ON COHOMOLOGY OF INFINITESIMAL NEIGHBOURHOODS OF COMPLEX MANIFOLDS

In this paper,we introduce the concept of the(k,l)-th C~∞ infinitesimal neighbour-hoods of complex manifold M and difine defferential modules _(M,k-p,l-q)~(p,q) and _(M,k,l)~r forthe(k,l)-th C~∞ infinitesimal neighbourhoods.We prove some isomorphism theorems ofcohomology and hyper cohomology concerning _(M,k-p)~p and _(M,k-p)~,as followsH~p(M,_(M,k-r)~r)≈H_~p(M,_(M,k-r,l-*)~(r,*),H~p(M,_(M,k,l)~*)≈H_(DR)~p(M,C)and for hyper cohomologyH~p(M,_(M,k-*)~*≈H~p(M,C),H~p(M,_(M,k-*)~*)≈H~p(M,O_M).

On Cohomology of Infinitesimal Neighbourhoods of Complex Manifolds

In this paper, we introduce the concept of the (k, l)-th C^\infinity infinitesimal neighbourhoods of complex manifold M and difine defferential modules $mathscr{A}^{p,q}_{M,k-p,l-q}$ and $mathscr{A}^\tau_{M,k,l}$ for the (k, l)-th C^\infinity infinitesimal neighbourhoods. We prove some isomorphism theorems of cohomology and hyper cohomology concerning $\tilde \Omega]^p_{M,k-q}$and $\Omega^p_{M,k-p}$ as follows $H^p(M,\tilde \Omega]^\tau_{M,k-r})\approxH^p-\varepsilon(M,\tilde mathscr{A}]^\tau_{M,k-r,l-*}),$ $H^p(M,\tilde mathscr{A}]^*_{M,k,l})\approxH^p_D^R(M,C)$ and for hyper cohomology $H^p(M,\tilde \Omega]^*_{M,k-*})\approxH^p(M,C),$ $H^p(M,\Omega^*_{M,k-*}\approxH^p(M,C_M).$