Study of unramified extensions, with emphasis on Jacobian problem

(1)证明了正特征Weyl代数与仿射空间上的矩阵丛有关，并且我们可以得到这个对象的‘几何量子化’的‘逆’过程。(2)利用Spec(Z)上的超滤，我们构造了一个特征为0的域Qu。证明了Qu上Weyl代数的代数自同态对应于仿射空间的辛同态。(3)证明了Dixmier型猜想可以演绎为Jacobian猜想。(4)Cartier算子的超滤子极限给出了一个新的积分理论。(5)给出了广义极化流形的第i-分段几何亏格和第i-Delta亏格的定义.(6)当L的完备线性系统|L|没有基点时，我们观察到第i-分段几何亏格和第i-Delta亏格具有与分段亏格和Delta亏格类似的性质.(7)给出了极化流形(X，L)的第i截面H-算术亏格_X^H_I(X，L)的定义.(8)期望G_2(X，L)和_X^H_2(X，L)(分别为)类似于几何亏格和算术亏格X(O)(分别)，我们提出了一些关于极化流形的问题，作为对已有的曲面理论结果的类比。

(1)We show that a Weyl algebra of positive characteristics is related to a matrix bundle on a affine space, and that we may obtain a process which is a 'inversion' of the 'geometric quantization' for this object.(2)Using ultra filter on Spec(Z), we construct a field Qu of characteristic 0. We showed that an algebra endomorphism of Weyl algebra over Qu corresponds to an symplectic morphism of affine space.(3)We show Dixmier conjecture may be deduced to Jacobian conjecture.(4)Ultra filter limit of Cartier operator gives a new integration theory. It gives an important step to the Jacobian problem.(5)We give a definition of the i-th sectional geometric genus and i-th delta genus of a generalized polarized manifold.(6)We observe that the i-th sectional geometric genus and i-th delta genus have analogous properties to those of sectional genus and delta genus when the complete linear system |L| of L has no base point. We studied further what happens when |L| has some base points.(7)We give a definition of the i-th sectional H-arithmetic genus _X^H_i(X,L) of a polarized manifold(X,L).(8)Expecting that g_2(X,L) and _X^H_2(X,L)(resp.) are analogous to geometric genus and arithmetic genus _X(O)(resp.), we propose some problems on polarized manifold as an analogy to the known results on the theory of surface.