and \(E(-C)[1]\) are \(\sigma\)-stable of the same slope so is \(E|_{C}\). So one should search for a wall \(W(E, E(-C)[1])\). Further they implement this strategy giving a rough approximation of the Gieseker wall (a set of stability conditions \(\sigma\) for which every \((H,D)\)-twisted Gieseker stable sheaf of Chern character \(\textbf{v}\) is \(\sigma\)-stable). More concretely, they show that \(E\) cannot be destabilized by a subobject \(A\) unless the readius of \(W(A,E)\) is smaller than an explicit bound.
For a sheaves on \(\mathbb{P}^{2}\) the sufficient criteria for a \(E|_{C}\), \(C \subset \mathbb{P}^2\) to be stable is given in a similar fashion. Here they show that the wall \(W(E, E(-C)[1])\) is outside the effective wall beyond which the Gieseker semistable bundle is \(\sigma\)-stable. The effective wall was previously computed in the work of I. Coskun et al. [J. Eur. Math. Soc. (JEMS) 19, No. 5, 1421–1467 (2017; Zbl 1373.14042)]. For this they use the theory of exceptional bundles on \(\mathbb{P}^2\) and a fact that every slope of arbitrary sheaf lies on the interval which directly depends on the discriminant \(\Delta_{\textbf{v}}\) of the exceptional sheaf.
The next question studied in the article is the extension of stable vector bundle on \(C\) to a stable vector bundle on \(X\). It is shown that the restriction map \(M_{X,H}(\textbf{v}) \dashrightarrow U_{C}(r, c_1 \cdot C)\) is generically finite and the dimension of the image equals the dimension of \(M_{X,H}(\textbf{v})\).
The cohomologies of vector bundles \(E|_{C}\) are studied in the last part of the article in cases \(X=\mathbb{P}^2\) and \(X\) is a Hirzebruch surface, using the theorems obtained earlier in the article. Firstly they study it on \(\mathbb{P}^{2}\) in the context of the Brill-Noether theory. The first case concerns the plane curves in \(\mathbb{P}^{2}\) using Göttsche-Hirschowitz result. Secondly they study the same questions on the Hirzebruch surfaces using the Coskun-Huizenga results on Brill-Noether theorems and vector bundles on Hirzebruch surfaces.