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Kodaira–Spencer forms form a differential graded Lie algebra

Statement

Let $X$ be a complex manifold of complex dimension $n$ with complex structure $J$ . Write $T_{X}^{C} : = T X \otimes_{R} C$ , with the complex-linear extension $J_{C}$ of $J$ , and define $T_{X}^{1, 0} : = \ker (J_{C} - i I), T_{X}^{0, 1} : = \ker (J_{C} + i I) .$ The holomorphic coordinate transitions give $T_{X}^{1, 0}$ its holomorphic tangent-bundle structure. For $0 \leq p \leq n$ , define the smooth vector-valued $(0, p)$ -forms by $A^{0, p} (X, T_{X}^{1, 0}) : = Γ^{\infty} ⁣ (X, Λ^{p} (T_{X}^{0, 1})^{*} \otimes_{C} T_{X}^{1, 0}), L^{p} : = A^{0, p} (X, T_{X}^{1, 0}),$ and set $L^{p} = 0$ outside this range. On scalar complex-valued forms, let $\overset{ˉ}{\partial}$ be the $(0, 1)$ -type component of the exterior differential. In a holomorphic frame $\partial_{1}, \dots, \partial_{n}$ of $T_{X}^{1, 0}$ , define $\overset{ˉ}{\partial} ⁣ (\sum_{i} α^{i} \otimes \partial_{i}) : = \sum_{i} (\overset{ˉ}{\partial} α^{i}) \otimes \partial_{i} .$ This is independent of the holomorphic frame because frame-transition coefficients are holomorphic. In holomorphic coordinates $(z^{1}, \dots, z^{n})$ , write $\partial_{a} = \partial / \partial z^{a}$ and define the bracket by $[α, β]^{i} = α^{a} \land \partial_{a} β^{i} - (- 1)^{p q} β^{a} \land \partial_{a} α^{i}, α \in L^{p}, β \in L^{q} .$ Then this bracket is coordinate-independent, has degree $p + q$ , is graded antisymmetric and satisfies graded Jacobi; with differential $\overset{ˉ}{\partial}$ , it satisfies $\overset{ˉ}{\partial} [α, β] = [\overset{ˉ}{\partial} α, β] + (- 1)^{p} [α, \overset{ˉ}{\partial} β], {\overset{ˉ}{\partial}}^{2} = 0.$ Thus $(L^{∙}, \overset{ˉ}{\partial}, [-, -])$ is a cohomological differential graded Lie algebra.

Statement references

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Voisin, §1.1

Facts & Assumptions

Given: A complex manifold $X$ with holomorphic tangent bundle $T_{X}^{1, 0}$ .

[F1]

Holomorphic coordinate changes have holomorphic transition maps and induce the holomorphic tangent-bundle transition maps (Complex manifolds, holomorphic vector bundles and Hermitian metrics).

Proof

technique · direct

1.1F1givenalgebra

For $α \in L^{p}$ , contraction with the $(1, 0)$ part of exterior differentiation defines an intrinsic degree- $p$ operator $D_{α} : A^{0, ∙} (X) \to A^{0, ∙ + p} (X)$ by $D_{α} ω = α^{i} \land \partial_{i} ω$ . Intrinsicity follows because both the tensor contraction and the $(1, 0)$ exterior derivative are coordinate-independent under the holomorphic changes in [F1]. The product rule for $\partial_{i}$ and moving the degree- $p$ form $α^{i}$ past a degree- $r$ form gives $D_{α} (ω \land η) = D_{α} ω \land η + (- 1)^{p r} ω \land D_{α} η$ ; hence $D_{α}$ is a graded derivation. Computing the graded commutator on a function cancels the symmetric second-derivative terms $\partial_{a} \partial_{i} f = \partial_{i} \partial_{a} f$ and leaves coefficient $α^{a} \land \partial_{a} β^{i} - (- 1)^{p q} β^{a} \land \partial_{a} α^{i}$ . Therefore $[D_{α}, D_{β}] = D_{[α, β]}$ with the stated formula; applying the equality to local coordinate functions shows the coefficient tensor is unique.

2.1F1step 1.1givenalgebra∎

In any associative graded endomorphism algebra the graded commutator is graded antisymmetric and satisfies graded Jacobi: expanding each nested commutator yields each of the six ordered triple products twice with opposite signs. Commutators of degree- $p$ and degree- $q$ derivations are derivations of degree $p + q$ , as follows by expanding both sides of the graded Leibniz rule and cancelling the two mixed terms. These facts transfer through the injective map $α \mapsto D_{α}$ to the bracket on $L$ . Finally, in holomorphic coordinates $\overset{ˉ}{\partial} \partial_{i} = \partial_{i} \overset{ˉ}{\partial}$ on coefficients, and the graded Leibniz rule gives $[\overset{ˉ}{\partial}, D_{α}] = D_{\overset{ˉ}{\partial} α}$ . The graded Jacobi identity for the commutator with $\overset{ˉ}{\partial}$ now yields $\overset{ˉ}{\partial} [α, β] = [\overset{ˉ}{\partial} α, β] + (- 1)^{p} [α, \overset{ˉ}{\partial} β]$ . Since ${\overset{ˉ}{\partial}}^{2} = 0$ on forms, all differential graded Lie algebra identities hold.

Remarks

The formula fixes the sign convention used in the Maurer–Cartan equation. It establishes the DGLA identities, but does not identify this DGLA's Maurer–Cartan groupoid with the analytic deformation groupoid; that comparison belongs to the Kuranishi argument.

Depends on

Complex manifolds, holomorphic vector bundles and Hermitian metrics