Joel Fine

Hyperbolic geometry and non-Kahler 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 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

The diversity of symplectic calabi-yau 6-manifolds

Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1 = 0, b2 > b, b3 > b and pi = G. In the simply connected case, we can also arrange for b3 = 0; in particular, these examples are not diffeomorphic to Kahler manifolds with c1 = 0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi- Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.

Quantization and the Hessian of Mabuchi energy

Let L→X be an ample bundle over a compact complex manifold. Fix a Hermitian metric in L whose curvature defines a Kahler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D*D on functions which arises in the study of scalar curvature. We quantize D*D by the Hessian P*k Pk of balancing energy, a function appearing in the study of balanced embeddings. P*k Pk is defined on the space of Hermitian endomorphisms of H0(X,Lk) endowed with the L2-inner product. We first prove that the leading order term in the asymptotic expansion of P*k Pk is D*D. We next show that if Aut (X,L)/ℂ* is discrete, then the eigenvalues and eigenspaces of P*k Pk converge to those of D*D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kahler metric. As consequences of our results we prove that an estimate of Phong and Sturm is sharp and give a negative answer to a question posed by Donaldson. We also discuss some possible applications to the study of Calabi flow.

A note on positivity of the CM line bundle

We show that positivity of the CM line associated to a family of polarised varieties is intimately related to the stability of its members. We prove that the CM line is nef on any curve which meets the stable locus, and that it is pseudoeffective (i.e. in the closure of the effective cone) as long as there is at least one stable fibre. We give examples showing that the CM line can be strictly negative or strictly positive on curves in the unstable locus.

Toric Anti-self-dual 4-manifolds Via Complex Geometry

Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Čech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kähler representatives are described; each such class contains a circle of Kähler metrics. This gives new local examples of scalar flat Kähler surfaces and generalises work of Joyce [Duke. Math. J. 77(3), 519–552 (1995)] who considered the case where the distribution orthogonal to the torus action is integrable.

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Klieners theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

Toric anti-self-dual Einstein metrics via complex geometry

Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (Donaldson and Fine in Math Ann 336(2):281–309, 2006). The results complement the work of Calderbank–Pedersen (J Differential Geom 60(3):485–521, 2002), who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.

The space of hyperkähler metrics on a 4-manifold with boundary

Let X be a compact 4-manifold with boundary. We study the space of hyperk¨ahler triples ω1, ω2, ω3 on X, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperk¨ahler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperk¨ahler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of SU(2).