Local study of ordinary double points

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Assume that <math>0</math> and <math>\infty</math> are regular values of <math>f</math>. Let <math>x_1, \ldots, x_k \in \mathbb{P}^1</math> be all critical points. Connect <math>0</math> with <math>x_1, \ldots ,x_k</math> by some real paths, and define <math>L_i</math> together with <math>\Delta_i</math> for <math>1 \leq i \leq k</math> to be the Lefschetz pencil and the Lefschetz thimble at <math>0</math>, respectively, with respect to the ordinary double point <math>p_i</math>. Then <math>\widetilde{X} \backslash X_{\infty}</math> is homotopic to <math>\bigcup X_0 \cup_{L_i} \Delta_i</math>.
 
Assume that <math>0</math> and <math>\infty</math> are regular values of <math>f</math>. Let <math>x_1, \ldots, x_k \in \mathbb{P}^1</math> be all critical points. Connect <math>0</math> with <math>x_1, \ldots ,x_k</math> by some real paths, and define <math>L_i</math> together with <math>\Delta_i</math> for <math>1 \leq i \leq k</math> to be the Lefschetz pencil and the Lefschetz thimble at <math>0</math>, respectively, with respect to the ordinary double point <math>p_i</math>. Then <math>\widetilde{X} \backslash X_{\infty}</math> is homotopic to <math>\bigcup X_0 \cup_{L_i} \Delta_i</math>.
 
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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 1.2

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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