arXiv subjects

Joel Fine

Publications and source records attributed to Joel Fine.

27 records · Page 2Linked to original sources

arXiv2011

The diversity of symplectic Calabi-Yau six-manifolds

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 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.

math.SG↗

arXiv2011

The Hamiltonian geometry of the space of unitary connections with symplectic curvature

Let L->M be a Hermitian line bundle over a compact manifold. Write S for the space of all unitary connections in L whose curvatures define symplectic forms on M and G for the group of unitary bundle isometries of L, which acts on S by pull-back. The main observation of this note is that S carries a G-invariant symplectic structure, there is a moment map for the G-action and that this embeds the components of S as G_0-coadjoint orbits (where G_0 is the component of the identity). Restricting to the subgroup of G which covers the identity on M, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map. When M is a Kähler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.

math.SG↗

arXiv2009

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 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.

math.SG↗

arXiv2009

Calabi flow and projective embeddings

Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle, considering embeddings via H^0(L^k) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson's theorem relating balanced embeddings to metrics with constant scalar curvature [JDG 59(3):479-522, 2001]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu-Ma [arXiv:math/0601260v2] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS\circ Hilb whose fixed points are balanced metrics.

math.DG↗

arXiv2008

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

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 Reznikov. 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.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells-Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen--Tian. One metric satisfying the curvature inequality is hyperbolic four-space H^4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw-yz=0 in C^4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H^4 acting symplectomorphically on the resolution and isometries of H^3 acting biholomorphically on the smoothing.

math.SG↗

arXiv2008

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 (math.DG/0602423). The results complement the work of Calderbank-Pedersen (math.DG/0105263), 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.

math.DG↗

arXiv2006

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 Cech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kahler representatives are described; each such class contains a circle of Kahler metrics. This gives new local examples of scalar-flat Kahler surfaces and generalises work of Joyce who considered the case where the distribution orthogonal to the torus action is integrable.

math.DG↗

arXiv2006

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

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kahler metric on the total space 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 Kahler manifolds of constant scalar curvature in every dimension, each with Kahler class arbitrarily far from the canonical class.

math.DG↗

arXiv2004

Constant scalar curvature Kahler metrics on fibred complex surfaces

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a 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.

math.DG↗