The d-c operator - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold, and write $d=\partial+\bar\partial$ for its Dolbeault decomposition. In this knowl the $d^c$ -operator is the real first-order differential operator

d^c=i(\bar\partial-\partial).

It raises total form degree by one and satisfies

dd^c=2i\,\partial\bar\partial,\qquad d^cd=-dd^c,\qquad (d^c)^2=0.

On a real-valued smooth function $f$ , one has $d^cf=-df\circ J$ , where $J$ is the complex structure on the real tangent bundle. The displayed normalization is part of the definition.

Structure and consequences

Complex conjugation interchanges $\partial$ and $\bar\partial$ , while also replacing $i$ by $-i$ ; hence $d^c$ preserves real forms. The identity $dd^c=2i\partial\bar\partial$ follows from $\partial^2=\bar\partial^2=0$ and $\partial\bar\partial=-\bar\partial\partial$ . Thus a real function determines a real $(1,1)$ -form $dd^cf$ .

In a holomorphic coordinate $z=x+iy$ ,

d^cf=\frac{\partial f}{\partial x}\,dy-\frac{\partial f}{\partial y}\,dx.

This local formula makes the relation $d^cf=-df\circ J$ explicit.

Conventions and scope

d^c_{\mathrm{Dem}}=\frac{1}{2\pi i}(\partial-\bar\partial) =\frac{1}{2\pi}d^c, \qquad dd^c_{\mathrm{Dem}}=\frac{i}{\pi}\partial\bar\partial

in Chapter III, §3. Other texts insert a factor $1/2$ or reverse the sign. Formulas involving $d^c$ , $dd^c$ , curvature, or Kähler potentials must therefore be interpreted with their stated convention.

References

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter III, §3, “Definition of Monge–Ampère Operators,” for an explicitly normalized $d^c$ -operator and the resulting $dd^c$ formula.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: §3.1 for $\partial\bar\partial$ -formulas in Kähler geometry.