We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.

https://www.ams.org/jams/2012-25-02/S0894-0347-2011-00717-0/S0894-0347-2011-00717-0.pdf

Canonical measures and Kähler-Ricci flow

The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly-spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS5 × S5. We also include several tools and tricks that have been useful throughout the literature.

/pdf/numerical-methods-for-finding-stationary-gravitational-j1rzf6xpk3.pdf

Numerical methods for finding stationary gravitational solutions

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $(X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $\mu$ for varieties and their subschemes; if $(X,L)$ is semistable then $\mu(Z)\le\mu(X)$ for all $Z\subset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.

A study of the Hilbert-Mumford criterion for the stability of projective varieties

The J-flow of S. K. Donaldson and X. X. Chen is a parabolic flow on Kahler manifolds with two Kahler metrics. It is the gradient flow of the J-functional that appears in Chen's formula for the Mabuchi energy. We find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the J-flow to a critical metric. We use this result to show that on manifolds with ample canonical bundle, the Mabuchi energy is proper on all Kahler classes in an open neighborhood of the canonical class defined by a positivity condition. This improves previous results of Chen and of the second author. We discuss the implications of this for the problem of the existence of constant-scalar-curvature Kahler metrics.



We also study the singularities of the J-flow and, under certain conditions (which always hold for dimension 2) derive some estimates away from a subvariety. We discuss the conjectural remark of Donaldson that if the J-flow does not converge on a Kahler surface, then it should blow up over some curves of negative self-intersection. © 2007 Wiley Periodicals, Inc.

On the convergence and singularities of the J-Flow with applications to the Mabuchi energy

A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kahler manifolds.

/pdf/constant-scalar-curvature-kahler-metrics-on-fibred-complex-erord0swg3.pdf

Constant scalar curvature Kähler metrics on fibred complex surfaces

We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C 4 : the smoothing is a natural S 3 ‐bundle over H 3 , its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S 2 ‐bundle over H 4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we find the first example of a simply connected symplectic 6‐manifold with c1D 0 that does not admit a compatible Kahler structure. We also find infinitely many distinct complex structures on 2.S 3 S 3 / #.S 2 S 4 / with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler “Fano” manifolds of dimension 12 and higher. 53D35, 32Q55; 51M10, 57M25

/pdf/hyperbolic-geometry-and-non-kahler-manifolds-with-trivial-3lzenwkco3.pdf

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S^2-bundle over H^4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we find the first example of a simply-connected symplectic 6-manifold with c_1=0 that does not admit a compatible Kahler structure. We also find infinitely many distinct complex structures on 2(S^3xS^3)#(S^2xS^4) with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler "Fano" manifolds of dimension 12 and higher.

/pdf/hyperbolic-geometry-and-non-kahler-manifolds-with-trivial-2e7fjssmnp.pdf

Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

Given an SO(3)-bundle with connection, the associated two- sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Rezn- ikov [34]. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c 1 =0 which are never Kahler; e.g., we produce such manifolds with b 1 = 0 = b 3 and also with c 2 [w]

/pdf/symplectic-calabi-yau-manifolds-minimal-surfaces-and-the-2e9fykixof.pdf

Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

Let $\pi \colon X \to B$ be a holomorphic submersion between compact K\"ahler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature K\"ahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature K\"ahler metric on $X$. The condition involves the $\CM$-line bundle---a certain natural line bundle on $B$---which is proved to be nef. Knowing this, the condition is then implied by $c_1(B) <0$. This provides infinitely many K\"ahler manifolds of constant scalar curvature in every dimension, each with K\"ahler class arbitrarily far from the canonical class.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2007/0014/0002/MRL-2007-0014-0002-a007.pdf

Joel Fine

Papers

Constant scalar curvature Kähler metrics on fibred complex surfaces

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle