Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 15 / 4 / Solution

The Hermitian metric induces pointwise Hermitian inner products on the exterior powers of the complex cotangent bundle, with different bidegrees orthogonal, and a positive volume form $d{V}_g$. Thus the global $L^2$ inner product on smooth differential forms of type $(p,q)$ isLet ${\bar{\partial }}^{\ast }$ be the formal adjoint of the Dolbeault operator, defined by integration by parts, and let ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. This Dolbeault Laplacian preserves bidegree, is elliptic and nonnegative, and hasOn a compact Hermitian manifold without boundary, the Dolbeault Hodge decomposition on a compact Hermitian manifold says that the smooth harmonic space ${\mathcal{H}}_{\bar{\partial }}^{p,q}=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional andMore precisely, there are the orthogonal harmonic projection $H$ and a Dolbeault Green operator $G$ taking smooth forms to smooth forms, with $G=0$ on harmonics and ${\Delta }_{\bar{\partial }}G=G{\Delta }_{\bar{\partial }}=I-H$. Elliptic regularity ensures these are statements about smooth forms, not only $L^2$ completions. The Green operator commutes with $\bar{\partial }$ and ${\bar{\partial }}^{\ast }$, since the Laplacian does and its inverse is unique on the orthogonal complement of the kernel.

Expanding $\alpha =H\alpha +{\Delta }_{\bar{\partial }}G\alpha$ givesThe three summands are orthogonal: harmonic forms are killed by both adjoints, and $⟨\bar{\partial }u,{\bar{\partial }}^{\ast }v⟩=⟨{\bar{\partial }}^{2}u,v⟩=0$. If $\alpha$ is $\bar{\partial }$-closed, its coexact component is orthogonal to every closed form; taking its inner product with $\alpha$ shows that component has norm zero. Conversely the first two components are closed. Thus $\boxed{\ker \bar{\partial }={\mathcal{H}}_{\bar{\partial }}^{p,q}\oplus \bar{\partial }{A}^{p,q-1}}$, with out-of-range form spaces taken to be zero.