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

Ricci flow and birational surgery
Jian Song http://arxiv.org/abs/1304.2607

Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata's base point free theorem
Jian Song http://arxiv.org/abs/1409.8374

Canonical measures and Kahler-Ricci flow, with G. Tian, J. Amer. Math. Soc. 25 (2012), no. 2, 303-353, arXiv:0802.2570

The Kahler-Ricci flow through singularities
Jian Song, Gang Tian http://lanl.arxiv.org/abs/0909.4898

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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(If it exists). In fact PDE surgery here means surgery by flow, like Kahler Ricci flow, because this type of flows smooth out singularities and resolve the singularities

Perelman used PDE surgery<===> Geometric surgery for solving Poincare conjecture.

But for finding Canonical Kahler metric huristicly you need 

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




Now I come back to your question

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

You need to know birational geometry in algebraic geometry for finding canonical metrics.

> One of connections of Algebraic geometry to canonical metric theory is
> the Minimal Model Program and a nice program introduced by Jian Song
> and Gang Tian, refes as Song-Tian program.

Now let you are facing with a variety which first Chern class is not definite, we prefer to find twisted Kahler-Einstein metric which is canonical metric(in fact finding generic Kahler Einstein make no sense here).

If $X$ be a projective variety(which first Chern class is not definite ) 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 

$$\kappa(X)=\limsup_{m\to \infty}\frac{\log dim H^0(X,K_X^m)}{\log m}$$


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.

**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)**

See papers of Jian Song about your question  https://www.math.rutgers.edu/~jiansong/