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I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is the setup:

All varieties are over the complex numbers. Given a smooth variety $X$, write $T^* X$ for its cotangent bundle, and $T^*_XX \subseteq T^*X$ for the zero section.

Let $f: X \to Y$ be a morphism of smooth varieties. Write $f_{\pi}: T^*Y \times_Y X \to T^*Y$ for the projection map, and let $f_d: T^*Y \times_Y X \to T^* X$ be the map dual to the derivative. Let $\Lambda \subseteq T^* Y$ be a closed $\mathbb{C}^* $ stable subvariety ($\mathbb{C}^*$ acting on $T^*Y$ in the evident way). Then $f$ is called non-characteristic for $\Lambda$ if

$f_{\pi}^{-1}(\Lambda) \cap f_d^{-1}(T^*_XX) \subseteq T^*_YY \times_Y X$

The `well known' fact: if $f$ is non-characteristic for $\Lambda$, then $f_d$ restricted to $f_{\pi}^{-1}(\Lambda)$ is finite.

I would be grateful if someone could explain the truth of this to me.

Some remarks:

a) The statement is actually an if and only if, but the converse is straightforward, since the fibres of $f_d$ are $\mathbb{C}^* $ stable.

b) I believe I understand how to show that $f_d$ restricted to $f_{\pi}^{-1}(\Lambda)$ is quasi-finite (using the the $\mathbb{C}^* $ stability and the upper semi-continuity of fibre dimension). But that's as far as I have got.

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