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Bigraded complex forms and the Dolbeault operators

Definition

Let $U \subseteq C^{n}$ be open. Write $Ω^{k} (U; C)$ for smooth complex-valued $k$ -forms. For increasing multi-indices $I = (i_{1} < \dots < i_{p})$ and $J = (j_{1} < \dots < j_{q})$ , write $d z^{I} = d z_{i_{1}} \land \dots \land d z_{i_{p}}$ and $d {\overset{ˉ}{z}}^{J} = d {\overset{ˉ}{z}}_{j_{1}} \land \dots \land d {\overset{ˉ}{z}}_{j_{q}}$ . The invertible change of cotangent basis $d z_{j} = d x_{j} + i d y_{j}$ , $d {\overset{ˉ}{z}}_{j} = d x_{j} - i d y_{j}$ gives the direct sum decomposition

$Ω^{k} (U; C) = ⨁_{p + q = k} Ω^{p, q} (U), Ω^{p, q} (U) = {\sum_{I, J} a_{I, J} (z) d z^{I} \land d {\overset{ˉ}{z}}^{J} : a_{I, J} \in C^{\infty} (U; C)},$

where $0 \leq p, q \leq n$ and every sum is finite. On a $(p, q)$ form $η = \sum_{I, J} a_{I, J} d z^{I} \land d {\overset{ˉ}{z}}^{J}$ , define

$\partial η = \sum_{I, J, j} (\partial_{z_{j}} a_{I, J}) d z_{j} \land d z^{I} \land d {\overset{ˉ}{z}}^{J}, \overset{ˉ}{\partial} η = \sum_{I, J, j} (\partial_{{\overset{ˉ}{z}}_{j}} a_{I, J}) d {\overset{ˉ}{z}}_{j} \land d z^{I} \land d {\overset{ˉ}{z}}^{J} .$

Repeated differentials vanish by alternation; components outside the range $0 \leq p, q \leq n$ are zero. These operators are the components of $d$ of bidegrees $(p + 1, q)$ and $(p, q + 1)$ .

Facts & Assumptions

Given: An open $U \subseteq C^{n}$ and a smooth complex-valued form on $U$ .

[F1]

A smooth differential $k$ -form is a smooth section of the exterior power of the cotangent bundle (A smooth differential $k$ -form).

[F2]

In a chart, every smooth form has a unique expansion in the increasing wedge basis (Local coordinate expression for a differential form).

[F3]

In local coordinates, $d (\sum_{I} a_{I} d x^{I}) = \sum_{I} d a_{I} \land d x^{I}$ (The local coordinate formula for the exterior derivative).

[F4]

The Wirtinger derivatives are $\partial_{z_{j}} = \frac{1}{2} (\partial_{x_{j}} - i \partial_{y_{j}})$ and $\partial_{{\overset{ˉ}{z}}_{j}} = \frac{1}{2} (\partial_{x_{j}} + i \partial_{y_{j}})$ (Wirtinger operators in $C^{m}$ ).

[F5]

The complex derivative of a composite of holomorphic maps is the composite of their complex derivatives (The composite of holomorphic maps is holomorphic and its complex Jacobian is the product).

[F6]

Exterior differentiation commutes with pullback: $d (Φ^{*} ω) = Φ^{*} (d ω)$ (The exterior derivative commutes with pullback).

Proof

technique · direct

1.1F1F2givenalgebra

At each point, the displayed change from $(d x_{j}, d y_{j})$ to $(d z_{j}, d {\overset{ˉ}{z}}_{j})$ is an invertible complex-linear change of cotangent basis. Its increasing wedges therefore form a basis of the complexified alternating cotensors. By [F1] and [F2], every smooth complex-valued form has a unique expansion in this basis, and its coefficient functions are smooth. Grouping the terms by the numbers $p$ and $q$ of holomorphic and antiholomorphic factors gives the stated direct sum.

2.1F3F4step 1.1algebra

For a coefficient function $a$ , the real-coordinate formula for $d a$ and [F4] give $d a = \sum_{j} (\partial_{z_{j}} a) d z_{j} + (\partial_{{\overset{ˉ}{z}}_{j}} a) d {\overset{ˉ}{z}}_{j}$ . Since $d (d z_{j}) = d (d {\overset{ˉ}{z}}_{j}) = 0$ , [F3] applied termwise to $a_{I, J} d z^{I} \land d {\overset{ˉ}{z}}^{J}$ splits $d η$ into exactly the two displayed sums. Their bidegrees differ, so projection onto those summands recovers the coefficient formulas and proves $d = \partial + \overset{ˉ}{\partial}$ .

3.1F5F6step 1.1step 2.1algebra∎

If $Φ$ is a holomorphic coordinate change, [F5] makes its differential complex-linear; hence $Φ^{*} d z_{j}$ is a linear combination of holomorphic differentials and $Φ^{*} d {\overset{ˉ}{z}}_{j}$ is the conjugate linear combination of antiholomorphic differentials. Thus pullback preserves each bidegree. By [F6], pullback also commutes with $d$ ; uniqueness of the bidegree decomposition from step 1.1 implies it commutes separately with its two projections $\partial$ and $\overset{ˉ}{\partial}$ . The definitions are therefore independent of holomorphic coordinates.

Depends on

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Sources

Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapters 4–5 (standard reference, not scraped)
Jabbari, Notes for Analysis and Geometry of Several Complex Variables, §3.2 (standard reference, not scraped)
Guillemin and Campbell, MIT 18.117 Lecture Notes, Lectures 1–4 (standard reference, not scraped)