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## 数学代写|黎曼几何代写Riemannian geometry代考|Complex Structures and the Dolbeault Double Complex

We start by reviewing the case of a classical complex manifold, where we can work locally in open subsets of $\mathbb{C}^m$ and use holomorphic maps to change between local coordinates. The most well-known nontrivial example is the Riemann sphere $\mathbb{C P}^1$ or $\mathbb{C}_{\infty}$. This has two open charts, $U=\mathbb{C}$ and $V=\mathbb{C}$, and the change of coordinate map is such that $z \in U \backslash{0}$ corresponds to $z^{-1} \in V \backslash{0}$. We shall write the local complex coordinates as $z^1, \ldots, z^m$ for a complex manifold $M$ of dimension $m$. Then $M$ also has the structure of a real $2 m$-dimensional manifold, as we have $z^j=x^j+\mathrm{i} y^j$ for real coordinates $x^1, \ldots, x^m, y^1, \ldots, y^m$.

Locally the complex-valued 1 -forms on $M$ have basis $\mathrm{d} z^j=\mathrm{d} x^j+\mathrm{id} y^j$ and its conjugate $\mathrm{d} \bar{z}^j=\mathrm{d} x^j-\operatorname{id} y^j$. Dually,
$$\frac{\partial}{\partial z^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}-\mathrm{i} \frac{\partial}{\partial y^j}\right), \quad \frac{\partial}{\partial \bar{z}^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}+\mathrm{i} \frac{\partial}{\partial y^j}\right)$$
for vector fields. It is common to abbreviate $\partial_j=\frac{\partial}{\partial z^j}$ and $\bar{\partial}_j=\frac{\partial}{\partial \bar{z}}$. The CauchyRiemann equations are the condition for a complex-valued function of a complex variable to be analytic. They are usually written in terms of splitting the function into real parts $u+\mathrm{i} v$ and the single complex variable $z=x+\mathrm{i} y$ as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$. These can be written using the $\bar{\partial}$ operators in several complex variables for a complex-valued function $f$ on $M$ as $\bar{\partial}_j f=0$ for all $j$. By writing the $n$ forms in terms of $\mathrm{d} z^j$ and $\mathrm{d} \bar{z}^j$, we split $\Omega^n(M)$ into the direct sum of $\Omega^{p, q}(M)$ for $p+q=n$. Here $p$ is the number of $\mathrm{d} z^j$ and $q$ is the number of $\mathrm{d} \bar{z}^j$, so $\mathrm{d} z^1 \wedge \mathrm{d} z^2 \wedge \mathrm{d} z^3$ is in $\Omega^{3,0}(M)$ and $\mathrm{d}^{-1} \wedge \mathrm{d} z^2 \wedge \mathrm{d}^{-3}$ is in $\Omega^{1,2}(M)$. Now define derivatives $\partial$ : $\Omega^{p, q}(M) \rightarrow \Omega^{p+1, q}(M)$ and $\bar{\partial}: \Omega^{p, q}(M) \rightarrow \Omega^{p, q+1}(M)$ by
$$\partial \xi=\mathrm{d} z^j \wedge \partial_j \xi, \quad \bar{\partial} \xi=\mathrm{d} \bar{z}^j \wedge \bar{\partial}_j \xi$$
summed over $j$. It is fairly easy to check that $\partial+\bar{\partial}=\mathrm{d}$ and that $\partial^2=\bar{\partial}^2=0$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Holomorphic Modules and Dolbeault Cohomology

Throughout this section, $(\Omega, \mathrm{d}, *, J)$ is an integrable almost complex structure on $A$. The noncommutative equivalent of the classical Cauchy-Riemann condition for $a \in A$ to be holomorphic is that $\bar{\partial} a=0$, and we call the collection of such holomorphic elements $A_{\mathrm{hol}}$. As $\bar{\partial}$ is a derivation, $A_{\mathrm{hol}}$ is a subalgebra of $A$. Similarly define $\Omega_{\text {hol }}^p$, the holomorphic $p$-forms, as the elements of $\xi \in \Omega^{p, 0}$ for which $\bar{\partial} \xi=0$. The holomorphic forms form a sub-DGA of the de Rham complex, the holomorphic de Rham complex,
$$0 \longrightarrow A_{\mathrm{hol}} \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^1 \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^2 \stackrel{\partial}{\longrightarrow} \cdots$$
For a complex analytic manifold $M$ it is natural to consider vector bundles with fibre $\mathbb{C}^n$ where the transition functions are holomorphic. In real differential geometry, the ‘obvious’ way to differentiate sections of a vector bundle is to take a trivialising open set, and then simply apply partial derivative $\frac{\partial}{\partial x^i}$ to the components of the section. Of course, this does not work globally as the derivatives of the transition functions enter, forcing us to use covariant derivatives and Christoffel symbols. However, in complex differential geometry, the ‘obvious’ thing to do works, with one condition. If we take the $\bar{\partial}$ derivatives of the components of the section, we get $\bar{\partial}_E: E \rightarrow \Omega^{0,1} \otimes_A E$ defined locally by
$$\bar{\partial}_E(v)=\mathrm{d} \bar{z}^i \otimes \frac{\partial v^j}{\partial \bar{z}^i} e_j,$$
where $e_j$ is the local basis of the vector bundle, $E$ is the sections of the bundle and $v=v^j e_j \in E$. Furthermore, this formula is perfectly well behaved under holomorphic change of basis as the $\bar{\partial}$ derivatives of the transition functions are zero, so we get a globally defined derivative. Thus, every complex vector bundle with holomorphic transition functions (we will just say holomorphic vector bundle) has a well-defined operator $\bar{\partial}_E$ satisfying the left $\bar{\partial}$-Leibniz rule, for $v \in E$ and $a \in A$,
$$\partial_E(a \cdot v)=\bar{\partial} a \otimes v+a \cdot \partial_E(v) .$$

## 数学代写|黎曼几何代写黎曼几何代考|复结构和Dolbeault双复复数

$$\frac{\partial}{\partial z^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}-\mathrm{i} \frac{\partial}{\partial y^j}\right), \quad \frac{\partial}{\partial \bar{z}^j}=\frac{1}{2}\left(\frac{\partial}{\partial x^j}+\mathrm{i} \frac{\partial}{\partial y^j}\right)$$

$$\partial \xi=\mathrm{d} z^j \wedge \partial_j \xi, \quad \bar{\partial} \xi=\mathrm{d} \bar{z}^j \wedge \bar{\partial}_j \xi$$

## 数学代写|黎曼几何代写黎曼几何代考|全纯模和Dolbeault上同调

$$0 \longrightarrow A_{\mathrm{hol}} \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^1 \stackrel{\partial}{\longrightarrow} \Omega_{\mathrm{hol}}^2 \stackrel{\partial}{\longrightarrow} \cdots$$

$$\bar{\partial}_E(v)=\mathrm{d} \bar{z}^i \otimes \frac{\partial v^j}{\partial \bar{z}^i} e_j,$$

$$\partial_E(a \cdot v)=\bar{\partial} a \otimes v+a \cdot \partial_E(v) .$$

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