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

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

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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.

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