Self-dual metrics on toric 4-manifolds: Extending the Joyce construction
نویسندگان
چکیده
Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, T . Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this result to find representatives which are scalar flat Kähler and self-dual Einstein respectively. We review some results concerning the topology of toric manifolds and the construction of Joyce metrics. We then extend this construction to give explicit complete scalar-flat Kähler and self-dual Einstein metrics on manifolds of infinite topological type, and to find a new family of Joyce metrics on open submanifolds of toric spaces. We then give two applications of these extensions — first, to give a large family of scalar flat Kähler perturbations of the Ooguri-Vafa metric, and second to search for a toric scalar flat Kähler metric on a neighbourhood of the origin in C2 whose restriction to an annulus on the degenerate hyperboloid {(z1, z2)|z1z2 = 0} is the cusp metric.
منابع مشابه
Toric Anti-self-dual 4-manifolds via Complex Geometry
Using the twistor correspondence, this article gives a oneto-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 class...
متن کاملToric 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 [7]. The results complement the work of Calderbank–Pedersen [6], who describe where the ...
متن کاملOn a Construction of the Twistor Spaces of Joyce Metrics, I
We explicitly construct the twistor spaces of some self-dual metrics with torus action given by D. Joyce. Starting from a fiber space over a projective line whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the desired twistor spaces. Especially an important role is played by Atiyah’s flop.
متن کاملOn a Construction of the Twistor Spaces of Joyce Metrics
We explicitly construct the twistor spaces of some self-dual metrics with torus action given by D. Joyce. Starting from a fiber space whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the desired twistor spaces. Especially an important role is played by flops, a useful operation in algebraic geometry.
متن کاملStony Brook University
of the Dissertation Self-Dual Metrics on 4-Manifolds by Mustafa Kalafat Doctor of Philosophy in Mathematics Stony Brook University 2007 Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case. Secondly we give an exa...
متن کامل