The structure of cohomology groups associated with the theta-zerovalues (Q2743990)

Let \(X_2\) denote the space of all symmetric \(2\times 2\) complex matrices \((t_{jk})_{1\leq j,k\leq 2}\). The authors define several sections \(P_1,P_2,Q_1,Q_2\) in \(M(3\times 3;D_X)\) (that is, \(3\times 3\) matrices whose coefficients are differential operators in \(t=(t_{ij})\)) such that \((\exp P_j-I)\bigg(\begin{smallmatrix} \vartheta(t)\\0\\0\end{smallmatrix}\bigg)=0\) and \((\exp Q_j-I)\bigg(\begin{smallmatrix} \vartheta(t)\\0\\0\end{smallmatrix}\bigg)=0\), \(j=1,2\). Let \(\mathcal N\) be the \(D_X\)-module \(D_Xu_0\oplus(D_Xu_1+D_Xu_2)\) with relations \((\partial_{11}\partial_{22}-\partial^2_{12})u_0=0\), \(\partial_{12}u_1=\partial_{11}u_2\) and \(\partial_{22}u_1=\partial_{12}u_2\). They define \(\mathcal N^\infty=D_X^\infty\otimes_{D_X}\mathcal N\) and associated to this set of data they construct a Koszul complex: \(K=(\cdots\to{\mathcal N}^\infty\to({\mathcal N}^\infty)^4\to({\mathcal N}^\infty)^6\to({\mathcal N}^\infty)^4\to({\mathcal N}^\infty)\to 0)\) and give an explicit description of its characteristic set in \(2\).NEWLINENEWLINEThe rest of the paper is devoted to studying the solution complex \(S={\mathbb R}{\Hom}_{D^\infty}(K,\mathcal O)\) and its cohomology groups. This paper is a report on results obtained by the authors in the general case \(X_n\), generalizing results proved in the case \(n=1\) by \textit{T. Aoki, M. Kashiwara} and \textit{T. Kawai} [Adv. Math. 62, 155--168 (1986; Zbl 0628.35003)].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00052].