Uniruled degenerations of abelian varieties

Question

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal projective variety of general type.

Assume that the general fiber of $f$ is an Abelian variety, I would like to say that either there is a component of some singular fiber of $f$ which is uniruled, or otherwise all singular fibers (if any) are just smooth Abelian varieties with multiplicity $m>1$.

If $\mathrm{dim} X=2$ this statement is true thanks to Kodaira's classification of singular fibers of elliptic surfaces. In this case the singular fibers are either $mI_0, m>1$, or else have a component which is a rational curve. I am looking for a higher-dimensional analog of this statement, any reference (or counterexample) is greatly appreciated.

Answer 1

Maybe this can be of some help.

In this paper by K. Oguiso, you can find at the end an appendix by N. Nakayama. In this appendix he gives a theorem which describes the local structure of a degeneration of abelian surfaces (in the spirit of Kodaira's classification). It is Theorem B.1.

It roughly states the following. Suppose that if you have a projective surjective morphism $\alpha\colon\mathcal X\to\Delta$, where $\Delta$ is the complex unit disc, such that the fibers over the punctured disc are abelian surfaces, that $K_{\mathcal X}\simeq\mathcal O_{\mathcal X}$, and that the central scheme-theoretic fiber contains no ruled surfaces. Then, the central fiber $X_0$ may be written as $X_0=mS$, where $S$ is an abelian surface in the case $m=1$ and a hyper-elliptic surface in the case $m>1$.

Answer 2

Some more comments:

If you allow semi-stable reduction and run Minimal Model program then we have the following result due to Fujino which can be applied to semi-stable degenerations of Abelian varieties, Calabi-Yau varieties

Let $f : X \to Y$ be a proper surjective morphism from a smooth quasi-projective variety $X$ to a smooth quasi-projective curve $Y$ with connected fibers. Let $P \in Y$ be a point. Assume that $Supp f^*P$ is a simple normal crossing divisor on $X$ and $f$ is smooth over $Y \setminus P$. We further assume that $K_{f^{-1}Q} \cong 0$, for every $Q\in Y\setminus P$. Then there exists a sequence of flips and divisorial contractions

$$X=X_0 \to K_{X_1} \to K_{X_2} \to \cdots \to K_{X_k} \cdots \to K_{X_m}$$

over $Y$ such that $X_m$ has only $\mathbb Q$-factorial terminal singularities and $K_{X_m} \cong_{Q,Y}0$.

Let $S=Supp f^*_mP $ be the special fiber of $f_m : X_m \to Y$ . If $S$ is reducible, then every irreducible component of $S$ is uniruled. If $S$ is irreducible, then $S$ is normal and has only canonical singularities if and only if $S$ is not uniruled. Note that $K_S\cong_ Q 0$ when $S$ is irreducible and has only canonical singularities.