Theorem:

unique decomposition where is harmonic. . This can also be written using the laplacian as .

. This can be inverted to get as long as there are no 0 mode oscillations.

- all 3 terms are 0. This can be shown by taking the inner product
of with itself.

Can define a projection operator . Then use projection operator to solve for , then show that you can invert to get .

- if closed there is then there is an equivalence .
- (isomorphism)

source

psfile jl@crush.caltech.edu index

kahler_Hodge_decomposition_theorem