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Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$.

Does exists a Higgs quotient sheaf $\mathcal{Q}$ of $\mathfrak{E}$ of rank $p$, for any $p\in\{1,\dotsc,r-1\}$?

For clarity: Higgs sheaf $\mathfrak{E}$ means a pair $(E,\varphi)$ where $E$ is a $\mathcal{O}_X$-coherent sheaf equipped with a morphism $\varphi\colon E\to E\otimes\Omega^1_X$ such that the composition $$ \varphi\wedge\varphi\colon E\xrightarrow{\varphi}E\otimes\Omega^1_X\xrightarrow{\varphi\otimes\operatorname{Id}}E\otimes\Omega^1_X\otimes\Omega^1_X\to E\otimes\Omega^2_X $$ vanishes. A Higgs subsheaf of a Higgs sheaf $(E,\varphi)$ is a $\varphi$-invariant subsheaf $G$ of $E$, i.e., $\varphi(G)\subset G\otimes \Omega_X^1$. A Higgs quotient of $\mathfrak{E}$ is a quotient of $E$ such that the corresponding kernel is $\varphi$-invariant. A Higgs bundle is a Higgs sheaf whose underlying coherent sheaf is locally free.

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    $\begingroup$ I think for a counterexample we can take $X =C \times \mathbb P^1$ where $C$ is a curve of higher genus and $\mathfrak E$ the pullback of a Higgs bundle on $C$ with irreducible spectral curve. Quotient sheaves will correspond to subsets of the spectral curve, hence either of the same rank or $0$. $\endgroup$
    – Will Sawin
    Commented Jan 14, 2022 at 17:20
  • $\begingroup$ Ok, thank you. In other words, a general HIggs Grassmannian $HGr_p(\mathfrak{E})\to X$ (see arXiv:math/0603509 [math.AG] section 2) has no global sections. Setting $n=r=2$, there is a curve $C$ on $X$ (of course): is it know a characterization of rank $2$ Higgs bundles over $C$ which spectral curve is irreducible? $\endgroup$
    – Armando j18eos
    Commented Jan 15, 2022 at 8:50
  • $\begingroup$ The only thing I know is on curves it's equivalent to the discriminant of the characteristic polynomial not being a perfect square. $\endgroup$
    – Will Sawin
    Commented Jan 15, 2022 at 13:07

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