## Non-characteristic maps (ala D-modules)

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

I am trying to understand a ``well known' fact (see Kashiwara's`Introduction 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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9122924208641052, "perplexity_flag": "head"}