Let $f : X^m \to Y^n$ be an algebraic fiber space (between projective manifolds) whose discriminant locus is denoted by $E$. Let $U$ be a polydisk in $\mathbb{C}^n$ (with coordinates $(y_1, ..., y_n)$) such that $U \backslash E \simeq (\Delta^{\ast})^{\ell} \times \Delta^{n-\ell}$, where $1 \leq \ell \leq n$. Let $x_0$ be a point in $U\backslash E$. Associated to $x_0$ is the monodromy operator $$T_k : H^{m-n}(f^{-1}(x_0), \mathbb{C}) \longrightarrow H^{m-n}(f^{-1}(x_0), \mathbb{C})$$ for a loop based at $x_0$ around the $k$th copy of $\Delta^{\ast}$, where $1 \leq k \leq \ell$.

The monodromy theorem states that $T_k$ is quasi-unipotent, i.e., there are positive integers $m_k$ and $d_k$ such that $$(T_k^{m_k} - I)^{d_k} =0.$$ Here, $m_k$ is the least common multiple of the multiplicities of the irreducible components over the generic point of $\{ y_k =0 \}$.

**Question:** **What does the quasi-unipotence of the monodromy transformation tell us qualitatively?**

orderedbasis of the cohomology so that monodromy on a basis element gives you back the basis element (multiplied by a root of unity from the multiplicity of the irr. component) plus only basis elements that comelaterin the list. Think of it as a generalization of Picard-Lefschetz theory (en.wikipedia.org/wiki/Picard%E2%80%93Lefschetz_theory for a very concrete calculation) $\endgroup$