[Math] Algebraic Geometry needed for Kähler-Einstein metric

Question

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic geometry (by Algebraic Geometry I DON'T mean Complex Analytic Geometry) I need to know if I want to study Kähler-Einstein / Extremal Kähler metrics in my PhD. Is the role of Algebraic Geometry in Kähler-Einstein metric something like black-box or do I need to have a thorough understanding of Algebraic Geometry? Frankly speaking, I dislike Algebraic Geometry and I don't understand it.

Answer 1

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 (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 K-stable Fano fibers

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/