Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology of $(A^\bullet,\partial)$.

Let $G$ be a complex algebraic group. We suppose that $(A^\bullet,\partial)$ is equipped with an action of $G$, i.e., $G$ acts on $A^\bullet$ by preserving the grading and commuting with $\partial$. Then, $G$ acts on $H^\bullet$ as well.

Let $M$ be a compact differential manifold. We consider the moduli space of flat principla $G$-bundles on $M$, i.e., \begin{equation} \mathcal{M} = \{ \rho : \pi_1(M) \rightarrow G\} \;, \end{equation} where $\rho$ is a morphism between groups. Then, $\mathcal{M}$ has an abvious structure of scheme (over $\mathrm{spec}(\mathbb{C})$).

For any $\rho$ in $\mathcal{M}$, we consider the induced flat vector bundles $A^\bullet_\rho$ on $M$, which inherit the action of $\partial$ in the obvious way. We may also consider the induced flat vector bundles $H^\bullet_\rho$.

Let $\Omega^\bullet(M,A^\bullet_\rho)$ be the space of differential forms with values in $A^\bullet_\rho$, let $d$ be the de Rham operator acting on $\Omega^\bullet(M,A^\bullet_\rho)$. Then, $\Omega^\bullet(M,A^\bullet_\rho)$ being equipped with the actions of $d$ and $\partial$ becomes a double complex. We consider the total complex associated to this double complex, i.e., its degree $k$ component is \begin{equation} \bigoplus_{p+q=k}\Omega^p(M,A^q_\rho) \;, \end{equation} and the derivative is given by \begin{equation} D = d+(-1)^p\partial \;. \end{equation}

We may apply the standard spectral sequence technique for calculating the cohomology of this total complex. The first term will be $\Omega^\bullet(M,H^\bullet_\rho)$, the second term will the de Rham cohomology of $M$ with values in $H^\bullet_\rho$.

Let $\mathcal{N} \subseteq \mathcal{M}$ be the set of those $\rho$, which make the spectral sequence degenerate at its second term.

Question : Is $\mathcal{N}$ a Zariski closed subset of $\mathcal{M}$ ?

If possible, any further results concerning this subset $\mathcal{N}\subseteq\mathcal{M}$ ?

For example, can we tell something about the dependence of $\mathcal{N}$ on the acting of $G$ on $(A^\bullet,\partial)$ ? What if we take the intersection of all such $\mathcal{N}$ ? By a finite dimensional version of Delign theorem on degeneration, it is obvious that the intersection contains all the $\rho$ whose image is contained in a compact subgroup of $G$, but are there other points in $\mathcal{N}$?

Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology of $(A^\bullet,\partial)$.

Let $G$ be a complex algebraic group. We suppose that $(A^\bullet,\partial)$ is equipped with an action of $G$, i.e., $G$ acts on $A^\bullet$ by preserving the grading and commuting with $\partial$. Then, $G$ acts on $H^\bullet$ as well.

Let $M$ be a compact differential manifold. We consider the moduli space of flat principla $G$-bundles on $M$, i.e., \begin{equation} \mathcal{M} = \{ \rho : \pi_1(M) \rightarrow G\} \;, \end{equation} where $\rho$ is a morphism between groups. Then, $\mathcal{M}$ has an obvious structure of scheme (over $\mathrm{spec}(\mathbb{C})$).

For any $\rho$ in $\mathcal{M}$, we consider the induced flat vector bundles $A^\bullet_\rho$ on $M$, which inherit the action of $\partial$ in the obvious way. We may also consider the induced flat vector bundles $H^\bullet_\rho$.

Let $\Omega^\bullet(M,A^\bullet_\rho)$ be the space of differential forms with values in $A^\bullet_\rho$, let $d$ be the de Rham operator acting on $\Omega^\bullet(M,A^\bullet_\rho)$. Then, $\Omega^\bullet(M,A^\bullet_\rho)$ being equipped with the actions of $d$ and $\partial$ becomes a double complex. We consider the total complex associated to this double complex, i.e., its degree $k$ component is \begin{equation} \bigoplus_{p+q=k}\Omega^p(M,A^q_\rho) \;, \end{equation} and the derivative is given by \begin{equation} D = d+(-1)^p\partial \;. \end{equation}

We may apply the standard spectral sequence technique for calculating the cohomology of this total complex. The first term will be $\Omega^\bullet(M,H^\bullet_\rho)$, the second term will the de Rham cohomology of $M$ with values in $H^\bullet_\rho$.

Let $\mathcal{N} \subseteq \mathcal{M}$ be the set of those $\rho$, which make the spectral sequence degenerate at its second term.

Question : Is $\mathcal{N}$ a Zariski closed subset of $\mathcal{M}$ ?

If possible, any further results concerning this subset $\mathcal{N}\subseteq\mathcal{M}$ ?

For example, can we tell something about the dependence of $\mathcal{N}$ on the acting of $G$ on $(A^\bullet,\partial)$ ? What if we take the intersection of all such $\mathcal{N}$ ? By a finite dimensional version of Deligne theorem on degeneration, it is obvious that the intersection contains all the $\rho$ whose image is contained in a compact subgroup of $G$, but are there other points in $\mathcal{N}$?

Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology of $(A^\bullet,\partial)$.

Let $G$ be a complex algebraic group. We suppose that $(A^\bullet,\partial)$ is equipped with an action of $G$, i.e., $G$ acts on $A^\bullet$ by preserving the grading and commuting with $\partial$. Then, $G$ acts on $H^\bullet$ as well.

Let $M$ be a compact differential manifold. We consider the moduli space of flat principla $G$-bundles on $M$, i.e., \begin{equation} \mathcal{M} = \{ \rho : \pi_1(M) \rightarrow G\} \;, \end{equation} where $\rho$ is a morphism between groups. Then, $\mathcal{M}$ has an abvious structure of scheme (over $\mathrm{spec}(\mathbb{C})$).

For any $\rho$ in $\mathcal{M}$, we consider the induced flat vector bundles $A^\bullet_\rho$ on $M$, which inherit the action of $\partial$ in the obvious way. We may also consider the induced flat vector bundles $H^\bullet_\rho$.

Let $\Omega^\bullet(M,A^\bullet_\rho)$ be the space of differential forms with values in $A^\bullet_\rho$, let $d$ be the de Rham operator acting on $\Omega^\bullet(M,A^\bullet_\rho)$. Then, $\Omega^\bullet(M,A^\bullet_\rho)$ being equipped with the actions of $d$ and $\partial$ becomes a double complex. We consider the total complex associated to this double complex, i.e., its degree $k$ component is \begin{equation} \bigoplus_{p+q=k}\Omega^p(M,A^q_\rho) \;, \end{equation} and the derivative is given by \begin{equation} D = d+(-1)^p\partial \;. \end{equation}

We may apply the standard spectral sequence technique for calculating the cohomology of this total complex. The first term will be $\Omega^\bullet(M,H^\bullet_\rho)$, the second term will the de Rham cohomology of $M$ with values in $H^\bullet_\rho$.

Let $\mathcal{N} \subseteq \mathcal{M}$ be the set of those $\rho$, which make the spectral sequence degenerate at its second term.

Question : Is $\mathcal{N}$ a Zariski closed subset of $\mathcal{M}$ ?

If possible, any further results concerning this subset $\mathcal{N}\subseteq\mathcal{M}$ ?

For example, can we tell something about the dependence of $\mathcal{N}$ on the acting of $G$ on $(A^\bullet,\partial)$ ? What if we take the intersection of all such $\mathcal{N}$ ? By a finite dimensional version of Delign theorem on degeneration, it is abvious that the intersection contains all the $\rho$ whose image is contained in a compact subgroup of $G$, but are there other points in $\mathcal{N}$?

Let $(A^\bullet,\partial)$ be a complexe of $\mathbb{C}$-vector spaces. We suppose that this complex is of finite length, and all $A^\bullet$ are finite dimensional. Let $H^\bullet$ be the cohomology of $(A^\bullet,\partial)$.

Let $G$ be a complex algebraic group. We suppose that $(A^\bullet,\partial)$ is equipped with an action of $G$, i.e., $G$ acts on $A^\bullet$ by preserving the grading and commuting with $\partial$. Then, $G$ acts on $H^\bullet$ as well.

Let $M$ be a compact differential manifold. We consider the moduli space of flat principla $G$-bundles on $M$, i.e., \begin{equation} \mathcal{M} = \{ \rho : \pi_1(M) \rightarrow G\} \;, \end{equation} where $\rho$ is a morphism between groups. Then, $\mathcal{M}$ has an abvious structure of scheme (over $\mathrm{spec}(\mathbb{C})$).

For any $\rho$ in $\mathcal{M}$, we consider the induced flat vector bundles $A^\bullet_\rho$ on $M$, which inherit the action of $\partial$ in the obvious way. We may also consider the induced flat vector bundles $H^\bullet_\rho$.

Let $\Omega^\bullet(M,A^\bullet_\rho)$ be the space of differential forms with values in $A^\bullet_\rho$, let $d$ be the de Rham operator acting on $\Omega^\bullet(M,A^\bullet_\rho)$. Then, $\Omega^\bullet(M,A^\bullet_\rho)$ being equipped with the actions of $d$ and $\partial$ becomes a double complex. We consider the total complex associated to this double complex, i.e., its degree $k$ component is \begin{equation} \bigoplus_{p+q=k}\Omega^p(M,A^q_\rho) \;, \end{equation} and the derivative is given by \begin{equation} D = d+(-1)^p\partial \;. \end{equation}

We may apply the standard spectral sequence technique for calculating the cohomology of this total complex. The first term will be $\Omega^\bullet(M,H^\bullet_\rho)$, the second term will the de Rham cohomology of $M$ with values in $H^\bullet_\rho$.

Let $\mathcal{N} \subseteq \mathcal{M}$ be the set of those $\rho$, which make the spectral sequence degenerate at its second term.

Question : Is $\mathcal{N}$ a Zariski closed subset of $\mathcal{M}$ ?

If possible, any further results concerning this subset $\mathcal{N}\subseteq\mathcal{M}$ ?

For example, can we tell something about the dependence of $\mathcal{N}$ on the acting of $G$ on $(A^\bullet,\partial)$ ? What if we take the intersection of all such $\mathcal{N}$ ? By a finite dimensional version of Delign theorem on degeneration, it is obvious that the intersection contains all the $\rho$ whose image is contained in a compact subgroup of $G$, but are there other points in $\mathcal{N}$?