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Let start with a definition

Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the function

$$\dim H^0(X_p, m K_{X/D}~_{|X_p})$$ is constant on $D$ (where now $X_p$ denotes the fibre over the point $p \in D$ and $D$ is a disc).

1) I am lokking for examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical.

2) Is there any approach for the following conjecture?

Conjecture: Every variety with a single pluricanonical form is birational to a variety with canonical singularities such that in addition the canonical divisor is nef

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    $\begingroup$ With regards to (1), it sounds like you are restricting the hypothesis to be over the Complex numbers. In this case, remark 4.9 in arxiv.org/abs/0901.0389 may be of interest. $\endgroup$
    – aegbert
    Apr 22, 2016 at 18:25

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