Set-up: Let $X$ be a complex manifold. Let $A^k$ be the sheaf of sections of the differential $k$-forms on a differentiable manifold, and let $A_{\mathbb{C}}^k$ be the sheaf of sections of $\Omega^k_{X,\mathbb{C}}=\bigwedge^k(\Omega_{X,\mathbb{R}}\otimes\mathbb{C})$. Let’s call $d,d_{\mathbb{C}}$ to exterior differentiation on $A^{\bullet}(X),A^{\bullet}_{\mathbb{C}}(X)$, respectively. Let $A^{p,q}$ be the sheaf of forms of type $(p,q)$ (induced by the complex structure), and denote as usual $\partial,\overline{\partial}$ the operators on $A^{\bullet,q}$, and $A^{p,\bullet}$.
Now, in Huybrechts‘ book $``$Complex Geometry”, in order to prove the comparison of Laplacians for Kähler manifolds, the Lefschetz decomposition of $(A^{\bullet}(X),d)$ (and not of $(A_{\mathbb{C}}(X),d_{\mathbb{C}})$) is used, in fact we still haven’t defined a metric on $A^{\bullet}_{\mathbb{C}}(X)$ with respect to which the Lefschetz decomposition is an orthogonal decomposition (this is done some pages later in the compact case).
I have two questions:
- It seems to me that what’s being tried to prove is $\Delta_d=2\Delta_{\partial}=2\Delta_{\overline{\partial}}$, instead of $\Delta_{d_{\mathbb{C}}}=2\Delta_{\partial}=2\Delta_{\overline{\partial}}$. This to me seems like nonsense, as $d,\partial,\overline{\partial}$ aren’t operators on the same space. (I think that this is what’s being proved because the Lefschetz decomposition being used has (at that point) only been proved to apply for $(A^k(X),d)$.) Am I missing something? What?
- Suppose we fix this. At the end of the day, we can put a metric on $A^k_{\mathbb{C}}(X)$ for which the Lefschetz decomposition would now be valid. However, in order to do this, we need to assume compactness (I think). Is this right? Is there a way to avoid this? (I know the Kähler identities hold true in any Kähler manifold (and have seen different proofs for it), but I’m not sure if the proof using the Lefschetz decomposition can actually prove it).
Thank you!
Edit: For completeness, let me include the following direct quotes from the book:
Proposition 1.2.30 (page 36): Let $(V,\langle,\rangle,I)$ be a Euclidean vector space with a compatible almost complex structure, and let $L,\Lambda$ be the associated Lefschetz operators:
Then there exists a direct sum decomposition $$\bigwedge^k V^*=\bigoplus_{i\geq 0}L^i(P^{k-2i}),$$ where $P^{k-2i}$ are the principal forms (i.e the kernel of $\Lambda$). Moreover, this decomposition is orthogonal with respect to $\langle,\rangle$.
Corollary 3.1.2 (page 115): Let $(X,g)$ be a hermitian manifold. Then there exists a direct sum decomposition of vector bundles $$\Omega^k_{X,\mathbb{R}}=\bigoplus_{i\geq 0}L^i(P^{k-2i}(X)).$$
Proposition 3.1.12 (page 120): Let $X$ be a complex manifold endowed with a Kähler metric $g$. Then the following idenitites hold true:
- $L,\Lambda$ commute with $\partial^*,\overline{\partial}^*$,
- $[\Lambda,\overline{\partial}]=-i\partial^*, [\Lambda,\partial]=i\overline{\partial}^* $
- $\Delta=2\Delta_{\partial}=2\Delta_{\overline{\partial}}$.
In the proof of (2) (which is then used to prove (3) formally) of the previous theorem (page 121) this is written: “Using the Lefschetz decomposition it’s enough to prove the assertion for forms of the type $L^j\alpha$, with $\alpha$ a primitive $k$-form. Then $d\alpha\in A^{k+1}(X)$, can again be written, according to the Lefschetz decomposition (Corollary 3.1.2) as […]”.
Finally, let me also point out this:
Definition 3.2.1 (page 125): Let $(X,g)$ be a compact hermitian manifold. Then one defines an hermitian product on $A^*_{\mathbb{C}}(X)$ by $$(\alpha,\beta):=\int_X g_{\mathbb{C}}(\alpha,\beta)*1.$$
Proposition 3.2.2: Let $(X,g)$ be a hermitian manifold, then the Lefschetz decomposition holds and is orthogonal with respect to $(,)$: $$A^k_{\mathbb{C}}(X)=\bigoplus_{i\geq 0}L^i(P^{k-2i}_{\mathbb{C}}(X)).$$
