Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth projective variety, central fiber $Y=f^{-1}(0)=\bigcup D\_i$, with reduced induced $Y\_{red}$ is normal crossing. 


Question: If the GCD of multiplicities of components $D\_i$ is 1, does this condition imply that the monodromy is unipotent? 


There is a detailed proof by steenbrink in his paper "limits of Hodge structures" proof of (2.20). However, in his book "Mixed Hodge structures", He also mentioned if the LCM is $n$, then T^n is unipotent. I am confused about the correct version of this theorem.



Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber
On what group are you studying the monodromy? Begin with $\mathcal{Y} = \Delta \times \mathbb{P}^n$, $n\geq 2$, with its projection $\text{pr}\_1$ to $\Delta$. The central fiber is smooth of multiplicity $1$. Let $d\geq 2$ be an integer. Let $Z\subset \mathcal{Y}$ be a closed submanifold such that the restriction $\text{pr}\_1|\_Z:Z\to \Delta$ is a $d$-to-$1$ branched cover of the disk by the disk totally ramified over the origin with ramification profile $(d\_1,\dots,d\_m)$, i.e., $m$ preimages of the origin with the pullback of a coordinate vanishing to order $d\_r$ at the $r^\text{th}$ preimage point ($d\_1+\dots +d\_m = d$). Let $\mathcal{X}'\to \mathcal{Y}$ be the blowing up along $Z$. There is a further blowing up $\mathcal{X}\to \mathcal{X}'$ with center in the special fiber so that $\mathcal{X}$ is smooth with normal crossings special fiber. 


The strict transform of the special fiber of $\mathcal{Y}$ is an irreducible component with multiplicity $1$ of the special fiber of $\mathcal{X}$. Thus the GCD of the multiplicities equals $1$. Yet the monodromy action of $\pi\_1(\Delta^\*,\text{point})$ on $H^{2r}(X)$ is non-unipotent for $1\leq r \leq n-1$. On the other hand, the LCM of the multiplicities is divisible by the LCM $\ell$ of $(d\_1,\dots,d\_m)$, and the $\ell^\text{th}$ power of a generator of $\pi\_1(\Delta^\*,\text{point})$ does act unipotently (in fact it acts as the identity, in this case).


