Blow ups and Characteristic varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:27:26Z http://mathoverflow.net/feeds/question/54494 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54494/blow-ups-and-characteristic-varieties Blow ups and Characteristic varieties YBL 2011-02-06T04:42:48Z 2011-02-15T01:05:40Z <p>Let $f:Y\to X$ be a morphism of smooth varieties (complex analytic or algebraic as you wish). We have $$T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X$$ and, for any holonomic $\mathcal{D}_X$-module, we have the estimate $$Ch(Lf^* \mathcal{M}) \subset f_d f_\pi^{-1} Ch(\mathcal{M} ).$$</p> <p>We know this is an equality when $f$ is non-characteristic for $\mathcal{M}$ meaning that $$(f_d)^{-1} ( T_Y^* Y) \cap (f_{\pi})^{-1} Ch(\mathcal{M}) \subset Y\times_X T^*_X X$$ </p> <p><strong>Question</strong> Is this an equality when $f$ is the blow-up of $X$ along a smooth sub-variety $A$?</p> <p>I have checked this in a few simple cases but I'm having trouble proving it in full generality.</p> <p>Edit: Here's another closely related question.</p> <p><strong>Question</strong> If $Z = {g=0}$ is a smooth hypersurface of $X$. What is a necessary and sufficient condition for the proprer transform of $Z$ to be non-characteristic to $Ch(Lf^*\mathcal{M})$ in terms of $Z$, $A$ and $Ch(\mathcal{M})$? </p> <p>A typical example would be $X= \mathbb{A}^2$, $A = {0}$, $Ch(\mathcal{M})$ the Lagrangian variety corresponding to the stratification of $\mathbb{A}^2$ by the axises $x=0$, $y=0$ (i.e. the union of the zero section of $T^*X$, the conormal bundles to the axises and the cotangent space at 0). Then the proprer transform of an hypersurface $Z$ will be non-characteristic iff $Z$ is clean with respect (i.e. non tangent) to the axis.</p>