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where $g_{WP}$ is the Weil-Petersson metric on moduli-space of Fano fibers

where $g_{WP}$ is the Weil-Petersson metric on moduli-space of K-stable Fano fibers

is negative then you can find canonical metric by applying Mori fiber space and some condition on Chow-Mumford line bundle and extending the result of Ross-Fine we have

$$Ric(g_{can})=g_{can}+g_{WP}$$

where $g_{WP}$ is the Weil-Petersson metric on moduli-space of Fano fibers of positive first Chern class which can be introduced by using Deligne pairing.

is negative then you can find canonical metric by applying Mori fiber space (by assuming base is of general type)and some condition on Chow-Mumford line bundle and extending the result of Ross-Fine we have

$$Ric(g_{can})=-g_{can}+g_{WP}$$

where $g_{WP}$ is the Weil-Petersson metric on moduli-space of Fano fibers of positive first Chern class which can be introduced by using Deligne pairing.

Note that if base is Fano K-stable then along Mori fibre space we have

$$Ric(g_{can})=g_{can}+g_{WP}$$

If base is Calabi-Yau variety and along Mori fibre space we have

$$Ric(g_{can})=g_{WP}$$

where $g_{WP}$ is the Weil-Petersson metric on moduli-space of Fano fibers

Some references :

Finding Kahler-Einstein metric on a Kahler Variety is related to MMP (Minimal model program) in algebraic geometry

Some references

Finding Kahler-Einstein metric on a Kahler Variety is related to MMP (Minimal model program) in algebraic geometry