arXiv subjects

Dominic Joyce

Publications and source records attributed to Dominic Joyce.

78 records · Page 5Linked to original sources

arXiv2000

Ruled special Lagrangian 3-folds in C^3

This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds in C^3, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko and Bryant. An important class of ruled SL 3-folds is the special Lagrangian cones in C^3. Each ruled SL 3-fold is asymptotic to a unique SL cone. We study the family of ruled SL 3-folds N asymptotic to a fixed SL cone N_0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection Sigma of N_0 with the unit sphere in C^3 is a Riemann surface, and construct a ruled SL 3-fold N asymptotic to N_0 for each holomorphic vector field w on Sigma. As corollaries of this we write down two large families of explicit SL 3-folds depending on a holomorphic function on C, which include many new examples of singularities of SL 3-folds. We also show that each SL T^2 cone N_0 can be extended to a 2-parameter family of ruled SL 3-folds asymptotic to N_0, and diffeomorphic to T^2 x R.

math.DG↗

arXiv2000

A theory of quaternionic algebra, with applications to hypercomplex geometry

In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic geometry" of noncompact hypercomplex manifolds. The basic building blocks of the theory are AH modules, which should be thought of "vector spaces" over the quaternions. An AH-module is a left module over the quaternions H, together with a real vector subspace. There are natural concepts of linear map and tensor product of AH-modules, which have many of the properties of linear maps and tensor products of vector spaces. However, the definition of tensor product of AH-modules is strange and has some unexpected properties. Let M be a hypercomplex manifold. Then there is a natural class of H-valued "q-holomorphic functions" on M, satisfying a quaternionic analogue of the Cauchy-Riemann equations, which are analogues of holomorphic functions on complex manifolds. The vector space of q-holomorphic functions A on M is an AH-module. Now some pairs of q-holomorphic functions can be multiplied together to get another q-holomorphic function, but other pairs cannot. So A has a kind of partial algebra structure. It turns out that this structure can be very neatly described using the quaternionic tensor product and AH-morphisms, and that A has the structure of an "H-algebra", a quaternionic analogue of commutative algebra.

math.DG↗

arXiv1999

A new construction of compact 8-manifolds with holonomy Spin(7)

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. In a previous paper (Invent. math. 123 (1996), 507-552) the author constructed the first examples of compact 8-manifolds with holonomy Spin(7), by resolving orbifolds T^8/G, where T^8 is the 8-torus and G a finite group of automorphisms of T^8. This paper describes a different construction of compact 8-manifolds with holonomy Spin(7). We start with a Calabi-Yau 4-orbifold Y with isolated singularities, and an isometric, antiholomorphic involution σof Y fixing only the singular points. Let Z=Y/<σ>. Then Z is an orbifold with isolated singularities, and a natural Spin(7)-structure. We resolve the singular points of Z to get a compact 8-manifold M, and show that M has holonomy Spin(7). Taking Y to be a hypersurface in a complex weighted projective space, we construct new examples of compact 8-manifolds with holonomy Spin(7), and calculate their Betti numbers b^k. The fourth Betti number b^4 tends to be rather large, as high as 11,662 in one example.

math.DG↗

arXiv1999

Asymptotically Locally Euclidean metrics with holonomy SU(m)

Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then C^m/G has an isolated quotient singularity at 0. Let X be a resolution of C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on C^m/G. In this paper we study Ricci-flat ALE Kahler metrics on X. We show that if G is a subgroup of SU(m) acting freely on C^m - 0, and X is a crepant resolution of C^m/G, then there is a unique Ricci-flat ALE Kahler metric in each Kahler class. This is proved using a version of the Calabi conjecture for ALE manifolds. We also show the metrics have holonomy SU(m). These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold. The paper has a sequel, "Quasi-ALE metrics with holonomy SU(m) and Sp(m)", math.AG/9905043, which studies Kahler metrics on resolutions of non-isolated singularities C^m/G.

math.AG↗

arXiv1999

Quasi-ALE metrics with holonomy SU(m) and Sp(m)

This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic condition, implying that g is asymptotic to the Euclidean metric on C^m/G away from its singular set. When C^m/G has an isolated singularity, QALE metrics are just ALE metrics. Our main interest is in Ricci-flat QALE Kahler metrics on X. We prove an existence result for Ricci-flat QALE Kahler metrics: if G is a subgroup of SU(m) and X a crepant resolution of C^m/G, then there is a unique Ricci-flat QALE Kahler metric on X in each Kahler class. This is proved using a version of the Calabi conjecture for QALE manifolds. We also determine the holonomy group of the metrics in terms of G. These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold.

math.AG↗

arXiv1998

On the topology of desingularizations of Calabi-Yau orbifolds

Let X/G be a 3-dimensional Calabi-Yau orbifold with codimension 2 singularities. The topology of crepant resolutions of X/G is described by the McKay correspondence (Reid, Ito). We study Calabi-Yau 3-folds Y that arise by deforming the complex structure of X/G. The McKay correspondence does not hold for such Y. We describe the topology of Y using the `Weyl group' of the singular set of X/G. Even in simple examples, this can give many different ways to desingularize X/G. It would be interesting to interpret these results in String Theory, which should lead to a generalization of the idea of orbifold CFT, similar to the idea of `discrete torsion' (Vafa, Witten).

math.AG↗