# Local study of ordinary double points

Analogously to real Morse theory, every holomorphic function around an ordinary double points is of the form

$f=\sum z_{i}^{2},$ where $z_{i}$ are suitable coordinates. Let us consider the fiber around a critical point $X_{t}=\{\sum z_{i}^{2}=t\}$ for some $t\in \mathbb {C}$ .

Definition

We define a "vanishing cycle" for $t=se^{i\phi }\in \mathbb {C}$ , where $s\in \mathbb {R}$ , to be the sphere

$L_{t}{\mathrel {\mathop {:} }}=\{(z_{1},\ldots ,z_{n})\;|\;z_{i}={\sqrt {s}}e^{i{\frac {\phi }{2}}},x_{i}\in \mathbb {R} ,\sum x_{i}^{2}=1\}\subseteq X_{t}.$ We define a "Lefschetz thimble"$\Delta _{i}$ to be the union of all vanishing spheres at some real path between $0$ and $t$ . Note that $\Delta _{i}$ depends on the choice of the path.

One can check that a vanishing cycle $L_{t}$ is Lagrangian in $X_{t}$ .

Proposition

Assume that $0$ and $\infty$ are regular values of $f$ . Let $x_{1},\ldots ,x_{k}\in \mathbb {P} ^{1}$ be all critical points. Connect $0$ with $x_{1},\ldots ,x_{k}$ by some real paths, and define $L_{i}$ together with $\Delta _{i}$ for $1\leq i\leq k$ to be the Lefschetz pencil and the Lefschetz thimble at $0$ , respectively, with respect to the ordinary double point $p_{i}$ . Then ${\widetilde {X}}\backslash X_{\infty }$ is homotopic to $\bigcup X_{0}\cup _{L_{i}}\Delta _{i}$ .

Proof (Sketch of the proof)

Let $p_{i}$ be the path connecting $0$ and $x_{i}$ . Define a skeleton $S{\mathrel {\mathop {:} }}=\bigcup p_{i}$ . Then, the standard "run a flow" argument implies that ${\widetilde {X}}\backslash X_{\infty }$ is homotopic to ${\widetilde {X}}|_{S}$ . Similarily, we retract every fiber $X_{t}$ over $t\in S$ to $\bigcup X_{0}\cup _{L_{i}}\Delta _{i}$ .

In the case when $X={\widetilde {X}}$ , the proposition above implies the Lefschetz hyperplane theorem. In general, one can easily reprove Lefschetz hyperplane theorem by a careful inductive argument comparing $X$ and ${\widetilde {X}}$ .