A counterexample to the curve semistability conjecture for Higgs bundles

Abstract

In this paper, we give a counterexample to the Bruzzo--Graña Otero conjecture on the curve semistability for Higgs bundles. Let $\Sigma$ be a very general smooth plane quintic curve and let $X={\Sigma }^{(2)}$ be its second symmetric product. Starting from the tautological bundle $F={\mathcal{O}}_{\Sigma }(1{)}^{[2]}$, we construct a rank four Higgs bundle ${\mathcal{E}}_s=(E_s,{\theta }_s)$ on $X$. Then, for every smooth projective curve $C$ and every morphism $f:C\to X$, the pullback Higgs bundle $f^{\ast }{\mathcal{E}}_s=(f^{\ast }{E}_s,f^{\ast }{\theta }_s)$ is semistable. However, on the other hand, $\det (E_s)\cong {\mathcal{O}}_X$ and ${\int }_X{c}_2(E_s)=10$. Consequently, the discriminant of $E_s$ does not vanish, and ${\mathcal{E}}_s$ provides the desired counterexample.

arXiv comment13 pages. Terminology fixed, comments are welcome!