This question has long history, In fact the idea for connecting Algebraic geometry to PDE comes from your vision to meaning of "surgery"

In fact philosophically

$$Algebraic \; surgery<===>PDE \; surgery<===> Geometric \; Surgery$$

Here algebraic surgery I mean flips and flops.

Now I come back to your question

One of fundamental conjectures in Kahler geometry is about finding canonical metrics(Kahler-Einstein metrics, twisted Kahler Einstein metrics, constant scalar curvature,...)

In fact if $X$ be a projective variety and by definition the canonical metric $g_{can}$ is the  metric which is attached to  canonical model $X_{can}=\text{Proj}\bigoplus_{m\geq 0}H^0(X,K_X^{m})
$.
Let the canonical Ring $$\bigoplus_{m\geq 0}H^0(X,K_X^{m})$$ is finitely generated and let the Kodaira dimension is positive then 



$$\pi:X\to X_{can}=\text{Proj}(\bigoplus_{m\geq 0}H^0(X,K_X^{m}))$$

gives a canonical metric which is twisted by Weil-Petersson metric on $X$ via

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

where $g_{WP}$ is the Weil-Petersson metric and corresponds to moduli space of Calabi-Yau fibers.


So

If the Kodaira dimension 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 fibers of positive first Chern class which can be introduced by using Deligne pairing.

**If the first Chern class be positive then we need the notion of K-stability of Tian-Donaldson for finding canonical metric(Kahler-Einstein metric)**

**If the first chern class be zero or negative then we can apply the result of Yau and Aubin for finding canonical metric(Kahler-Einstein metric in this case)**