arXiv subjects

Dominic Joyce

Publications and source records attributed to Dominic Joyce.

At least 73 records · Page 4Linked to original sources

arXiv2004

U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

This is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a U(1)-invariant SL 3-fold. Then |z_1|^2-|z_2|^2=2a on N for some real a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a. When a is nonzero, u,v are smooth and N is nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. The first paper math.DG/0111324 studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The second paper math.DG/0111326 extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicity n>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in C^3, including singular fibres.

math.DG↗

arXiv2003

Singularities of special Lagrangian submanifolds

We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities. That is, near each singular point x, X is modelled on an SL cone C in C^m with isolated singularity at 0. We also discuss directions for future research, and give a list of open problems.

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arXiv2003

Lectures on special Lagrangian geometry

We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject. Special Lagrangian m-folds in C^m are defined, and ways of constructing them described. 'Almost Calabi-Yau manifolds' (a generalization of Calabi-Yau manifolds useful in special Lagrangian geometry) are introduced, and the deformation theory, obstruction theory, and moduli spaces of compact special Lagrangian m-folds in (almost) Calabi-Yau m-folds are explained. Then we consider singular special Lagrangian submanifolds which are locally modelled on special Lagrangian cones with an isolated singularity at 0. Compact singular special Lagrangian submanifolds of this type have a well-behaved deformation theory, and can often be realized as limits of families of compact, nonsingular special Lagrangian submanifolds. Applications of this to the SYZ Conjecture and Mirror Symmetry of Calabi-Yau 3-folds are discussed.

math.DG↗

arXiv2003

Singularities of special Lagrangian fibrations and the SYZ Conjecture

The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One of the main problems in proving the SYZ Conjecture (or even in finding the right statement of it) is that the singularities of special Lagrangian 3-folds and fibrations are poorly understood. This paper studies the singularities of special Lagrangian fibrations. Our main rigorous results are the construction of examples of special Lagrangian fibrations on open subsets of C^3. The simplest are given explicitly, and the rest are constructed using analytic existence results from the author's three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 on U(1)-invariant special Lagrangian 3-folds in C^3. We then argue, without full proofs, that some features of our examples should also hold for special Lagrangian fibrations f : M --> B of (almost) Calabi-Yau 3-folds, especially in the generic case. In particular, f will not be smooth but only piecewise-smooth, and the discriminant (set of singular fibres) of f will be of codimension 1 in B, and will typically be composed of 'ribbons'. Finally we draw some conclusions on the SYZ Conjecture, which contradict some stronger statements of it.

math.DG↗

arXiv2003

Special Lagrangian submanifolds with isolated conical singularities. I. Regularity

This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272, which surveys the series, gives examples, and applies the results to prove some conjectures. This first paper lays the foundations for the series, giving definitions and proving auxiliary results in symplectic geometry and asymptotic analysis that will be needed later. We also prove results on the regularity of X near its singular points. We show that X converges to the cone C_i near x_i with all its derivatives, at rates determined by the eigenvalues of the Laplacian on the intersection of C_i with the unit sphere. We show that if X is a special Lagrangian integral current with a tangent cone C at x satisfying some conditions, then X has an isolated conical singularity at x in our sense. We also prove analogues of many of our results for Asymptotically Conical SL m-folds in C^m. The sequel math.DG/0211295 studies the deformation theory of compact SL m-folds X in M with conical singularities. The third and fourth papers math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X as a limit of a family N^t of compact, nonsingular SL m-folds in M.

math.DG↗

arXiv2003

Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces

This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and proves some conjectures. In this paper we study the deformation theory of compact SL m-folds X in M with conical singularities. We define the moduli space M_X of deformations of X in M, and construct a natural topology on it. Then we show that M_X is locally homeomorphic to the zeroes of a smooth map Φ: I --> O between finite-dimensional vector spaces. Here the infinitesimal deformation space I depends only on the topology of X, and the obstruction space O only on the cones C_1,...,C_n at x_1,...,x_n. If the cones C_i are "stable" then O is zero and M_X is a smooth manifold. We also extend our results to families of almost Calabi-Yau structures on M. The first paper math.DG/0211294 laid the foundations for the series, and studied the regularity of X near its singular points. The third and fourth papers math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X as the limit of a family N^t of compact, nonsingular SL m-folds in M.

math.DG↗

arXiv2003

Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case

This is the third in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302356, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first two papers math.DG/0211294, math.DG/0211295 studied the regularity of X near its singular points, and the moduli space of deformations of X. In this paper and the fourth math.DG/0302356 we construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds \tilde N^t in M for small t>0. Suppose L_1,...,L_n are Asymptotically Conical SL m-folds in C^m, with L_i asymptotic to the cone C_i at infinity. We shrink L_i by a small t>0, and glue tL_i into X at x_i for i=1,...,n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N^t for small t>0. Then we show using analysis that when t is sufficiently small we can deform N^t to a compact, nonsingular SL m-fold \tilde N^t via a small Hamiltonian deformation. This \tilde N^t depends smoothly on t, and as t --> 0 it converges to the singular SL m-fold X, in the sense of currents. This paper studies the simpler cases, where by topological conditions on X and L_i we avoid various obstructions to existence of \tilde N^t. The sequel math.DG/0302356 will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi-Yau m-folds.

math.DG↗

arXiv2003

Special Lagrangian submanifolds with isolated conical singularities. IV. Desingularization, obstructions and families

This is the fourth in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first paper math.DG/0211294 studied the regularity of X near its singular points, and the second math.DG/0211295 the moduli space of deformations of X. The third paper math.DG/0302355 and this one construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds \tilde N^t in M for small t>0. Let L_1,...,L_n be Asymptotically Conical SL m-folds in C^m, with L_i asymptotic to C_i at infinity. We shrink L_i by t>0, and glue tL_i into X at x_i for i=1,...,n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N^t for small t>0. Then we show using analysis that for small t we can deform N^t to a compact, nonsingular SL m-fold \tilde N^t via a small Hamiltonian deformation. As t --> 0 this \tilde N^t converges to X, in the sense of currents. The third paper math.DG/0302355 studied simpler cases, where by topological conditions on X and L_i we avoid obstructions to existence of \tilde N^t. This paper considers more complex cases when these obstructions are nontrivial, and also desingularization in smooth families of almost Calabi-Yau m-folds M^s for s in F, rather than a single almost Calabi-Yau m-fold M.

math.DG↗

arXiv2003

Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications

This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with this paper. We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the existence of special Lagrangian connected sums N_1 # ... # N_k of SL m-folds N_1,...,N_k in M. We also study SL 3-folds with T^2-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds and the SYZ Conjecture. Let X be a compact SL m-fold with isolated conical singularities x_i and cones C_i for i=1,...,n. The first paper math.DG/0211294 studied the regularity of X near its singular points, and the the second paper math.DG/0211295 the moduli space of deformations of X. The third and fourth papers math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds N^t in M for small t>0. Let L_i be an Asymptotically Conical SL m-fold in C^m asymptotic to C_i at infinity. We make N^t by gluing tL_i into X at x_i for i=1,...n.

math.DG↗

arXiv2002

Lectures on Calabi-Yau and special Lagrangian geometry

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry. `Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in special Lagrangian geometry) are explained and the deformation theory and moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau m-folds is described. In the final part we consider isolated singularities of special Lagrangian m-folds, focussing mainly on singularities locally modelled on cones, and the expected behaviour of singularities of compact special Lagrangian m-folds in generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described.

math.DG↗

arXiv2002

U(1)-invariant special Lagrangian 3-folds in C^3 and special Lagrangian fibrations

This is a survey of the author's series of three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 using analysis to investigate special Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, and their sequel math.DG/0011179 on special Lagrangian fibrations and the SYZ Conjecture. We briefly present the main results of these four long papers, giving some explanation and motivation, but no proofs. The aim is to make the results and ideas accessible to String Theorists and others who have an interest in special Lagrangian 3-folds and fibrations, but have no desire to read pages of technical analysis. Let N be an SL 3-fold in C^3 invariant under the U(1)-action above. Then |z_1|^2-|z_2|^2=2a on N for some real number a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a=0 the equations may have singular points where u,v are not differentiable, which leads to analytic difficulties. We prove existence and uniqueness results for solutions u,v on domains S in R^2 with boundary conditions, including singular solutions. We study their singularities, giving a rough classification by multiplicity and type. We prove the existence of large families of fibrations of open subsets of C^3 by U(1)-invariant SL 3-folds, including singular fibres. Finally, we use these fibrations as local models to draw conclusions about the SYZ Conjecture on Mirror Symmetry of Calabi-Yau 3-folds.

math.DG↗

arXiv2002

Constructing compact manifolds with exceptional holonomy

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. There are also more complicated constructions which begin with a Calabi-Yau manifold or orbifold. All the material in this paper is covered in much more detail in the author's book, "Compact manifolds with special holonomy", Oxford University Press, 2000.

math.DG↗

arXiv2002

Constant Scalar Curvature Metrics on Connected Sums

Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant scalar curvature. We form the connected sum M' # M'' of M' and M'' by removing small balls from M' and M'' and joining the S^{n-1} boundaries together. In this paper we use analysis to construct metrics with constant scalar curvature on M' # M''. Our description is quite explicit, in contrast to the general Yamabe case when one knows little about what the metric looks like. There are 9 cases, depending on the signs of the scalar curvature on M' and M'' (positive, negative, or zero). We show that the constant scalar curvature metrics either develop small "necks" separating M' and M'', or one of M', M'' is crushed small by the conformal factor. When both have positive scalar curvature, we construct three different metrics with scalar curvature 1 in the same conformal class.

math.DG↗

arXiv2001

On counting special Lagrangian homology 3-spheres

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class. In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M. As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood. We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C). Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds.

hep-th↗

arXiv2001

Evolution equations for special Lagrangian 3-folds in C^3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P in R^n, and constructed a family of special Lagrangian m-folds N in C^m, which are swept out by the image of P under a 1-parameter family of linear or affine maps phi_t : R^n -> C^m, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C^3. We find a 1-1 correspondence between sets of evolution data with m=3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C^3. Our main results are a number of new families of special Lagrangian 3-folds in C^3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R^3 or S^1 x R^2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind. We hope these 3-folds will be helpful in understanding singularities of compact special Lagrangian 3-folds in Calabi-Yau 3-folds. This will be important in resolving the SYZ conjecture in Mirror Symmetry.

math.DG↗

arXiv2001

Special Lagrangian m-folds in C^m with symmetries

This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to understand Mirror Symmetry and the SYZ conjecture.

math.DG↗

arXiv2001

Special Lagrangian 3-folds and integrable systems

This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-folds in C^3 involving three commuting o.d.e.s. Now special Lagrangian cones in C^3 are linked to the theory of harmonic maps and integrable systems. Harmonic maps from a Riemann surface into complex projective space CP^n are an integrable system, and can be studied and classified using loop group techniques. If N is a special Lagrangian cone in C^3, then N is the cone on the image of a conformal harmonic map ψ: S --> S^5 for some Riemann surface S, and the projection of ψto CP^2 is also conformal harmonic. Our examples of special Lagrangian cones in C^3 yield conformal harmonic maps ψ: R^2 --> CP^2. We work through the integrable systems theory for these examples, showing that they are superconformal of finite type, and calculating their harmonic sequences, Toda and Tzitzeica solutions, algebra of polynomial Killing fields and spectral curves. We also study the double periodicity conditions for ψ, and so find families of superconformal tori in CP^2. We finish by asking whether our more general construction of special Lagrangian 3-folds can also be derived from a higher-dimensional integrable system, and whether the special Lagrangian equations themselves are in some sense integrable.

math.DG↗

arXiv2001

Constructing special Lagrangian m-folds in C^m by evolving quadrics

This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lagrangian 3-folds in C^3. This paper describes a construction of special Lagrangian m-folds in C^m which are fibred by (m-1)-submanifolds which are quadrics in Lagrangian planes R^m in C^m. Generically they have only discrete symmetry groups. Some of our examples have been previously constructed by Lawlor and Harvey, using different methods. The principal motivation for these papers is to lay the foundations for the study of singularities of compact special Lagrangian m-folds in Calabi-Yau m-folds. Understanding such singularities will be important in resolving the SYZ conjecture on Mirror Symmetry of Calabi-Yau 3-folds. The special Lagrangian m-folds in C^m we construct here include many cones on S^a x S^b x S^1 for a+b=m-2, which are local models for singularities of special Lagrangian m-folds in Calabi-Yau m-folds.

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