Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.