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 $\mathcal{N}$?

  • $\begingroup$ Could you provide a specific reference to the theorem of Deligne that you mention in the question? $\endgroup$
    – Dan Ramras
    Mar 3 '15 at 2:10
  • $\begingroup$ I read the Deligne Theorem from Claire Voisin 'Théorie de Hodge et géométrie algébrique complexe' chap 16. The original version due to Deligne comes from 'Théorème de Lefschetz et critères de dégénérescence de suites spectrales' $\endgroup$ Mar 3 '15 at 2:47

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