# à la Shafarevich conjecture for the moduli space of Calabi-Yau manifolds

Suppose that $\pi: X \to C$ is a family of polarized Calabi-Yau manifolds over a Riemann surfaces $C$. Suppose that the family of fibers $\pi$ contains a point $τ_0 \in C$ such that around it the monodromy operator $T$ has large complex structure limit, i.e maximal index of unipotency, i.e. $(T^N − id)^{n+1} = 0$ and $(T^N − id)^n\neq 0$. Then the family is rigid.

Now assume that $\pi: X \to B$ is a family of polarized Calabi-Yau manifolds over a compact Kahler manifold $B$ of dimension bigger than one, then under which condition we can get rigidity?