\(\overline\partial\)-coherent sheaves on complex manifolds (Q2433978)

The author generalizes in a natural but highly nontrivial way a classification result of \textit{J.-L. Koszul} and \textit{B. Malgrange} [Arch Math. 9, 102--109 (1958; Zbl 0083.16705)] which says that a differential vector fiber bundle over a complex manifold, admitting a connexion of type \((0, 1)\), \(\partial\) such that \(\overline\partial^2 = 0\) has a structure of holomorphic vector bundle. Let \((X, J)\) be a complex manifold, \(J\) the tensor of the almost complex structure, supposed to be integrable. Let \({\mathcal E}_X\) be the sheaf of \(C^\infty\) (complex valued) functions in \(X\), \({\mathcal E}_X^{0, q}\) the sheaf of \((0, q)\) forms and \(\partial_j\) the \((0, 1)\) component of the differential. Consider on \(X\) a sheaf \({\mathcal G}\) of \({\mathcal E}_X\)-modules, endowed with a connection \(\overline\partial : {\mathcal G} \otimes_{{\mathcal E}_X} {\mathcal E}_X^{0, 1}\) of type \((0, 1)\) with \(\overline\partial^2 = 0\); if \({\mathcal G}\) admits local \({\mathcal E}\)-resolutions of finite length, then \({\mathcal G}\) is called a \(\bar\partial\)-coherent sheaf. The main result is: The sheaf of the \({\mathcal O}_X\)-module \(\operatorname{Ker}\overline\partial \subset {\mathcal G}\) is an analytic coherent sheaf, \({\mathcal G} = (\operatorname{Ker}\overline\partial)\). \({\mathcal E}_X \simeq (\operatorname{Ker}\overline\partial)\otimes_{{\mathcal O}_X} {\mathcal E}_X\) and the \(\overline\partial\)-connection coincides (modulo a canonical isomorphism) with the natural extension \(\overline\partial_{\ker\overline\partial}\) associated with the coherent sheaf \(\ker\overline\partial\). We have also that for any finite local \(\mathcal{E}\)-resolution of \(\mathcal{G}\) over an open set \(U \subset X\), with morphisms \(\varphi_k\), the sets \(Z_l (\varphi_k) = \{ x \in U\), \(\text{rank}_{{\mathcal C}} \varphi_k(x) \leq l\}\) are analytic sets for any integer \(l \geq 0\) and \(k = 1, \dots, m\) (\(m=\) the length of the resolution). The important fact is that there is an exact equivalence between the category of analytic coherent sheaves and the category of \(\bar\partial\)-coherent sheaves. The essential point here is that the ring of germs of \(C^\infty\) complex valued functions is \({\mathcal O}_X\)-faithfully flat (\({\mathcal O}_X=\) sheaf of germs of holomorphic functions). The proof has several steps. The first step is to find an local expression of the integrability condition \(\overline\partial^2 = 0\), which is realized through a recursion procedure. Then the author introduces the notion of recalibration, which generalize the notion of change of gauge for the local forms of an integrable connection of \((0, 1)\) type on locally free sheaf. The third step consists in a formulation of the differential form of the local expression. In fact, the difficulty consists in showing that for any local resolution of the sheaf \({\mathcal G}\) one can find, in the neighborhood of any point of the open set on which the local resolution is defined, another local resolution where elements are holomorphic matrices. This is shown to be equivalent to the existence of solution of a certain quasilinear differential system. Finally, after a suitable choice of norm for the Leray-Koppelman operator of the ball of radius \(r\), the author uses a Nash-Moser scheme (rapidly converging) which solves the existence problem for the differential system considered above. This requires a very careful analysis in order to obtain the precise estimates which ensure the convergence of the procedure. An application of the main results is a method (caled \(\overline\partial\)-stability) which allows to find analytic structures by \(C^\infty\)-deformation of other analytic structures.