Action of the monodromy on the cycle made of the real points

Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients. Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t > 0$ small enough, is compact and has dimension $n-1$. It defines a $(n-1)$-cycle, say $\gamma$, in $X_t$.

If we follow a loop in $\Bbb C$ around zero, this induces a map on the homology group $H_{n-1}(X_t)$. Is it true that $\gamma$ is a fixed point of this map? (Or that the orbit under the action of this map is finite?)

• Not if $n$ is even. Look for "Picard-Lefschetz formula". – abx May 10 '17 at 14:28