Friday, October 14, 2016

"Elementary Geometry"

This entry defines what my professors literally refer to as "elementary geometry". I am beginning to suspect that anything that is not cutting-edge mathematics is deemed elementary. It is either that or these concepts become elementary after constant perusal.

Let $D\subseteq\mathbb C^n$ be open. Then $f:D\to\mathbb C$ is holomorphic if it has a local power series representation at each point of $D$.

Holomorphic functions are uniquely determined by their behavior on open sets: if $f$ and $g$ are holomorphic on a domain $D$ (where a domain is a set that is open and connected), then $f=g$ on $D$. This is because the largest open subset of $D$ on which $f=g$ is also closed (relative to $D$) since the partial derivatives which determine the power series expansion are continuous. Therefore, since $D$ is connected, it must be the case that this set is $D$.

Say that $f$ is holomorphic at $z\in\mathbb C^n$ if it is holomorphic on some neighborhood of $z$. Let $A_z$ be the collection of functions that are holomorphic at $z$. Then $A_z$ is an algebra over $\mathbb C$ in which addition and multiplication are defined pointwise such that if $f:U\to\mathbb C$ and $g:V\to\mathbb C$, then $f+g:U\cap V\to\mathbb C$ and $fg:U\cap V\to\mathbb C$.

Let $I_z$ be the ideal in $A_z$ consisting of the functions of $A_z$ that vanish on some neighborhood of $z$.

The algebra of germs of holomorphic functions at $z$ is defined to be the quotient algebra $O_z:=A_z/I_z$. Thus a germ of a holomorphic function is an element $f+I_z$ of $O_z$, where $f$ is holomorphic at $z$. Denote this germ by $[f]_z$. We identify functions which belong to the same germ due to the uniqueness property mentioned above.

Define the stalk space (espace étalé) of germs of holomorphic functions to be $S=\{(z,[f]_z):f$ is holomorphic at $z\in\mathbb C^n\}$ together with $\rho:S\to\mathbb C^n$ defined by $(z,[f]_z)\mapsto z$. Call $\rho^{-1}(z)$ the stalk at $z$ (this is a copy of $O_z$). Define the stalk space to be the disjoint union of these stalks.

Let $\{U_\alpha\}$ be an open cover of $\mathbb C^n$ and define $V(f_\alpha,U_\alpha):=\{(z,[f_\alpha]_z):z\in U_\alpha\}$, where $f_\alpha$ is holomorphic on $U_\alpha$. Then $\{V(f_\alpha,U_\alpha)\}$ is a basis for a topology on $S$. Moreover, relative to this topology, $\rho$ is a local homeomorphism.

This topological space together with $\rho$ is called the sheaf of germs of holomorphic functions over the base space $\mathbb C^n$.

It turns out that $S$ is Hausdorff. In fact, $S$ is an analytic manifold that contains a few surprising properties.