Isomorphic IC sheaves induced from different locally closed subvarieties

Question

Let us work with varieties over $\mathbb{C}$ and $D^{b}_{c}(X)$ the bounded constructible derived category of sheaves of $\mathbb{Q}$ vector spaces. Say $X$ and $Y$ are smooth locally closed subvarieties in some algebraic variety $W$ and we have that $\overline{X} = \overline{Y} = Z$. So we can think of $j_{X}: X \to Z$ and $j_{Y}:Y \to Z$ as two different smooth open dense subvarieties in $Z$.

Now take $F = IC_{Y}(L)$ for some local system $L$ on $Y$. Here $IC_{Y}(L)$ is the Intersection Complex perverse sheaf.

My question is: Is there a criteria for when we can find a local system $L’$ on $X$ such that $F \cong IC_X(L’)$?

One idea is that one can restrict $L$ to the intersection $X \cap Y$ with inclusions $j^{'}_{X}:X \cap Y \to X $ and $j^{'}_{Y}: X \cap Y \to Y$. $X \cap Y$ is also smooth open dense in $Z$. Then we should have $$IC_{Y}(L)=j_{Y, !*}\circ j^{'}_{Y, !*} L|_{X \cap Y} = j_{X, !*} \circ j^{'}_{X,!* } L|_{X \cap Y}$$ However, I am not sure if $j_{X \cap Y, !*}L|_{X \cap Y}$ is a local system on $X$.

Answer 1

It's not hard to see that the constructtion you give is the unique way to do it. It's not hard to check that the intermediate extension of an intermediate extension is an intermediate extension, and the restriction of an intermediate extension recovers the original perverse sheaf so there is a unique way to write a perverse sheaf as an intermediate extension from a given set.

Thus, the question is exactly what you say at the end - is $j_{X \cap Y !} L\mid_{X \cap Y}$ a local system on $Y$.

This is a purely local question - it suffices for this to be true in a small neighborhood of each point $y$ of $Y$.

If $j_{X \cap Y !} L\mid_{X \cap Y}$ is a local system then restricted to some neighborhood of $y$ it is constant. It follows that $L$, restricted to the intersection of $X$ with some neighborhood of $y$ in $Y$, is constant - i.e. it has no local monodromy around $y$.

Conversely, if $L$ restricted to the intersection of $X$ with some neighborhood of $y$ in $Y$ is constant, then $j_{X \cap Y !} L\mid_{X \cap Y}$ is constant on that neighborhood since the intermediate extension of a constants sheaf from an open subset of a smooth variety is constant.

So this local monodromy condition is necessary and sufficient.