Deformation of line bundles on coisotropic subvarieties (Q2907033)

Let \(X\) be a smooth algebraic variety and let \(\mathcal{A}_{2}\) be a second order deformation of the structure sheaf \(\mathcal{O}_{X}\). Given a smooth subvariety \(Y\subset X\) and a line bundle \(L \longrightarrow Y\) (regarded as a coherent \(\mathcal{O}_{X}\)-module), the authors prove conditions for \(L\) to admit a \(\mathbb{C}[\epsilon]/\epsilon^{2}\) flat second-order deformation to an \(\mathcal{A}_{2}\)-module \(\mathcal{L}_{2}\).NEWLINENEWLINEFor the first order deformation problem, which they consider in first place, three obstructions arise in \(H^{0}(Y, \bigwedge^{2} N), H^{1}(Y,N)\) and \(H^{2}(Y,\mathcal{O}_{Y})\) respectively, where \(N\) is the normal bundle of \(Y\) in \(X\). The vanishing of the first obstruction is equivalent to requiring \(Y\) to be coisotropic. The authors prove the following theorem, which we quote here:NEWLINENEWLINESuppose that \(X\) is a smooth algebraic variety with a bivector \(P\) and \(\beta^{X}_{1}\in H^{1}(X,T_{X})\) and \(Y\) is a smooth coisotropic subvariety with a line bundle \(L\). Denoting by \(\beta_{1}^{X}|_{Y}\) the image of \(\beta_{1}^{X}\) in \(H^{1}(Y,N)\), \(L\) admits a first order deformation if: NEWLINE\[NEWLINE c_{N}(L^{\otimes 2}\otimes \det N^{\vee})+2\beta^{X}_{1}|_{Y}=0 NEWLINE\]NEWLINE in \(H^{1}(Y,N)\), where \(c_{N}(L)=p^{\ast}(c_{1}(L))\) and \(p^{\ast}\) is the adjoint of the morphism \(p:N^{\vee} \rightarrow T_{Y}\) given by the coisotropness of \(Y\). If \(H^{2}(Y,\mathcal{O}_{Y})=0\), the condition is also sufficient for existence of first-order deformation.NEWLINENEWLINEThe authors also analyze the second-order deformation case and give sufficient and necessary conditions for its existence, which involve more mathematical tools, such as the normal de Rham complex \(\mathcal{N}^{\bullet}\) and a cohomology class \((\mathcal{Q},\beta_{1}^{X})\mid_{Y}\) defined in terms of the anti-symmetrization \(\mathcal{Q}\) of the local operator \(\alpha^{X}_{2}\) of the second order deformation \(\mathcal{A}_{2}\). The necessary condition for the existence of the second-order deformation is: NEWLINE\[NEWLINE c_{\mathcal{N}}(L^{\otimes2}\otimes \det N^{\vee}) = 2(\mathcal{Q},\beta_{1}^{X})\mid_{Y} NEWLINE\]NEWLINE in \(H^{2}(Y,\mathcal{N}^{\geq 1})\). The condition is also sufficient if \(H^{1}(Y,N)=H^{2}(Y,\mathcal{O}_{Y})=0\).