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\section*{nonabelian Hodge theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\begin{quote}%
\textbf{under construction}


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{nonabelian_hodge_theorems}{Nonabelian Hodge theorems}\dotfill \pageref*{nonabelian_hodge_theorems} \linebreak
\noindent\hyperlink{nonabelian_harmonic_sections}{Nonabelian harmonic sections}\dotfill \pageref*{nonabelian_harmonic_sections} \linebreak
\noindent\hyperlink{LocalSystemsAndHiggsBundles}{K\"a{}hler case: Equivalence between Local systems and Higgs bundles}\dotfill \pageref*{LocalSystemsAndHiggsBundles} \linebreak
\noindent\hyperlink{relation_to_the_abelian_hodge_theorem}{Relation to the abelian Hodge theorem}\dotfill \pageref*{relation_to_the_abelian_hodge_theorem} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{generalizations_to_twisted_bundles}{Generalizations to twisted bundles}\dotfill \pageref*{generalizations_to_twisted_bundles} \linebreak
\noindent\hyperlink{relation_to_geometric_langlands}{Relation to geometric Langlands}\dotfill \pageref*{relation_to_geometric_langlands} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Nonabelian Hodge theory generalizes aspects of [[Hodge theory]] from abelian cohomology ([[abelian sheaf cohomology]]) to [[nonabelian cohomology]].

\hypertarget{nonabelian_hodge_theorems}{}\subsection*{{Nonabelian Hodge theorems}}\label{nonabelian_hodge_theorems}

\hypertarget{nonabelian_harmonic_sections}{}\subsubsection*{{Nonabelian harmonic sections}}\label{nonabelian_harmonic_sections}

Notice or recall (for instance from [[generalized universal bundle]] and [[action groupoid]]) the following equivalent description of [[section]]s of [[associated bundle]]s:

for $G$ a [[group]] with [[action]] $\rho$ on an object $V$ witnessed by the [[action groupoid]] sequence

\begin{displaymath}
V \to V//G \to \mathbf{B}G
\end{displaymath}
the $\rho$-[[associated bundle]] $E \to X$ to a $G$-[[principal bundle]] $P \to X$ classified by an [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} Y  \to \mathbf{B}G$ is the [[pullback]]

\begin{displaymath}
\itexarray{
    E &\to& V//G
    \\
    \downarrow && \downarrow
    \\
    Y &\to& \mathbf{B}G
  }
  \,.
\end{displaymath}
Since this is a [[pullback]] diagram by definition, a glance at a pasting diagram of the form

\begin{displaymath}
\itexarray{
    && E &\to& V//G
    \\
    & \nearrow & \downarrow && \downarrow
    \\
    Y &\stackrel{=}{\to}& Y &\to& \mathbf{B}G
  }
\end{displaymath}
shows that [[section]]s

\begin{displaymath}
\itexarray{
     && E
     \\
     & {}^{\sigma}\nearrow & \downarrow
     \\
     Y &\stackrel{=}{\to}& Y
  }
\end{displaymath}
are in bijection with maps $Y \to V//G$ that make

\begin{displaymath}
\itexarray{
    Y &\to& V//G
    \\
    \downarrow^= && \downarrow
    \\
    Y &\to& \mathbf{B}G
  }
\end{displaymath}
commute.

In the special case that $X$ is a connected [[manifold]] and $G$ a discrete group we can without restriction take $Y = \hat X//\pi_1(X)$ be the [[action groupoid]] of the [[universal cover]] by the [[fundamental group|homotopy group]], so that the classifying map $Y \to \mathbf{B}G$ is the same as a [[group]] homomorphism

\begin{displaymath}
\rho : \pi_1(X) \to G
  \,.
\end{displaymath}
In that case the above says that a section of the associated bundle is a $\rho$-equivariant map

\begin{displaymath}
\phi : \hat X \to V
  \,.
\end{displaymath}
This is the way these sections are formulated usually in the literature. The above description has the advantage that it works more generally in [[nonabelian cohomology]] for [[principal bundle]]s generalized to [[principal ∞-bundle]]s.

Next consider furthermore the special case that $V = G/K$ is the [[coset]] [[homogeneous space]] of $G$ quotiented by a subgroup $K$. Then if $G$ is a [[Lie group]] or [[algebraic group]] consider moreover a choice of $G$-invariant metric on the quotient $G/K$. Also consider a [[Riemannian manifold]] structure on $X$.

Then

\begin{defin}
\label{}\hypertarget{}{}
The \textbf{energy} of a [[section]] $\sigma$ of an associated $G/K$-bundle as above is the real number

\begin{displaymath}
E(\phi) := \int_X |d \phi|^2
  \,.
\end{displaymath}
\end{defin}
Here

\begin{itemize}%
\item $\phi$ is the $\rho$-equivariant map describing the section as above,


\item the [[norm]] is taken with respect to the chocen invariant [[metric]] on $G/K$


\item and the [[integral]] is taken with respect to the [[Riemannian metric]] on $X$.



\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
Such a $\phi$ is called \textbf{harmonic} if it is a [[critical point]] of $E(-)$.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Corlette, generalizing Eells-Sampson)}

If $\rho : \pi_1(X) \to G$ is a representation with

\begin{itemize}%
\item $G$ a [[reductive group|reductive]] [[algebraic group]]


\item $K$ is a [[maximal compact subgroup]]


\item $\rho(\pi_1(X))$ is

\begin{itemize}%
\item Zariski-dense in $G$


\item or its Zariski-closure is itself reductive



\end{itemize}


\end{itemize}
then there exists a \emph{harmonic} section $\phi$ in the above sense.

\end{theorem}
This is due to (\hyperlink{Corlette88}{Corlett 88}). A version of the proof is reproduced in \hyperlink{Simpson96}{Simpson 96, p. 8}

\hypertarget{LocalSystemsAndHiggsBundles}{}\subsubsection*{{K\"a{}hler case: Equivalence between Local systems and Higgs bundles}}\label{LocalSystemsAndHiggsBundles}

The \emph{nonabelian Hodge theorem} due to (\hyperlink{Simpson92}{Simpson 92}) establishes, for $X$ a [[compact topological space|compact]] [[Kähler manifold]], an [[equivalence]] between (irreducible) [[flat vector bundles]] on $X$ and (stable) [[Higgs bundles]] with vanishing [[first Chern class]].

\hypertarget{relation_to_the_abelian_hodge_theorem}{}\paragraph*{{Relation to the abelian Hodge theorem}}\label{relation_to_the_abelian_hodge_theorem}

The sense in which the nonabelian Hodge theorem of (\hyperlink{Simpson92}{Simpson 92}) generalizes the abelian [[Hodge theorem]] is the following (\hyperlink{Simpson92}{Simpson 92, Introduction}).

The abelian [[cohomology group]] $H^1(X,\mathbb{C}_{disc})$ classifies [[flat vector bundle|flat]] [[complex line bundles]] whose underlying [[line bundle]] is trivial, hence closed [[differential 1-forms]] modulo 0-forms. The abelian [[Hodge theorem]] gives for this hence the decomposition

\begin{displaymath}
H^1(X,\mathbb{C}_{disc}) 
  \simeq
  H^1(X, \mathcal{O}_X) \oplus H^0(X, \Omega^1_X)
  \,.
\end{displaymath}
It is this kind of relation which is generalized by the nonabelian Hodge theorem. Here one starts instead with the [[nonabelian cohomology]] set $H^1(X, GL_n(\mathbb{C})_{disc})$ which classifies [[flat vector bundle|flat]] [[rank]]-$n$ [[vector bundles]] on $X$, for $n \in \mathbb{N}$. The equivalence to [[Higgs bundles]] gives now a decomposition of these structures into a [[holomorphic vector bundle]] classified by $H^1(X, GL_n(\mathcal{O}_X))$ and a differential 1-form with values in endomorphisms of that, subject to some conditions.

\hypertarget{statement}{}\paragraph*{{Statement}}\label{statement}

A quick review of the theorem in (\hyperlink{Simpson92}{Simpson 92}) is for instance in (\hyperlink{Raboso14}{Raboso 14, section 1.2}). An elegant abstract reformulation in terms of [[differential cohesion]]/[[D-geometry]], following (\hyperlink{Simpson96}{Simpson 96}) is in (\hyperlink{Raboso14}{Raboso 14, section 4.2.1}):

Analogous to how the [[de Rham stack]] $\int_{inf} X = X_{dR}$ of $X$ is the ([[homotopy quotient|homotopy]]) [[quotient]] of $X$ by the first order [[infinitesimal neighbourhood]] of the [[diagonal]] in $X \times X$, so there is a space ([[stack]]) $X_{Dol}$ which is the formal competion of the 0-section of the [[tangent bundle]] of $X$ (\hyperlink{Simpson96}{Simpson 96}).

Now a [[flat vector bundle]] on $X$ is essentially just a [[vector bundle]] on the [[de Rham stack]] $X_{dR}$, and a [[Higgs bundle]] is essentially just a [[vector bundle]] on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear [[algebraic group]] over $\mathbb{C}$)

\begin{displaymath}
\mathbf{H}(X_{dR}, \mathbf{B}G)
  \simeq
  \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0}
  \,,
\end{displaymath}
where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing [[first Chern class]] (see \hyperlink{Raboso14}{Raboso 14, theorem 4.2}).

\hypertarget{generalizations_to_twisted_bundles}{}\paragraph*{{Generalizations to twisted bundles}}\label{generalizations_to_twisted_bundles}

A generalization of the nonabelian Hodge theorem of (\hyperlink{Simpson92}{Simpson 92}) to [[twisted bundles]] in discussed in (\hyperlink{Raboso14}{Raboso 14}).

\hypertarget{relation_to_geometric_langlands}{}\subsection*{{Relation to geometric Langlands}}\label{relation_to_geometric_langlands}

Nonabelian Hodge theory is closely related to the [[geometric Langlands correspondence]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hodge structure]]


\item [[noncommutative Hodge theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes on nonabelian Hodge theory include:

\begin{itemize}%
\item [[Ron Donagi]], [[Tony Pantev]], \emph{Lectures on the geometric Langlands conjecture and non-abelian Hodge theory}, 2009 (\href{http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf}{pdf})


\item [[Alberto García Raboso]], [[Steven Rayan]], \emph{Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure}, Fields Inst. Monogr. 34 (2015), 131--171 (\href{https://arxiv.org/abs/1406.1693}{arXiv}) (\href{http://link.springer.com/chapter/10.1007/978-1-4939-2830-9_5}{Springer})



\end{itemize}
Corlette's nonabelian Hodge theorem can be found in:

\begin{itemize}%
\item K. Corlette, \emph{Flat $G$-bundles with canonical metric}, J. Diff Geometry 28 (1988)

\end{itemize}
Works by [[Carlos Simpson]] on nonabelian Hodge theory include:

\begin{itemize}%
\item [[Carlos Simpson]], \emph{Higgs bundles and local systems}, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 75, 5\{95. MR 1179076 (94d:32027) (\href{http://www.numdam.org/item?id=PMIHES_1992__75__5_0}{numdam})


\item [[Carlos Simpson]], \emph{The Hodge filtration on nonabelian cohomology}, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217\{281. MR 1492538 (99g:14028) (\href{http://arxiv.org/abs/alg-geom/9604005}{arXiv:9604005})


\item [[Carlos Simpson]], \emph{Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology} (\href{http://arxiv.org/abs/alg-geom/9712020}{arXiv:9712020})


\item [[Carlos Simpson]], \emph{Algebraic aspects of higher nonabelian Hodge theory} (\href{http://arxiv.org/abs/math/9902067}{arXiv:9902067})


\item [[Carlos Simpson]], [[Tony Pantev]], [[Ludmil Katzarkov]], \emph{Nonabelian mixed Hodge structures} (\href{http://arxiv.org/abs/math/0006213}{arXiv})



\end{itemize}
The nonabelian Hodge theorem of (\hyperlink{Simpson92}{Simpson 92}) is generalized to [[twisted bundles]] in:

\begin{itemize}%
\item [[Alberto García Raboso]], \emph{A twisted nonabelian Hodge correspondence}, PhD thesis 2014 (\href{http://arxiv.org/abs/1501.05872}{arXiv:1501.05872}, \href{http://www.math.toronto.edu/agraboso/files/TwistedNAHT_Talk_Handout.pdf}{pdf slides})

\end{itemize}
[[!redirects nonabelian Hodge theorem]] [[!redirects nonabelian Hodge theorems]]



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