I'm reading the Bruzzo and Graña Otero's paper *Semistable and Numerically Effective Principal (Higgs) Bundles*; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, reductive, (affine) algebraic group; $E$ is a principal $G$-bundle on $X$.


A *Higgs field* $\varphi$ on $E$ is a global section of $Ad(E)\otimes\_{\mathcal{O}\_X}\Omega\_X^1$ such that $[\varphi,\varphi]=0$. A *principal Higgs* $G$*-bundles on* $X$ is a pair $\mathfrak{E}=(E,\varphi)$.


Let $K$ be a closed subgroup of $G$ and let $\sigma:X\to E\_{\displaystyle/K}$ be a reduction of the structure group of $E$ to $K$; that is, there exists a principal $K$-bundle $F\_{\sigma}$ on $X$ and an injective $K$-equivariant bundle morphism $i\_{\sigma}:F\_{\sigma}\to E$. Let $\Pi\_{\sigma}:Ad(E)\otimes\_{\mathcal{O}\_X}\Omega^1\_X\to\left(Ad(E)\_{\displaystyle/Ad(F\_{\sigma})}\right)\otimes\_{\mathcal{O}\_X}\Omega^1\_X$ be the induced projection; $\sigma$ is called a *Higgs reduction* of $\mathfrak{E}$ if and only if $\varphi\in\ker\Pi\_{\sigma}$.


Let $E\_{K}$ denotes the principal $K$-bundle $E\to E\_{\displaystyle/K}$. Because:
\begin{gather\*}
T\_{E/K,X}\cong E\_{\displaystyle/K}\left(Ad\_K,\mathfrak{g}\_{\displaystyle/\mathfrak{k}}\right),\\
\pi\_K^{\*}Ad(E)=\pi\_K^{\*}\left(E(Ad\_G,\mathfrak{g})\right)=E\_{\displaystyle/K}(Ad\_K,\mathfrak{g}),
\end{gather\*}
there exists a natural morphism $\eta:\pi\_K^{\*}Ad(E)\to T\_{E/K,X}$ and $\varphi$ determines a global section $\eta(\varphi)\equiv(\eta\otimes Id)\left(\pi\_K^{\*}\varphi\right)$ of $T\_{E/K,X}\otimes\_{\mathcal{O}\_{E/K}}\Omega^1\_{E/K}$; the zero locus $\mathfrak{R}\_K(\mathfrak{E})$ of $\eta(\varphi)$ is called *scheme of Higgs reduction of* $\mathfrak{E}$ (definition 3.5).


They state (page 6):



> 
> By construction, $\sigma:X\to E\left(G\_{\displaystyle/K}\right)\cong E\_{\displaystyle/K}$ is a Higgs reduction if and only if it takes value in the subscheme $\mathfrak{R}\_K(\mathfrak{E})\subset E\_{\displaystyle/K}$.
> 
> 
> 


But I had check, via computation, that for any reduction $\sigma$ of $E$ one has $\mathfrak{R}\_K(\mathfrak{E})$ is contained in the scheme-theoretic image of $\sigma$. Am I wrong?



Scheme of Higgs reductions
Generalising t3suji's comment, let us consider $GL(n)$ Higgs bundles, i.e., rank $n$ Higgs vector bundles $\mathcal E$. The Higgs Grassmannian of rank $m$ quotients is a closed subscheme of the usual Grassmannian bundle; in general it is a proper subset, as there are quotients of $\mathcal E$ that are not Higgs quotients. The images of corresponding sections of the Grassmannian bundle are not contained in the Higgs Grassmannian. This is what the paper states. (In this case $K$ is a suitable parabolic subgroup of $GL(n)$).


