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Motivation: Unipotent reduction condition is very important for study of family of algebraic varieties. For example for algebraic fiber space if we have such condition then the direct image of dualizing sheaf is locally free.

When we are working on algebraic fiber space $f:X\to Y$ with $dim Y=1$, a semi-stable reduction provides a simple special fiber(central fiber to be snc due to Mumford), which is useful from many points of view, such as Hodge theory, period maps, and monodromy

Now, when $dim Y>1$, semi-stable reduction theorem is not known, and Kawamata suggested Unipotent reduction theorem by inspiring of the works of Deligne

So the study of analytical equivalency of such URC is important. For example works of P. Deligne

Unipotent reduction condition: Let $\pi : X \to Y$ be an algebraic fiber space, we say $\pi$ satisfies the unipotent reduction condition(URC) if and only if the following conditions holds:

1.There is a Zariski open dense subset $Y_0$ of $Y$ such that $D = Y − Y_0$ is a divisor of normal crossing on $Y$

  1. $\pi_0:X_0\to Y_0$ is smooth where $X_0=\pi^{-1}(Y_0)$

  2. The local monodromies of $R^d{π_{0}}_∗\mathbb C_{X_0}$ around $D$ are unipotent where $d = dim X − dim Y $.

If $\pi : X \to S$ is a generically smooth CY family of complex projective varieties, when $dim S=1$, we have semi-stable reduction theorem, but when $dim S>1$ then semi-stable reduction theorem has not been proven yet. But Kawamata instead gave Unipotent Reduction theorem when the dimension of base is bigger than one.

Kawamata, Y. Characterization of abelian varieties. Compositio Math. 43 (1981), no. 2, 253–276

From Analytical point of view it is known that, allowing semi-stable reduction condition is equivalent with boundedness of Weil-Petersson distance near central fiber ( this easily can be derived from paper of Fujino. He showed base change+MMP gives central fiber be CY)(which is equivalent to klt singularites from algebraic view due to the work of Iishi ).

What is the Analytical point of view of Unipotent Reduction Condition?

(Note that semi-stable reduction gives Unipotent Reduction )

I have conjecture saying that if the base of algebraic fiber space which has dimension bigger than one admit canonical metric, then if the sum of canonical hermitian metric of base (which correspond to Kahler metric of base) and canonical hermitian metric correspond to moduli part (correspond to relative canonical bundle) be bounded, then we can allow semi-stable reduction and vise versa

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