Local study of ordinary double points

 
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where <math>z_i</math> are suitable coordinates. Let us consider the fiber around a critical point <math>X_t = \{ \sum z_i^2 = t\}</math> for some <math>t \in \mathbb{C}</math>.
 
where <math>z_i</math> are suitable coordinates. Let us consider the fiber around a critical point <math>X_t = \{ \sum z_i^2 = t\}</math> for some <math>t \in \mathbb{C}</math>.
  
{{BeginDefinition|title=|number=1.2|anchor=1_2}}
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{{BeginDefinition|title=}}
 
We define a "vanishing cycle" for <math>t = se^{i \phi} \in \mathbb{C}</math>, where <math>s \in \mathbb{R}</math>, to be the sphere
 
We define a "vanishing cycle" for <math>t = se^{i \phi} \in \mathbb{C}</math>, where <math>s \in \mathbb{R}</math>, to be the sphere
 
<math display="block">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.</math>
 
<math display="block">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.</math>

Latest revision as of 10:52, 14 December 2016

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

where are suitable coordinates. Let us consider the fiber around a critical point for some .

Definition

We define a "vanishing cycle" for , where , to be the sphere

We define a "Lefschetz thimble" to be the union of all vanishing spheres at some real path between and . Note that depends on the choice of the path.

 

One can check that a vanishing cycle is Lagrangian in .


Proposition

Assume that and are regular values of . Let be all critical points. Connect with by some real paths, and define together with for to be the Lefschetz pencil and the Lefschetz thimble at , respectively, with respect to the ordinary double point . Then is homotopic to .

 


Proof (Sketch of the proof)

Let be the path connecting and . Define a skeleton . Then, the standard "run a flow" argument implies that is homotopic to . Similarily, we retract every fiber over to .

 

In the case when , the proposition above implies the Lefschetz hyperplane theorem. In general, one can easily reprove Lefschetz hyperplane theorem by a careful inductive argument comparing and .

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