Boundary value problems and Index Theorem for D Modules

我们找到了用d -模表示微分方程组椭圆边值问题并构造其特征环（或微局部欧拉类）的一种思路。我们考虑一个流形上的微分方程组M，流形的边界为n，设M_<tan>表示它对边界的拉回。给定一个边界上的微分方程组N和一个D_N线性态射α: N→M_<tan>。根据定义，如果a从ε_N[〇！]×] N到ε_N [N !]的相干商模M^+（tan）x] M^<tan>从边界微局部定义。（ε_N为边界上微微分算子的环束。）在这个朴素的集合中构造一个特征环仍然是困难的，我们想把这个BVP集合转化为环上的一个模（或模的派生范畴的对象）。如果我们引入环BD为BD = D_M[〇！d (n, m)[〇！+]D_N，我们可以得到派生范畴D^ B (D_M[〇！x] B)，其中B为2次的上半三角形矩阵环。我们可以定义一个特征循环（或微局部欧拉类）与D_M[〇！x] b模和一个b模（Z_M, Z_N），在α的椭圆性条件下。我们期望人们可以（通过追逐图）用这里定义的特征类来证明边值问题的指标定理，因为它们的构造几乎完全是泛函的。

We have found an idea to formulate elliptic boundary value problems for systems of differential equations in terms of D-Modules and to construct their characteristic cycles (or microlocal Euler classes). Let us consider a system of differential equations M on a manifold M with boundary N. Let M_<tan> denote its pull back to the boundary. Given a system of differential equations N on the boundary and a D_N linear morphism α: N → M_<tan>. By definition, this boundary value problem is said to be elliptic if a induces an isomorphism from ε_N [○!×] N to a coherent quotient module M^+(tan)of ε_N [○!×] M^<tan> defined microlocally from the boundary. (ε_N is the sheaf of rings of microdifferential operators on the boundary.) It is still difficult to construct a characteristic cycle in this naive setting, and we want to translate this setting of BVP as a module (or an object of a derived category of modules) over some ring. If we introduce the ring BD as BD = D_M [○!+] D(N,M) [○!+]D_N, we can get an object B (M, N) of the derived category D^b(D_M [○!×] B), with B the ring of upper half triangle matrices of degree 2. One can possibly define a characteristic cycle (or a microlocal Euler class) associated to the pair of a D_M [○!×]B-module and a B-module (Z_M, Z_N) by the diagonal argument under the condition of ellipticity of α. We expect that one can prove (by chasing diagrams) an index theorem for boundary value problems in terms of characteristic classes defined here, since their construction is almost totally functorial.