Study of unramified extensions, with emphasis on Jacobian problem

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(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.