Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
In the particular case of a locally free sheaf \(\mathcal E\) of \(\mathcal O_X\)-modules on a complex manifold \(X\), the Kodaira-Spencer description of deformations of \(\mathcal E\) is given in terms of the Čech functor \(H^1(X;\, \text{exp} \, \mathcal End(\mathcal E))\). A deformation of \(\mathcal E\) is reduced to a deformation of the gluing data of its local charts, and the compatibility condition these gluing data have to satisfy is precisely expressed by the cocycle condition in the Čech functor. Also, deformations of \(\mathcal E\) are controlled by the Differential Graded Lie Algebra (DGLA) \(A^{0,\ast}_X(\mathcal End(\mathcal E))\) of \((0,\ast)\)-forms on \(X\) with values in the sheaf of endomorphisms of \(\mathcal E\). The equivalence of these two descriptions can be understood by moving to groupoid-valued deformation functors: The groupoid \(\text{Def}_{\mathcal E|_U}\) of \(\mathcal E\) over \(U\) defines a stack over \(\text\mathbf{Top}_X\). That is, the global deformations of \(\mathcal E\) are the same thing as the descent data for its local deformations: \(\text{Def}_{\mathcal E}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\text{Def}_{\mathcal E|_U},\) where \(\Delta_{\mathcal U}\) is the semisimplicial object in \(\text\mathbf{Top}_X\) associated with an open cover \(\mathcal U\) of \(X\). One sees that locally the groupoid of deformations of \(\mathcal E|_U\) is equivalent to the Deligne groupoid of \(\mathcal End(\mathcal E)(U)\), and since these equivalences are compatible with restriction maps, one has an equivalence of semicosimplicial groupoids. Finally, the Deligne groupoid commutes with homotopy limits of DGLA concentrated in nonnegative degree so that NEWLINE\[CARRIAGE_RETURNNEWLINE\text{Def}_{\mathcal E}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\text{Del}_{\mathcal End(\mathcal E)(U)}\simeq\text{Del}_{\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End(\mathcal E)(U)}CARRIAGE_RETURNNEWLINE\]NEWLINE proving that the problem of infinitesimal deformations of \(\mathcal E\) is controlled by the DGLA \(\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End(\mathcal E)(U)\). Then there is a quasi isomorphism of DGLA's \(\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End(\mathcal E)(U)\simeq A^{0,\ast}_X(\mathcal End(\mathcal E)).\)NEWLINENEWLINEThe authors turn to deformations of a coherent sheaf \(\mathcal F\) of \(\mathcal O_X\)-modules on a complex manifold or an algebraic variety \(X\). This deformation theory is based on a locally free resolution \(\mathcal E^{\cdot}\rightarrow\mathcal F\rightarrow 0\). Then the deformation theory might be controlled by the deformations of \(\mathcal E^\cdot\) with appropriate gluing conditions. On the DGLA side, one proves that infinitesimal deformations of \(\mathcal F\) are controlled by the DGLA of global sections of an acyclic resolution of \(\mathcal End^\ast(\mathcal E^\cdot)\). To see why this result holds, the authors go one step further, from groupoid-valued to \(\infty\)-groupoid-valued deformation functors, and think of this as \(\infty\)-stacks. From the \(\infty\)-groupoid point of view it is natural to expect that the stack \(\text{Def}_{\mathcal F}\) is locally homotopy equivalent to the \(\infty\)-stack \(\mathbf{MC}_{\bullet}(\mathcal End^\ast(\mathcal E)).\) Arguing more or less as in the case with bundles, the following holds: NEWLINE\[CARRIAGE_RETURNNEWLINE \text{Def}_{\mathcal F}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\text{Def}_{\mathcal F|_U}\simeq\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathbf{MC}_{\bullet}(\mathcal End^\ast(\mathcal E^\cdot)(U))\simeq\text\textbf{MC}_{\bullet}(\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End^\ast(\mathcal E^\cdot)(U)).CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINEThe homotopy limit \(\underset{U\in\Delta_{\mathcal U}} {\text{holim}}\mathcal End^\ast(\mathcal E^\cdot)(U)\) is quasi-isomorphic to the DGLA of global sections of an acyclic resolution of \(\mathcal End^\ast(\mathcal E^\cdot)\) which therefore controls the infinitesimal deformations of \(\mathcal F\). The aim of the paper is to give a direct proof of this fact at the level of set-valued deformation functors.NEWLINENEWLINEThe work is done over an algebraically closed field of characteristic \(0\). The authors give a thorough introduction to infinitesimal deformations and sheaves of DGLAs, given by the Maurer-Cartan equations. They give the definition of the Thom-Withney DGLA.There are some actions (\(\ast\) and \(\bullet\)) defined in the references, but which could be defined in the article to improve the impression. Besides this, the article is well written and gives nice and important results. Having different DGLA's to control deformations give different, but equivalent, ways to compute deformations.
scientific article; zbMATH DE number 6023340
scientific article