Suppose $X,C$ are smooth varieties. Let $X \to C$ be a fiberation, that is $\dim C < \dim X$ (the example in my mind is a fiber contraction of an $(K_X + D)$-extremal ray), suppose the anti-canonical divisor $-K_X$ of $X$ is big, is it true that the anti-canonical divisor $-K_C$ of $C$ is still big?
I feel there might be counterexamples in ruled surface over an elliptic curve, but have not figured out yet.
However, I feel the result should be true in toric variety.
Any suggestion are welcome!!