An introduction to d-manifolds and derived differential geometry

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". For brevity, this survey concentrates mostly on d-manifolds without boundary. A longer and more detailed summary of the book is given in arXiv:1208.4948. Much of differential geometry extends very nicely to d-manifolds and d-orbifolds -- immersions, submersions, submanifolds, transverse fibre products, orientations, etc. Compact oriented d-manifolds and d-orbifolds have virtual classes. There are truncation functors to d-manifolds and d-orbifolds from essentially every geometric structures on moduli spaces used in enumerative invariant problems in differential geometry or complex algebraic geometry, including Fredholm sections of Banach vector bundles over Banach manifolds, the "Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer, Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect obstruction theories in algebraic geometry. Thus, results in the literature imply that many important classes of moduli spaces are d-manifolds or d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic geometry. D-manifolds and d-orbifolds will have applications in symplectic geometry, and elsewhere.Comment: 45 pages. (v3) Minor changes, references update

Hypercomplex Algebraic Geometry

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C. The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry

Constant Scalar Curvature Metrics on Connected Sums

The Yamabe problem (proved in 1984) guarantees the existence of a metric of constant scalar curvature in each conformal class of Riemannian metrics on a compact manifold of dimension $n \geq 3$, which minimizes the total scalar curvature of this conformal class. Let $(M',g')$ and $(M'',g'')$ be compact Riemannian $n$-manifolds. We form their connected sum $M'\#M''$ by removing small balls of radius $\epsilon$ from $M'$, $M''$ and gluing together the $S^{n-1}$ boundaries, and make a metric $g$ on $M'\#M''$ by joining together $g'$,$g''$ with a partition of unity. In this paper we use analysis to study metrics with constant scalar curvature on $M'\#M''$ in the conformal class of $g$. By the Yamabe problem, we may rescale $g'$ and $g''$ to have constant scalar curvature 1, 0 or -1. Thus there are 9 cases, which we handle separately. 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 sides have positive scalar curvature we find three metrics with scalar curvature 1 in the same conformal class

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.Comment: 66 pages, LaTeX, uses packages amstex and amssym

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 \Phi : 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.Comment: 49 pages. (v2) minor changes, new references. (v3) Changed notatio

Configurations in abelian categories. II. Ringel-Hall algebras

This is the second in a series math.AG/0312190, math.AG/0410267, math.AG/0410268 on configurations in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite collection of objects \sigma(J) and morphisms \iota(J,K) or \pi(J,K) : \sigma(J) --> \sigma(K) in A satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper math.AG/0312190 defined configurations and studied moduli spaces of (I,<)-configurations in A, using the theory of Artin stacks. It proved well-behaved moduli stacks Obj_A, M(I,<)_A of objects and configurations in A exist when A is the abelian category coh(P) of coherent sheaves on a projective K-scheme P, or mod-KQ of representations of a quiver Q. Write CF(Obj_A) for the vector space of constructible functions on Obj_A. Motivated by Ringel-Hall algebras, we define an associative multiplication * on CF(Obj_A) using pushforwords and pullbacks along 1-morphisms between the M(I,<)_A, making CF(Obj_A) into an algebra. We also study representations of CF(Obj_A), the Lie subalgebra CF^ind(Obj_A) of functions supported on indecomposables, and other algebraic structures on CF(Obj_A). Then we generalize these ideas to stack functions SF(Obj_A), a universal generalization of constructible functions on stacks introduced in math.AG/0509722, containing more information. Under extra conditions on A we can define (Lie) algebra morphisms from SF(Obj_A) to some explicit (Lie) algebras, which will be important in the sequels on invariants counting t-(semi)stable objects in A.Comment: 66 pages, LaTeX. (v4) Minor changes, now in final for

A generalization of manifolds with corners

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under much weaker conditions than in $\bf Man^c$. This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of $J$-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners. Our manifolds with g-corners are related to the 'interior binomial varieties' of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874), and to the 'positive log differentiable spaces' of Gillam and Molcho in arXiv:1507.06752.Comment: 97 pages, LaTeX. (v3) final version, to appear in Advances in Mathematic

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.Comment: 44 pages, LaTeX; (v4) minor corrections and improvement