Some remarks on papers: math.DG/0211091, math.DG/0306187, math.CA/0312090
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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.
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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.
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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.
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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.
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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.
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U(1)-invariant special Lagrangian 3-folds I. Nonsingular solutions
This is the first of three papers math.DG/0111326, math.DG/0204343 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 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 is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N. This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with two kinds of boundary conditions. In the sequels we extend these results to the singular case a=0. The next paper math.DG/0111326 proves existence and uniqueness of continuous weak solutions to the two Dirichlet problems when a=0. This gives existence and uniqueness of a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions. The final paper math.DG/0204343 studies the nature of the singularities that arise, and constructs U(1)-invariant special Lagrangian fibrations of open sets in C^3.
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U(1)-invariant special Lagrangian 3-folds II. Existence of singular solutions
This is the second of three papers math.DG/0111324, math.DG/0204343 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. If N is such a 3-fold 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. The first paper math.DG/0111324 studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in C^3, with boundary conditions. In this paper and its sequel math.DG/0204343 we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in C^3, with boundary conditions.
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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.
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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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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.
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Constructing Associative 3-folds by Evolution Equations
This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG/0012060 on special Lagrangian 3-folds in C^3. The two methods described involve the use of an affine evolution equation with affine evolution data and the area of ruled submanifolds. We first give a derivation of an evolution equation for associative 3-folds from which we derive an affine evolution equation using affine evolution data. We then use this on an example of such data to construct a 14-dimensional family of associative 3-folds. One of the main result of the paper is then an explicit solution of the system of differential equations generated in a particular case to give a 12-dimensional family of associative 3-folds. We also find that there is a straightforward condition that ensures that the associative 3-folds constructed are closed and diffeomorphic to S^1xR^2, rather than R^3. In the final section we define ruled associative 3-folds and derive an evolution equation for them. This then allows us to characterise a family of ruled associative 3-folds using two real analytic maps that must satisfy two partial differential equations. We finish by giving a means of constructing ruled associative 3-folds M from r-oriented two-sided associative cones M_0 such that M is asymptotically conical to M_0 with order O(r^{-1}).
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Some remarks on Hermitian manifolds satisfying Kähler-like conditions
We study Hermitian metrics whose Bismut connection $\nabla^B$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$, for every tangent vectors $x,y,z,w$, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with Kähler-like curvature, to appear in Commun. Anal. Geom., arXiv:1809.02632 [math.DG]], [Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604 [math.DG]], [S. T. Yau, Q. Zhao, F. Zheng, On Strominger Kähler-like manifolds with degenerate torsion, arXiv:1908.05322 [math.DG]]. Using the characterization of SKT almost abelian Lie groups in [R. M. Arroyo, R. Lafuente, The long-time behavior of the homogeneous pluriclosed flow, Proc. London Math. Soc. (3), 119, (2019), 266-289], we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.
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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.
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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.
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The classification problem for pseudo-Riemannian symmetric spaces
Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this survey article we want to explain these results and some of their applications. Among other things, the material developed in our previous papers math.DG/0312243, math.DG/0408249, and math.DG/0503220 is presented in a unified way.
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Spectral curves and Nahm transform for doubly-periodic instantons
We explore the role played by the spectral curves associated with Higgs pairs in the context of the Nahm transform of doubly-periodic instantons defined in "Construction of doubly-periodic instantons" (math.DG/9909069) and "Nahm transform for doubly-periodic instantons" (math.DG/9910120). More precisely, we show how to construct a triple consisting of an algebraic curve plus a line bundle with connection over it from a doubly-periodic instanton, and that these coincide with the Hitchin's spectral data associated with the Nahm transformed Higgs bundle.
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Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 3
This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double submersions.
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