Type decomposition of the complexified tangent bundle

Definition

Let $M$ carry an almost-complex structure $J$ . Extend $J$ complex-linearly to $T_{\mathbb C}M=TM\otimes_{\mathbb R}\mathbb C$ . Since $J^2=-1$ , its only eigenvalues are $i$ and $-i$ , and the complexified tangent bundle splits as

T_{\mathbb C}M=T^{1,0}M\oplus T^{0,1}M, \qquad T^{1,0}M=\ker(J-i),\quad T^{0,1}M=\ker(J+i).

Complex conjugation interchanges the two summands. This is the type decomposition of the complexified tangent bundle; it exists for every almost-complex structure and does not require integrability.

Projection operators

The summands are the images of the smooth bundle projections

P^{1,0}=\tfrac12(1-iJ),\qquad P^{0,1}=\tfrac12(1+iJ).

For a real tangent vector $X$ , its components are $X^{1,0}=\frac12(X-iJX)$ and $X^{0,1}=\frac12(X+iJX)$ . These formulas also show directly that the summands are complex vector bundles of complex rank $\frac12\dim_{\mathbb R}M$ .

Dual decomposition and forms

Dualizing gives

T_{\mathbb C}^*M=T^{*1,0}M\oplus T^{*0,1}M.

Exterior powers then decompose complex-valued differential forms into $(p,q)$ -types. This algebraic decomposition is the starting point for the operators $\partial$ and $\bar\partial$ on a complex manifold Wells, Chapter I, §2.

Integrability

The splitting itself is pointwise linear algebra. The additional statement that sections of $T^{0,1}M$ are closed under the complexified Lie bracket is equivalent to integrability. Only under that condition do the summands agree locally with the tangent directions generated by holomorphic and antiholomorphic coordinates.

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Springer DOI record. Relevant: Chapter I, §2.
D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Springer DOI record. Relevant: complexified tangent and cotangent bundles and forms of type $(p,q)$ .