Canonical Metrics in Complex Geometry

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Flatland (Edwin A. Abbott) Intrinsic Geometry and Curvature Flatlanders can measure Length, Angles, Area, Curvature...

Circumference of a circle of radius r = Question

Given a complete, smooth surface. Can we “deform” it into a surface whose curvature is constant?

The Uniformisation Theorem (Poincare, Kobe 1907)

Any complete, smooth surface is conformally equivalent to a surface of constant curvature. The Gauss-Bonnet Theorem Let M be a compact smooth surface and the Euler Characteristic of M. Then Cone Points

Question

Given points and angles can we deform into a manifold with constant curvature and, for each i, asymptotic to a cone of angle at p_i. The Gauss-Bonnet Theorem with Cone Points Let M be a compact surface, smooth on and for each i asymptotic to a cone of angle at . Then The Troyanov Conditions

Theorem ● Assume Then is conformally equivalent to a manifold with constant curvature and, for each i, asymptotic to a cone of angle at p_i. (Troyanov 1987 )

● Assume Then is conformally equivalent to a manifold with constant curvature and, for each i, asymptotic to a cone of angle at p_i if and only if

(Troyanov 1991, Luo-Tian 1992) Kähler Manifolds

M smooth manifold of real dimension 2n (M,J,g) J complex structure g Riemannian metric

Main example: Projective manifolds, i.e. smooth submanifolds of complex projective space, cut out by homogeneous polynomials.

Remark: The Kähler form is a closed (1,1)-form, i.e. defines a class Curvature in higher dimensions

The Riemann Curvature Tensor

Curvature Ricci curvature

Scalar curvature

Definition A Kähler-Einstein metric is a Kähler metric such that

for some Question

Let M be a projective manifold. Does M admit a Kähler-Einstein metric?

Topological condition:

Theorem <0, =0, >0 ● Assume or Then M admits a Kähler-Einstein metric (Aubin, Yau 1978)

● Assume Then M admits a Kähler-Einstein metric if and only if M is K-stable. (Chen-Donaldson-Sun 2015) Constant scalar curvature metrics

Question

Let (M,L) be a polarised manifold. When does contain a Kähler metric such that

is constant? (Calabi 1962) Coupled Kähler-Einstein metrics

Definition (H - Witt Nyström 2017) A coupled Kähler-Einstein metric is a k-tuple of Kähler metrics such that for some

Question

Let M be a projective manifold. Does M admit a coupled Kähler-Einstein metric? Theorem (H - Witt Nyström 2017) ● Let M be a projective manifold and be positive classes such that

Then there exist a unique coupled Kähler-Einstein metric such that for all i.

● Assume and is a coupled Kähler-Einstein metric on M. Assume also for some line bundles over M. Then is K-stable.