# Higgs quotient sheaf of a Higgs bundle

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.

• 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 $0$. Jan 14 at 17:20
• 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? Jan 15 at 8:50
• The only thing I know is on curves it's equivalent to the discriminant of the characteristic polynomial not being a perfect square. Jan 15 at 13:07