show/hide this revision's text 5 latex adjustment

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 $$ Ch(Lf^* \mathcal{M}) \subset f_d f_\pi^{-1} Ch(\mathcal{M} ). $$

We know this is an equality when $f$ is non-characteristic for $\mathcal{M}$ meaning that $$ (f_d)^{-1} ( T^T_Y^* YY) \cap (f{\pi})^{-1} f_{\pi})^{-1} Ch(\mathcal{M}) \subset Y\times_X T^*_X X $$

Question Is this an equality when $f$ is the blow-up of $X$ along a smooth sub-variety $A$?

I have checked this in a few simple cases but I'm having trouble proving it in full generality.

Edit: Here's another closely related question.

Question 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 $Ch(\mathcal{M})$?

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.
show/hide this revision's text 4 added 9 characters in body

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 $$ Ch(Lf^* \mathcal{M}) \subset f_d f_\pi^{-1} Ch(\mathcal{M} ). $$

We know this is an equality when $f$ is non-characteristic for $\mathcal{M}$ meaning that $$ f_d^{-1}(T^(f_d)^{-1} ( T^*Y Y) \cap (f\pi^{-1}Ch(\mathcal{M}) {\pi})^{-1} Ch(\mathcal{M}) \subset Y\times_X T^*_X X $$

Question Is this an equality when $f$ is the blow-up of $X$ along a smooth sub-variety $A$?

I have checked this in a few simple cases but I'm having trouble proving it in full generality.

Edit: Here's another closely related question.

Question If $Z = {f=0}$ 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 $Ch(\mathcal{M})$?

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.
show/hide this revision's text 3 give it appropriate tags. Spelling error in body.

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 $$ Ch(Lf^* \mathcal{M}) \subset f_d f_\pi^{-1} Ch(\mathcal{M} ). $$

We know this is an equality when $f$ is non-characteristic for $\mathcal{M}$ meaning that $$ f_d^{-1}(T^*Y Y) \cap f\pi^{-1}Ch(\mathcal{M}) \subset Y\times_X T^*_X X $$

Question Is this an equality when $f$ is the blow-up of $X$ along a smooth sub-variety $A$?

I have checked this in a few simple cases but I'm having trouble proving it in full generality.

Edit: Here's another closely related question.

Question If $Z = {f=0}$ 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 $Ch(\mathcal{M})$?

A typical example would be $X= \mathbb{A}^2$, $A = {0}$, $Ch(\mathcal{M})$ the lagrangian 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 axisesaxis.
show/hide this revision's text 2 added 731 characters in body; added 7 characters in body
show/hide this revision's text 1