Welcome to MathOverflow

Question

4

2

Let p:X \to Y be a map of smooth algebraic varieties. Let $C \subset T^* X$ be a (locally closed) submanifold. Denote by $p_*(C) \subset T^* Y$ the following set:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ {$(y,v) \in T^*(Y)|\exists x \in p^{-1}(y) \text{ with } (x,(d_x(p))^*(v)) \in C$ } .

This opreration can describe (to some extent) what happens to the singular support of a D-module when one takes its direct image.

My question is: when can one claim that $p_*$ of a (conic) Lagrangian manifold is Lagrangian? I heard it is not true in general. Is it true when p is proper? What are the counter examples?

Thank you very much, Rami

edited Jun 29 2011 at 18:06

asked Jun 26 2011 at 18:18

Rami
701●4●8

Rami · Accepted Answer · 2012-04-12 22:34:43Z UTC

I've got a negative answer from Thomas Bitoun:

http://www.springerlink.com/content/k8306r7717672442/ ("Sur les images directes de D-modules")

It is at the very end of the paper, (6.3) page 23 of the electronic version. It is in French and Malgrange attributes the example to Kashiwara.

Direct image of Lagrangian subspaces of the co-tangent bundle:

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.