and trivial at infinity, so that $H^2\left({\mathcal{E}}_{\alpha,\beta}^*\otimes{\mathcal{E}}_{\alpha,\beta}\right)=0$ by Lemma \[lm3\] and ${\mbox{\rm d}}\zeta$ has maximal rank everywhere on ${L_{\vec{k}}}$. Since the singular locus of ${L_{\vec{k}}}$ coincides with the set of points in which ${\mbox{\rm d}}\zeta$ fails to have maximal rank, ${L_{\vec{k}}}$ is smooth.

The variety ${P_{\vec{k}}}$
---------------------------

Let us introduce the varieties ${\Upsilon}=\Sigma_n\times{L_{\vec{k}}}$ and ${\Upsilon_\infty}={\ell_\infty}\times{L_{\vec{k}}}$, together with the canonical projections shown in the following diagram: $$\xymatrix{
{\ell_\infty}\ar@{^{(}->}[r] & \Sigma_n\\
{\Upsilon_\infty}\ar@{^{(}->}[r] \ar[u]_-{{\breve{\mathfrak{u}}}_1} \ar[dr]_{{\breve{\mathfrak{u}}}_2} & {\Upsilon}\ar[u]_{{\breve{\mathfrak{t}}}_1} \ar[d]^{{\breve{\mathfrak{t}}}_2}\\
 & {L_{\vec{k}}}\,.
}
\label{eqdT}$$ On ${\Upsilon}$ we define the complex $$\xymatrix@C+1em{
{\breve{\mathbb{M}}_{\vec{k}}}:\qquad {\breve{\mathfrak{t}}}_1^*{{\mathcal{U}}_{\vec{k}}}\ar[r]^-{f_A} & {\breve{\mathfrak{t}}}_1^*{{\mathcal{V}}_{\vec{k}}}\ar[r]^-{f_B} & {\breve{\mathfrak{t}}}_1^*{{\mathcal{W}}_{\vec{k}}}}$$ where $f_A$ and $f_B$ are the defined by the formulas $$\begin{array}{rccl}
f_{A}\colon&({\breve{\mathfrak{t}}}_1^*{{\mathcal{U}}_{\vec{k}}})_{(x,\alpha,\beta)}&\longrightarrow&({\breve{\mathfrak{t}}}_1^*{{\mathcal{U}}_{\vec{k}}})_{(x,\alpha,\beta)}\\
&u&\longmapsto&\alpha(u)\\[7pt]
f_{B}\colon&({\breve{\mathfrak{t}}}_1^*{{\mathcal{V}}_{\vec{k}}})_{(x,\alpha,\beta)}&\longrightarrow&({\breve{\mathfrak{t}}}_1^*{{\mathcal{V}}_{\vec{k}}})_{(x,\alpha,\beta)}\\
&v&\longmapsto&\beta(v)
\end{array}\,;$$ here $u$ and $v$ are points in the fibres over $(x,\alpha,\beta)\in \Sigma_n\times{L_{\vec{k}}}$. This complex is actually a monad by Lemma \[lmInjSurj\]: let ${\breve{\mathfrak{E}}_{\vec{k}}}$ be its cohomology. This sheaf satisfies condition $\vec{k}$, and more precisely one has the following isomorphism for all points $(\alpha,\beta)\in{L_{\vec{k}}}$: $$\left({\breve{\mathfrak{E}}_{\vec{k}}}\right)_{(\alpha,\beta)}\simeq{\mathcal{E}}_{\alpha,\beta}\,.$$ The restriction ${\breve{\mathbb{M}}_{\vec{k},\infty}}$ of ${\breve{\mathbb{M}}_{\vec{k}}}$ to ${\Upsilon_\infty}$ is isomorphic to the monad $$\xymatrix@C+1em{
0 \ar[r] & {\breve{\mathfrak{u}}}_1^*{{\mathcal{U}}_{\vec{k},\infty}}\ar[r]^-{g_A} & {\breve{\mathfrak{u}}}_1^*{{\mathcal{V}}_{\vec{k},\infty}}\ar[r]^-{g_B} & {\breve{\mathfrak{u}}}_1^*{{\mathcal{W}}_{\vec{k},\infty}}\ar[r] & 0\,.
}
\label{eqMoi}$$ where $g_A$ and $g_B$ are the defined by the formulas $$\begin{array}{rccl}
g_{A}\colon&({\breve{\mathfrak{u}}}_1^*{{\mathcal{U}}_{\vec{k},\infty}})_{(x,\alpha,\beta)}&\longrightarrow&({\breve{\mathfrak{u}}}_1^*{{\mathcal{U}}_{\vec{k},\infty}})_{(x,\alpha,\beta)}\\
&u&\longmapsto&\alpha|_{{\ell_\infty}}(u)\\[7pt]
g_{B}\colon&({\breve{\mathfrak{u}}}_1^*{{\mathcal{V}}_{\vec{k},\infty}})_{(x,\alpha,\beta)}&\longrightarrow&({\breve{\mathfrak{u}}}_1^*{{\mathcal{V}}_{\vec{k},\infty}})_{(x,\alpha,\beta)}\\
&v&\longmapsto&\beta|_{{\ell_\infty}}(v)
\end{array}\,.$$ The cohomology of ${\breve{\mathbb{M}}_{\vec{k},\infty}}$ will be denoted 