The Dolbeault operator $\bar\partial : \Omega^{p,q}(M) \to \Omega^{p,q+1}(M)$ on a complex manifold $M$, here applied to smooth forms of total degree $k$, producing one of degree $k+1$.
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The Dolbeault operator squares to zero: $\bar\partial^2 = 0$.
Additivity of $\bar\partial$: $\bar\partial(\alpha + \beta) = \bar\partial\alpha + \bar\partial\beta$.
$\mathbb{C}$-linearity of $\bar\partial$: $\bar\partial(c \cdot \alpha) = c \cdot \bar\partial\alpha$.
The formal adjoint $\bar\partial^*$ of the Dolbeault operator with respect to the $L^2$ inner product, lowering the form degree from $k+1$ to $k$.
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Additivity of $\bar\partial^*$: $\bar\partial^*(\alpha + \beta) = \bar\partial^*\alpha + \bar\partial^*\beta$.
$\mathbb{C}$-linearity of $\bar\partial^*$: $\bar\partial^*(c \cdot \alpha) = c \cdot \bar\partial^* \alpha$.
The holomorphic Dolbeault operator $\partial : \Omega^{p,q}(M) \to \Omega^{p+1,q}(M)$ on a complex manifold, increasing the total degree by one.
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The formal $L^2$-adjoint $\partial^*$ of the operator $\partial$, lowering the form degree from $k+1$ to $k$.
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Stokes' theorem for $\bar\partial$ on a compact complex manifold $M$ (without boundary): the integral $\int_M \bar\partial \omega = 0$ for any smooth form $\omega$ of top-1 degree.
The Leibniz rule for $\bar\partial$ applied pointwise to a wedge product $\alpha \wedge (*\bar\beta)$: $\bar\partial(\alpha \wedge *\bar\beta) = \bar\partial\alpha \wedge *\bar\beta + (-1)^k \alpha \wedge \bar\partial(*\bar\beta)$.
Sign formula relating the second term in the Leibniz expansion to the formal adjoint: $(-1)^k \alpha \wedge \bar\partial(*\bar\beta) = -\alpha \wedge *\overline{\bar\partial^* \beta}$, the algebraic identity underlying the construction of $\bar\partial^*$.
The pointwise scalar associated with a wedge product of smooth forms on a compact manifold is integrable with respect to any measure $\mu$.
Stokes' theorem for $\partial$ on a compact complex manifold $M$ (without boundary): $\int_M \partial \omega = 0$ for any smooth form $\omega$.
The Leibniz rule for $\partial$ applied pointwise to the wedge product $\alpha \wedge *\bar\beta$, splitting into $\partial\alpha \wedge *\bar\beta + (-1)^k \alpha \wedge \partial(*\bar\beta)$.
Sign formula relating the Leibniz remainder $(-1)^k \alpha \wedge \partial(*\bar\beta)$ to $-\alpha \wedge *\overline{\partial^* \beta}$, the algebraic identity behind $\partial^*$.
Pointwise identity combining the Leibniz rule and the sign formula: $\bar\partial(\alpha \wedge *\bar\beta) = \bar\partial\alpha \wedge *\bar\beta - \alpha \wedge *\overline{\bar\partial^* \beta}$.
The $L^2$ adjointness defect of $\bar\partial$ and $\bar\partial^*$ equals an exact boundary integral: $\langle \bar\partial\alpha, \beta \rangle_{L^2} - \langle \alpha, \bar\partial^* \beta \rangle_{L^2} = \int_M \bar\partial(\alpha \wedge *\bar\beta)$.
Pointwise identity combining the Leibniz rule and the sign formula for $\partial$: $\partial(\alpha \wedge *\bar\beta) = \partial\alpha \wedge *\bar\beta - \alpha \wedge *\overline{\partial^* \beta}$.
The $L^2$ adjointness defect of $\partial$ and $\partial^*$ equals an exact boundary integral: $\langle \partial\alpha, \beta \rangle_{L^2} - \langle \alpha, \partial^* \beta \rangle_{L^2} = \int_M \partial(\alpha \wedge *\bar\beta)$.
$L^2$ adjointness of the Dolbeault operators: on a compact complex manifold $M$, $\bar\partial^*$ is the $L^2$-adjoint of $\bar\partial$ and $\partial^*$ is the $L^2$-adjoint of $\partial$, i.e. $\langle \bar\partial\alpha, \beta\rangle = \langle\alpha, \bar\partial^*\beta\rangle$ and $\langle \partial\alpha, \beta\rangle = \langle\alpha, \partial^*\beta\rangle$.
The Dolbeault Laplacian (or $\bar\partial$-Laplacian) $\Delta_{\bar\partial} = \bar\partial \bar\partial^* + \bar\partial^* \bar\partial$ acting on smooth $(p,q)$-forms.