Asymptotic Betti bounds for hypersurfaces in a singular variety

We show that for any degree $d$ hypersurface $Y\subset X$ in a possibly singular projective variety $X\subset {\mathbf{P}}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n+C\cdot d^{n-1}$ for some explicit constant $C>0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n+C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves.