Let $\phi: \mathcal{X}\rightarrow B$ be a family of complex manifolds i.e. $\phi$ is a proper submersive holomorphic morphism, i write $X_b$ for the fiber of $b\in B$.

Suppose $dim_{\mathbb{C}}X_b=2$ and $X_b$ is kahler. Suppose that $R^2\phi_*\mathbb{Z}$ is a local system, then i can consider the monodromy action

$\pi_1(B,b)\rightarrow Aut(H^2(X_b,\mathbb{Z}))$

I have read that the monodromy action actually is a subgroup of $O(H^2(X_0,\mathbb{Z}))$ i.e. it preserves the intersection pairing. Why is that?

notact on the cohomology of $X_0$ (that is, fiber over zero, which contains critical points of $f$). It acts on the cohomology of $X_c=f^{-1}(c)$, where $c\ne 0$. $\endgroup$ – Serge Lvovski Jun 26 '13 at 15:06