Moduli of rank 2 Higgs sheaves on elliptic surfaces
type: Item
instance of: scholarly article
title: Moduli of rank 2 Higgs sheaves on elliptic surfaces
author: Vitantonio Peragine
author: Ugo Bruzzo
published in: Geometriae Dedicata
publication date: 2022-12-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/2110.03383
review text: 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 op…
reviewed by: Souradeep Majumder
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 7637012
zbMATH Keywords: semistable Higgs sheaves
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: moduli of HIggs sheaves
MaRDI profile type: Publication
OpenAlex ID: W3202814025
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DOI: 10.1007/S10711-022-00756-W
description: scientific article; zbMATH DE number 7637012