88 research outputs found
Deformations of glued G_2-manifolds
We study how a gluing construction, which produces compact manifolds with
holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds,
behaves under deformations. We show that the gluing construction defines a
smooth map from a moduli space of gluing data to the moduli space of
torsion-free G_2-structures on the glued manifold, and that this is a local
diffeomorphism. We use this to partially compactify the moduli space of
torsion-free G_2-structures, including it as the interior of a topological
manifold with boundary. The boundary points are equivalence classes of matching
pairs of torsion-free asymptotically cylindrical G_2-structures.Comment: 13 pages; minor corrections, numbering changed to match print versio
The classification of 2-connected 7-manifolds
We present a classification theorem for closed smooth spin 2-connected
7-manifolds M. This builds on the almost-smooth classification from the first
author's thesis. The main additional ingredient is an extension of the
Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether
the spin characteristic class p_M in the fourth integral cohomology of M is
torsion. In addition we determine the inertia group of 2-connected M -
equivalently the number of oriented smooth structures on the underlying
topological manifold - in terms of p_M and the torsion linking form.Comment: Corrected the definition of pseudo-isotopy of almost diffeomorphisms,
56 pages. To appear in Proc. Lond. Math. So
New invariants of G_2-structures
We define a Z/48-valued homotopy invariant nu of a G_2-structure on the
tangent bundle of a closed 7-manifold in terms of the signature and Euler
characteristic of a coboundary with a Spin(7)-structure. For manifolds of
holonomy G_2 obtained by the twisted connected sum construction, the associated
torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples
constructed by Joyce by desingularising orbifolds have odd nu.
We define a further homotopy invariant xi of G_2-structures such that if M is
2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy
and diffeomorphism. The class of a G_2-structure is determined by nu on its own
when the greatest divisor of p_1(M) modulo torsion divides 224; this sufficient
condition holds for many twisted connected sum G_2-manifolds.
We also prove that the parametric h-principle holds for coclosed
G_2-structures.Comment: 26 pages, 1 figure. v3: Defined further invariant, strengthened
classification results, changed titl
An analytic invariant of G_2 manifolds
The first and third authors have constructed a defect invariant
in Z/48 for -structures on a closed 7-manifold. We describe the
nu-invariant using -invariants and Mathai-Quillen currents on M and show
that it can be refined to an integer-valued invariant for
-holonomy metrics. As an example, we determine the -invariants of
twisted and extra twisted connected sums a la Kovalev,
Corti-Haskins-Nordstr\"om-Pacini, and Nordstr\"om. In particular, we exhibit
examples of 7-manifolds where the moduli space of -holonomy metrics has at
least two connected components. In one of these examples, the underlying
-structures are homotopic, in another one, they are not.Comment: pdfLaTeX, 26 pages, 2 figures (tikz). v2: 28 pages, 2 figures.
Reorganised the contents for clarity. Comments welcome
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Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy
In Berger's classification of Riemannian holonomy groups
there are several infinite families and two exceptional cases:
the groups Spin(7) and G_2.
This thesis is mainly concerned with 7-dimensional manifolds
with holonomy G_2.
A metric with holonomy contained in G_2 can be defined in terms of
a torsion-free G_2-structure, and a G_2-manifold is a 7-dimensional manifold
equipped with such a structure.
There are two known constructions of compact manifolds with holonomy
exactly G_2. Joyce found examples by resolving singularities of
quotients of flat tori.
Later Kovalev found different examples by gluing pairs of exponentially
asymptotically cylindrical (EAC) G_2-manifolds (not necessarily with holonomy
exactly G_2) whose cylinders match. The result of this gluing construction
can be regarded as a generalised connected sum of the EAC components, and has
a long approximately cylindrical neck region.
We consider the deformation theory of EAC G_2-manifolds and show, generalising from
the compact case, that there is a smooth moduli space of torsion-free EAC
G_2-structures.
As an application we study the deformations of the gluing construction for
compact G_2-manifolds, and find that the glued torsion-free G_2-structures form an open
subset of the moduli space on the compact connected sum. For a fixed pair of
matching EAC G_2-manifolds the gluing construction provides a path of torsion-free
G_2-structures on the connected sum with increasing neck length.
Intuitively this defines a boundary point for the moduli space on the connected
sum, representing a way to `pull apart' the compact G_2-manifold into a pair of EAC
components. We use the deformation theory to make this more precise.
We then consider the problem whether compact G_2-manifolds constructed by Joyce's
method can be deformed to the result of a gluing construction.
By proving a result for resolving singularities of EAC G_2-manifolds we show
that some of Joyce's examples can be pulled apart in the above sense.
Some of the EAC G_2-manifolds that arise this way satisfy a necessary and
sufficient topological condition for having holonomy exactly G_2.
We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8
manifolds with holonomy contained in Spin(7). On such manifolds there is
a smooth moduli space of torsion-free EAC Spin(7)-structures.
Generalising a result of Wang for compact manifolds we show that for
EAC G_2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.This work was supported by the EPSRC and the Gates Cambridge Trust
Asymptotically cylindrical Calabi-Yau manifolds
Let be a complete Ricci-flat Kahler manifold with one end and assume that
this end converges at an exponential rate to for some
compact connected Ricci-flat manifold . We begin by proving general
structure theorems for ; in particular we show that there is no loss of
generality in assuming that is simply-connected and irreducible with
Hol SU, where is the complex dimension of . If we
then show that there exists a projective orbifold and a divisor
in with torsion normal bundle such that is
biholomorphic to , thereby settling a long-standing
question of Yau in the asymptotically cylindrical setting. We give examples
where is not smooth: the existence of such examples appears not to
have been noticed previously. Conversely, for any such pair we give a short and self-contained proof of the existence and
uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on
.Comment: 33 pages, various updates and minor corrections, final versio
Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau
3-folds starting with (almost) any deformation family of smooth weak Fano
3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau
3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We
pay particular attention to a subclass of weak Fano 3-folds that we call
semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing
theorems and enjoy certain topological properties not satisfied by general weak
Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike
Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to
compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.
We introduce some general methods to compute the basic topological invariants
of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a
small number of representative examples in detail. Similar methods allow the
computation of the topology in many other examples.
All the features of the ACyl Calabi-Yau 3-folds studied here find application
in arXiv:1207.4470 where we construct many new compact G_2-manifolds using
Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds
constructed from semi-Fano 3-folds are particularly well-adapted for this
purpose.Comment: 107 pages, 1 figure. v3: minor corrections, changed formattin
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