On very stablity of principal \(G\)-bundles. (English) Zbl 1430.14070
Let \(X\) be a smooth irreducible projective curve over \(\mathbb{C}\) of genus \(g\geq2\). The aim of this paper is to generalize the main result of C. Pauly and A. Peón-Nieto [Geom. Dedicata 198, 143–148 (2019; Zbl 1409.14060)] to principal \(G\)-bundles for any reductive linear algebraic group \(G\). The proof is purely algebraic and is independent of the geometry of the moduli space of semistable \(G\)-Higgs pairs. After defining very stability of principal \(G\)-bundles, the author shows that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. In the last section, he also studies the ad-stable \(\mathrm{SL}_2\)-bundles on \(X\), i.e., principal \(\mathrm{SL}_2\)-bundles such that the associated adjoint vector bundle is stable. This study concerns the relation between very stability and other stability conditions in the case of \(\mathrm{SL}_2\)-bundles. This result classifies all ad-stable \(\mathrm{SL}_2\)-bundles. In particular, it allows easily to construct examples of stable non ad-stable \(\mathrm{SL}_2\)-bundles which simplifies the results of D. Hyeon and D. Murphy [Proc. Am. Math. Soc. 132, No. 8, 2205–2213 (2004; Zbl 1049.14025)]. This also allows to construct examples of very stable \(\mathrm{SL}_2\)-bundle which is not ad-stable and ad-stable \(\mathrm{SL}_2\)-bundle which is not very stable.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 14H70 | Relationships between algebraic curves and integrable systems |
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