Suppose $f:X\to Y$ is a proper smooth morphism of $\mathbb C$-varieties, and $y\in Y$ is a point. I want to get an action of $\pi_1(y,Y)$ (topological fundamental group) on $H^i_{sing}(X_y,\mathbb C)$.
An idea is to get an action on the fiber directly. But it seems hard, because we don't know whether $X$ is a vector bundle over $Y$(in that case we can use paralle transport by defining a connection).
Is there any other ways to achieve it? Thanks ahead.