Abstract:

The exterior differential d on complex-valued differential forms of complex manifolds decomposes into the Cauchy-Riemann operator and its complex conjugate. Meanwhile on almost complex manifolds, the exterior d in general has two extra components, thus decomposes into four operators. In this talk, I will introduce these operators and discuss the structure of the (graded) associative algebra generated by these four components of d, subject to relations deduced from d squaring to zero. Then I will compare this algebra to the corresponding one in the complex (i.e. integrable) case, we shall see they are very different strictly speaking but similar in a weak sense (quasi-isomorphic). This is based on joint work with Shamuel Aueyung and Jin-Cheng Guu.

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