Hitchin Functional - Stable Forms

Stable Forms

Action functionals often determine geometric structure on and geometric structure are often characterized by the existence of particular differential forms on that obey some integrable conditions.

If an m-form can be written with local coordinates

and

,

then defines symplectic structure.

A p-form is stable if it lies in an open orbit of the local action where n=dim(M), namely if any small perturbation can be undone by a local action. So any 1-form that don't vanish everywhere is stable; 2-form (or p-form when p is even) stability is equivalent to nondegeneratacy.

What about p=3? For large n 3-form is difficult because the dimension of, grows more firstly than the dimension of, . But there are some very lucky exceptional case, namely, when dim, dim . Let be a stable real 3-form in dimension 6. Then the stabilizer of under has real dimension 36-20=16, in fact either or .

Focus on the case of and if has a stabilizer in then it can be written with local coordinates as follows:

where and are bases of . Then determines an almost complex structure on . Moreover, if there exist local coordinate such that then it determines fortunately an complex structure on .

Given the stable :

.

We can define another real 3-from

.

And then is a holomorphic 3-form in the almost complex structure determined by . Furthermore, it becomes to be the complex structure just if i.e. and . This is just the 3-form in formal definition of Hitchin functional. These idea induces the generalized complex structure.