# the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n

we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the Cech-cohomology of the sheaf.Then i think that plurisubharmonic functions is in some sense a weaker function than holomorphic functions.So we may get some beautiful theorems as the case of holomorphic case, for example if we can proof that for a domain in C^n,the first Cech-cohomology of the sheaf of germs of plurisubharmonic functions vanishes ,we then can choose any good plurisubharmonic functions as we want. What i want to ask is that have you ever considered such a question ,and i don't know whether this is a good question ? I want to hear some suggestions.

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What is the "sheaf of germs of plurisubharmonic functions"? –  Petya Mar 7 '10 at 2:11

## 1 Answer

As Petya has pointed out, plurisubharmonic functions on an open set do not form a group, so when one sheafifies, one gets a sheaf of sets, not groups; it has H^0, but no higher cohomology.

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yes,you are right,i have not considering about the sheaf of sets but i notice that there are some sheaf theory of semigroups ,if in this sense , can we consider the cohomology ? –  HKSHLZW Mar 7 '10 at 4:31
wangzw -- the first Cech cohomology (set) can be defined for sheaves of not necessarily abelian groups, but you really need inverses. –  algori Mar 7 '10 at 4:59