We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf ${\mathcal F}$ are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf $\Eps nd^*(\Eps^\cdot)$, where $\Eps^\cdot$ is any locally free resolution of ${\mathcal F}$. In particular, one recovers the well known fact that the tangent space to $\Def_{\mathcal F}$ is $\Ext^1({\mathcal F},{\mathcal F})$, and obstructions are contained in $\Ext^2({\mathcal F},{\mathcal F})$. \par The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA ${\mathfrak g}^\Delta$, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor $H^1_{\rm sc}(\exp {\mathfrak g}^\Delta)$.
Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
IACONO, Donatella;
2012-01-01
Abstract
We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf ${\mathcal F}$ are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf $\Eps nd^*(\Eps^\cdot)$, where $\Eps^\cdot$ is any locally free resolution of ${\mathcal F}$. In particular, one recovers the well known fact that the tangent space to $\Def_{\mathcal F}$ is $\Ext^1({\mathcal F},{\mathcal F})$, and obstructions are contained in $\Ext^2({\mathcal F},{\mathcal F})$. \par The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA ${\mathfrak g}^\Delta$, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor $H^1_{\rm sc}(\exp {\mathfrak g}^\Delta)$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
