关于希格斯丛的Bruzzo曲线半稳定猜想的一个反例
A counterexample to the curve semistability conjecture for Higgs bundles
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中文总结 AI 辅助
本文针对希格斯丛的Bruzzo曲线半稳定猜想构造出一个反例，验证了该猜想不成立。
中文摘要 AI 辅助
本文给出了希格斯丛的Bruzzo曲线半稳定猜想的一个反例。设Σ为非常一般的光滑平面五次曲线，X=Σ⁽²⁾为其第二对称积。从重言丛F=𝒪_Σ(1)^[2]出发，在X上构造了一个秩为4的希格斯丛ℰₛ=(Eₛ,θₛ)。对任意光滑射影曲线C及任意态射f:C→X，拉回希格斯丛f*ℰₛ=(f*Eₛ,f*θₛ)均为半稳定的；但另一方面，det(Eₛ)≅𝒪_X且∫_X c₂(Eₛ)=10，故Eₛ的判别式非零，ℰₛ即为所需的反例。
英文摘要
In this paper, we give a counterexample to the Bruzzo--Graña Otero conjecture on the curve semistability for Higgs bundles. Let $Σ$ be a very general smooth plane quintic curve and let $X=Σ^{(2)}$ be its second symmetric product. Starting from the tautological bundle $F=\mathcal{O}_Σ(1)^{[2]}$, we construct a rank four Higgs bundle $\mathcal{E}_s=(E_s,θ_s)$ on $X$. Then, for every smooth projective curve $C$ and every morphism $f:C\to X$, the pullback Higgs bundle $f^*\mathcal{E}_s=(f^*E_s,f^*θ_s)$ is semistable. However, on the other hand, $\mathrm{det}(E_s)\cong\mathcal{O}_X$ and $\int_Xc_2(E_s)=10$. Consequently, the discriminant of $E_s$ does not vanish, and $\mathcal{E}_s$ provides the desired counterexample.
