Non-characteristic maps (ala D-modules) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:40:47Z http://mathoverflow.net/feeds/question/97777 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97777/non-characteristic-maps-ala-d-modules Non-characteristic maps (ala D-modules) Reladenine Vakalwe 2012-05-23T17:25:29Z 2012-05-23T17:25:29Z <p>I am trying to understand a <code>well known' fact (see Kashiwara's</code>Introduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is the setup: </p> <p>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.</p> <p>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 <em>non-characteristic</em> for $\Lambda$ if</p> <p>$f_{\pi}^{-1}(\Lambda) \cap f_d^{-1}(T^*_XX) \subseteq T^*_YY \times_Y X$</p> <p>The well known' fact: if $f$ is non-characteristic for $\Lambda$, then $f_d$ restricted to $f_{\pi}^{-1}(\Lambda)$ is finite.</p> <p>I would be grateful if someone could explain the truth of this to me.</p> <p>Some remarks:</p> <p>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.</p> <p>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.</p>