30 research outputs found
A new construction of compact torsion-free -manifolds by gluing families of Eguchi-Hanson spaces
We give a new construction of compact Riemannian 7-manifolds with holonomy
. Let be a torsion-free -manifold (which can have holonomy a
proper subgroup of ) such that admits an involution preserving
the -structure. Then is a -orbifold, with
singular set an associative submanifold of , where the singularities are
locally of the form . We resolve
this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a
nonvanishing closed and coclosed -form on . Much of the
analytic difficulty lies in constructing appropriate closed -structures
with sufficiently small torsion to be able to apply the general existence
theorem of the first author. In particular, the construction involves solving a
family of elliptic equations on the noncompact Eguchi-Hanson space,
parametrized by the singular set . We also present two generalizations of
the main theorem, and we discuss several methods of producing examples from
this construction.Comment: 83 pages. Version 3: Fixed three grammar mistakes and added a missing
parenthesis. Final version to appear in Journal of Differential Geometr
Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps
We study the moduli space of torsion-free G2-structures on a fixed compact
manifold, and define its associated universal intermediate Jacobian J. We
define the Yukawa coupling and relate it to a natural pseudo-Kahler structure
on J. We consider natural Chern-Simons type functionals, whose critical points
give associative and coassociative cycles (calibrated submanifolds coupled with
Yang-Mills connections), and also deformed Donaldson-Thomas connections. We
show that the moduli spaces of these structures can be isotropically immersed
in J by means of G2-analogues of Abel-Jacobi maps.Comment: 31 pages. Version 2: added a reference and some remarks. Version 3:
Incorporated the referee's suggestions. Final version to appear in
Proceedings of the London Mathematical Societ
Bundle Constructions of Calibrated Submanifolds in R^7 and R^8
We construct calibrated submanifolds of R^7 and R^8 by viewing them as total
spaces of vector bundles and taking appropriate sub-bundles which are naturally
defined using certain surfaces in R^4. We construct examples of associative and
coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This
construction is a generalization of the Harvey-Lawson bundle construction of
special Lagrangian submanifolds of R^{2n}.Comment: 22 pages; for Revised Version: Minor changes, improved notation,
streamlined expositio
Cohomologies on almost complex manifolds and the -lemma
We study cohomologies on an almost complex manifold , defined using
the Nijenhuis-Lie derivations and induced from
the almost complex structure and its Nijenhuis tensor , regarded as
vector-valued forms on . We show how one of these, the -cohomology
, can be used to distinguish non-isomorphic non-integrable
almost complex structures on . Another one, the -cohomology
, is familiar in the integrable case but we extend its
definition and applicability to the case of non-integrable almost complex
structures. The -cohomology encodes whether a complex manifold satisfies the
-lemma, and more generally in the non-integrable case
the -cohomology encodes whether satisfies the -lemma, which we introduce and motivate in this paper. We discuss
several explicit examples in detail, including a non-integrable example. We
also show that is finite-dimensional for compact integrable ,
and use spectral sequences to establish partial results on the
finite-dimensionality of in the compact non-integrable case.Comment: 23 pages. Version 3: one harmless sign error was corrected. Final
version, to appear in "Asian Journal of Mathematics