Moduli of rank 2 Higgs sheaves on elliptic surfaces
The article under review deals with torsion-free, (semi)stable rank \(2\) Higgs sheaves on genus one fibered surfaces. In [\textit{U. Bruzzo} and \textit{V. Peragine}, Adv. Geom. 22, No. 2, 151--169 (2022; Zbl 1487.14080)] the authors consider the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on a certain class of elliptic surfaces, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to the surface. More precisely it was shown that a semistable Higgs bundle with non-zero discriminant is unstable when pulled back to a suitable curve. In this paper the authors continue this study in a more general setting, namely, it is assumed that \(\pi : X \to B\) is a genus one fibered surface i.e. the generic fiber \(X_{\eta}\) is a genus one curve (not necessarily smooth) over the function field of the generic point of \(B\). Let \(c = (r, c_1, c_2) \in \mathbb{Z} \times Num(X) \times \mathbb{Z}\), where \(Num(X)\) is the Picard group of \(X\) modulo numerical equivalence. A coherent sheaf \(F\) on \(X\) is said to have class \(c\) if \(F\) has rank \(r\), determinant \(c_1\) and second Chern number \(c_2\). There is a notion of \(c\)-suitable polarization following Friedman, R.A. and O'Grady, K.G. Suppose \(H\) is one such polarization on \(X\). It is known that for every torsion-free sheaf \(F\) of class \(c\) on \(X\) , if \(F\) is \(\mu_H\)-semistable then \(F_{\eta}\) is semistable. The authors extend this result to the case of rank \(2\) torsion-free Higgs sheaves where the base field is arbitrary. Let \(M(c) := M(X,H,c), M_{\mathrm{Higgs}}(c) := M_{\mathrm{Higgs}}(X,H,c)\) be the moduli schemes of \(H\)-semistable sheaves, Higgs sheaves on \(X\) of class \(c\) and \(M^s(c), M^s_{\mathrm{Higgs}}(c)\) be the respective open subschemes of stable sheaves. Under the hypothesis that the base field is algebraically closed and the fiber degree of the elliptic surface is odd, the open immersions \(M^s(c) \hookrightarrow M(c) , M^s_{\mathrm{Higgs}}(c) \hookrightarrow M_{\mathrm{Higgs}}(c)\) are isomorphisms. Further, the natural forgetful map \(p : M_{\mathrm{Higgs}}(c) \to M(c)\) is surjective. When the base field is of characteristic \(\neq 2\) and \(\pi\) is elliptic and non-isotrivial, the article demonstrates that the fibration \(p\) is trivial with fibre \(\mathbb{A}(H^0(\Omega_X))\). Thus \(M_{\mathrm{Higgs}}(c)\) has a canonical identification with the product \(M(c) \times \mathbb{A}(H^0(\Omega_X))\) and hence it is non-singular. Finally, the authors consider the case of non-isotrivial elliptic surfaces with a section where the base field is algebraically closed of characteristic \(0\). Let \((F,\phi)\) be a torsion-free Higgs sheaf of arbitrary rank on \(X\). If \(F\) has reduced Friedman spectral curve, then \(\phi\) takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface. The same result holds with non-reduced spectral cover if \(F\) is of rank \(2\) and regular on a general fiber of \(\pi\).
scientific article; zbMATH DE number 7637012
scientific article