Complex and Kahler Geometry

Lecture Notes on Complex Geometry and Kahler Geometry

Abstract. Our goals of this lecture notes are
  • Calabi-Yau Theorem
  • Existence of Hermitian-Einstein metric (Donaldson-Uhlenbeck-Yau Theorem or Hitchin-Kobayashi Correspondence)
  • The Existence of Kahler-Einstein Metric (related with stability)
 1.  Basic Concepts in Complex Geometry
   1.1.  Holomorphic Map
   1.2.  Complex Manifold and Examples
   1.3.  Almost Complex Structure and Almost Complex Manifolds
   1.4.  Newlander-Nirenberg Theorem
   1.5.  Complex and Holomorphic Vector Bundle
   1.6.  Hermitian Metric and Kahler Metric on Complex Manifolds
   1.7.  Curvature Tensor of Kahler Manifolds
   1.8.  Laplace Operator and Harmonic Form
   1.9.  Sheaf Cohomology Theory
     1.9.1.  De Rahm Theorem
     1.9.2.  Dolbeault Theorem
     1.9.3.  Hodge Theorem
   1.10.  Chern Class
   1.11.  Vanishing Theorem
 2.  Hermitian-Einstein Metric
   2.1.  Stability by Mumford
   2.2.  Hermitian-Einstein Connection
   2.3.  Donaldson Functional
   2.4.  Donaldson Heat Flow and Yang-Mills Flow
   2.5.  The Proof of Donaldson-Uhlenbeck-Yau Theorem
 3.  Calabi-Yau Theorem
   3.1.  Complex Monge-Ampere Equation
   3.2.  $C^0$-Estimate
   3.3.  $C^2$-Estimate
   3.4.  Kahler-Einstein Metric and Kahler-Ricci Flow
1. Basic Concepts in Complex Geometry 1.1. Holomorphic Map
Definition 1. A complex-valued function $f(z)$ defined on a connected open domain $W\subseteq\C^n$ is called holomorphic if for each $a=(a_1,\ldots, a_n)\in W$, $f(z)$ can be represented as a power series \[ \sum_{k_1\geq0,\cdots,k_n\geq0}^\infty C_{k_1\cdots k_n}(z_1-a_1)^{k_1}\cdots (z_n-a_n)^k_n \] in some neighbourhood of $a$.
We have the following equivalent definition
Definition 2. Let $f(z)$ be a (continuously) differentiable function on an open set $W\subseteq\C^n$. Then $f(z)$ is holomorphic iff $\frac{\pt{f}}{\pt\bar z_\nu}=0$, $1\leq\nu\leq n$.
Proof . The proof is based on the Cauchy integral theorem and called Osgood Theorem”, cf. [1] for detail.
Recall that a complex-valued function of $n$ complex variables can be considered as a function of $2n$ real variables, since $\C^n\simeq \R^{2n}$, related by $z_\nu=x_\nu+i y_\nu$, $i=\sqrt{-1}$, $x_\nu,y_\nu\in\R$. We have the following \begin{alignat*}{2} \rd z_\nu&=\rd x_\nu+i\rd y_\nu, &\qquad \rd \bar z_\nu&=\rd x_\nu-i\rd y_\nu \ppt{z_\nu}&=\frac{1}{2}\left(\ppt{x_\nu}-i\ppt{y_\nu}\right), &\qquad \ppt{\bar z_\nu}&=\frac{1}{2}\left(\ppt{x_\nu}+i\ppt{y_\nu}\right) \rd x_\nu&=\frac{1}{2}(\rd z_\nu+\rd\bar z_\nu), &\qquad \rd y_\nu&=\frac{1}{2i}(\rd z_\nu-\rd\bar z_\nu) \ppt{x_\nu}&=\ppt{z_\nu}+\ppt{\bar z_\nu}, &\qquad \ppt{y_\nu}&=i\left(\ppt{z_\nu}-\ppt{\bar z_\nu}\right). \end{alignat*} With the help of the above relation, the total differential of a complex-$n$ valued function $f$ is \begin{align*} \rd f &=\frac{\pt f}{\pt x_\nu}\rd x_\nu+\frac{\pt f}{\pt y_\nu}\rd y_\nu &=\left( \frac{\pt f}{\pt z_\nu}+\frac{\pt f}{\pt \bar z_\nu}\right)\frac{1}{2}(\rd z_\nu+\rd\bar z_\nu)+ i\left( \frac{\pt f}{\pt z_\nu}-\frac{\pt f}{\pt \bar z_\nu}\right) \frac{1}{2i}(\rd z_\nu-\rd\bar z_\nu) &=\frac{\pt f}{\pt z_\nu}\rd z_\nu+\frac{\pt f}{\pt \bar z_\nu}\rd \bar z_\nu &\eqdef \pt f+\bar\pt f. \end{align*} We shall call $\pt f$ and $\bar\pt f$ the holomorphic part and Anti-holomorphic part of $f$, respectively.
Remark 1. $\bar\pt f=0$ is the Cauchy-Riemann Equation.
1.2. Complex Manifold and Examples 1.3. Almost Complex Structure and Almost Complex Manifolds 1.4. Newlander-Nirenberg Theorem 1.5. Complex and Holomorphic Vector Bundle 1.6. Hermitian Metric and Kahler Metric on Complex Manifolds 1.7. Curvature Tensor of Kahler Manifolds 1.8. Laplace Operator and Harmonic Form 1.9. Sheaf Cohomology Theory 1.9.1. De Rahm Theorem1.9.2. Dolbeault Theorem1.9.3. Hodge Theorem1.10. Chern Class 1.11. Vanishing Theorem 2. Hermitian-Einstein Metric 2.1. Stability by Mumford 2.2. Hermitian-Einstein Connection Existence of H-E => semi-stable or Direct sum of stable(Kobayashi), we will use Donaldson’s heat flow method to proof this. Note that Uhlenbeck-Yau has show this with the method of continuous. 2.3. Donaldson Functional 2.4. Donaldson Heat Flow and Yang-Mills Flow 2.5. The Proof of Donaldson-Uhlenbeck-Yau Theorem 3. Calabi-Yau Theorem 3.1. Complex Monge-Ampere Equation 3.2. $C^0$-Estimate 3.3. $C^2$-Estimate 3.4. Kahler-Einstein Metric and Kahler-Ricci Flow the case of Chern class $<0$ is solved, $>0$ is still open. Note also that we can use Kahler-Ricci Flow to prove Calabi-Yau Theorem, refer Huai-Dong, Cao‘s work.

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