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.

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$is clean with respect (i.e. non tangent) to the axis. 3 give it appropriate tags. Spelling error in body. Let$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.