# Local study of ordinary double points

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

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

Definition

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

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

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

Proposition

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

Proof (Sketch of the proof)

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

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