conditions (c1) and (c2), the cohomology ${\mathcal{E}}$ of the monad $M(\alpha,\beta)$ is trivial at infinity if and only if the morphisms $(\alpha,\beta)$ satisfy conditions (c3) and (c4).

Condition (c3) is equivalent to the local freeness of ${\mathcal{E}}|_{{\ell_\infty}}$. As for condition (c4), the display of $M(\alpha,\beta)$ produces the exact sequence $$\xymatrix{
H^0\left({\mathcal{E}}|_{{\ell_\infty}}(-1)\right) \ar@{>->}[r] & H^0\left({\mathcal{A}}|_{{\ell_\infty}}(-1)\right) \ar[r]^-\Phi & H^0\left({{\mathcal{W}}_{\vec{k},\infty}}(-1)\right) \ar@{->>}[r] & H^1\left({\mathcal{E}}|_{{\ell_\infty}}(-1)\right)\,.
}$$ Condition (c4) is equivalent to the vanishing of $H^i\left({\mathcal{E}}|_{{\ell_\infty}}(-1)\right)$, $i=0,1$. The thesis follows easily.

This result enables us to identify $L_{\vec k}$ with the subset of ${\mathbb{V}_{\vec{k}}}$ whose points correspond to cohomology sheaves ${\mathcal{E}}_{\alpha,\beta}$ that are torsion-free and trivial at infinity.

\[6lm10\] The variety ${L_{\vec{k}}}$ is smooth of dimension $\dim{L_{\vec{k}}}=\dim{\mathbb{V}_{\vec{k}}}-\dim{\mathbb{W}_{\vec{k}}}$, where ${\mathbb{W}_{\vec{k}}}={\operatorname{Hom}}\left({{\mathcal{U}}_{\vec{k}}},{{\mathcal{W}}_{\vec{k}}}\right)$.

We define the map $$\begin{array}{rccl}
\zeta:&{\mathbb{V}_{\vec{k}}}&\longrightarrow&{\mathbb{W}_{\vec{k}}}\\
&(\alpha,\beta)&\longmapsto&\beta\alpha.
\end{array}$$ So $\overline{L}_{\vec{k}}$ is the set $\{\zeta=0\}$. The differential ${\mbox{\rm d}}\zeta$ at the point $(\alpha_0,\beta_0)\in{L_{\vec{k}}}$ is the linear map $$\begin{array}{rccl}
({\mbox{\rm d}}\zeta)|_{(\alpha_0,\beta_0)}:&{\mathbb{V}_{\vec{k}}}&\longrightarrow&{\mathbb{W}_{\vec{k}}}\\
&(\alpha,\beta)&\longmapsto&\beta_0\alpha+\beta\alpha_0.
\end{array}$$ The rank of this map is equal to $d=\dim{\mathbb{V}_{\vec{k}}}-\dim{L_{\vec{k}}}$ on the (non empty) nonsingular locus of ${L_{\vec{k}}}$, and outside of this set is bounded above by $d$ (see [@Har pp. 31-33]).

One can prove as in [@Ok Lemma II.4.1.7] that for any point $(\alpha,\beta)\in{L_{\vec{k}}}$, there is an isomorphism $${\operatorname{coker}}({\mbox{\rm d}}\zeta)|_{(\alpha,\beta)}\simeq H^2\left({\mathcal{E}}_{\alpha,\beta}^*\otimes{\mathcal{E}}_{\alpha,\beta}\right).
\label{eqdezeta}$$ Now, ${\mathcal{E}}_{\alpha,\beta}^*\otimes{\mathcal{E}}_{\alpha,\beta}$ is torsion-free 