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Let $X$ be a connected smooth complex projective variety.

A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \in H^0(X, End(E)\otimes\Omega_X^1)$ with $\theta \wedge \theta = 0$ in $H^0(X, End(E)\otimes\Omega^2_X)$.

A holomorphic Higgs bundle $(E, \theta)$ is said to have a structure of a system of Hodge bundles if $E$ has a holomorphic direct sum decomposition $E = \bigoplus\limits_{i=0}^n E_i$ such that $\theta(E_i) \subseteq E_{i-1}\otimes\Omega_X^1$, for all $1 \leq i \leq n$.

Now let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$.

A holomorphic principal $G$-Higgs bundle on $X$ is a pair $(E_G, \theta)$, where $E_G$ is a holomorphic principal $G$-bundle on $X$ and $\theta \in H^0(X, \text{ad}(E_G)\otimes\Omega_X^1)$ such that $\theta\wedge\theta = 0$ in $H^0(X, \text{ad}(E_G)\otimes\Omega_X^2)$; here $\text{ad}(E_G) := E_G \times^G \mathfrak{g}$ is the adjoint vector bundle associated to the adjoint representaion $ad : G \longrightarrow End(\mathfrak{g})$, and $\mathfrak{g}$ is the Lie algebra of $G$.

For example, when $G = GL_n$, $\text{ad}(E_G) = End(E_{GL_n})$, where $E_{GL_n}$ is the vector bundle corresponding to the $GL_n$-bundle $E_G$.

Question: Is there any analogue of system of Hodge bundles for the case of principal $G$-Higgs bundles?

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  • $\begingroup$ This looks like a $\theta$-compatible reduction of structure group from $G$ to a fixed Borel subgroup of $G$. $\endgroup$
    – Jason Starr
    Commented Nov 27, 2018 at 13:06
  • $\begingroup$ I think, $\theta$-compatible reduction of structure group to a Borel subgroup is similar to give a full-flag filtration by subbundles which is preserved by $\theta$. But we need one place shift of indices! $\endgroup$
    – user124771
    Commented Nov 27, 2018 at 13:12
  • $\begingroup$ Wait, how do you define "$\theta$-compatible"? $\endgroup$
    – Jason Starr
    Commented Nov 27, 2018 at 13:19
  • $\begingroup$ A reduction of structure group of $E_G$ to a parabolic subgroup $P \subset G$ is given by a section $\eta$ of the principal $G/P$-bundle $E_G\times^G (G/P)$ over $X$ obtained by the extension of structure group of $E_G$ by the surjective homomorphism $G \to G/P$. Equivalently, we get a principal $P$-bundle $E_P$ whose total space is a subvariety of $E_G$ and the principal $G$-bundle $E_P\times^P G$ is isomorphic to $E_G$. Such reduction is called $\theta$-compatible if $\theta \in H^0(X, ad(E_P)\otimes\Omega_X^1) \subseteq H^0(X, ad(E_G)\otimes\Omega_X^1)$. $\endgroup$
    – user124771
    Commented Nov 27, 2018 at 13:58
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    $\begingroup$ You probably want a reduction to a Levi subgroup $L$ of $P$. Then you may take an $sl_2$-triple $(e_-,h,e)$ such that $L$ is the centralizer of $h$. Let $\mathfrak g_1$ be the 1-eigenspace of $h$. Then you can require that $\theta$ comes from a section of $(E_L\times^L\mathfrak g_1)\otimes\Omega_X^1$. The question is what non-abelian Hodge theory statement you get. $\endgroup$
    – Roman Fedorov
    Commented Nov 29, 2018 at 11:49

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