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:


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


This operation 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?



Direct image of Lagrangian subspaces of the co-tangent bundle
I've got a negative answer from Thomas Bitoun:


*Malgrange, Bernard*, [**Sur les images directes de ${\mathcal D}$-modules**](https://doi.org/10.1007/BF01168827), Manuscr. Math. 50, 49–71 (1985). [Zbl 0572.32014](https://zbmath.org/0572.32014).


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.


