1,185 research outputs found
Holomorphic curves in exploded manifolds: Compactness
This paper establishes compactness results for the moduli stack of
holomorphic curves in suitable exploded manifolds. This result together with
the analysis in arXiv:0902.0087 allows the definition of Gromov Witten
invariants of these exploded manifolds.Comment: 76 pages. In v2, compactness is proved using more practical to verify
assumptions, and there is more focus on \dbar-log compatible almost complex
structures. arXiv admin note: text overlap with arXiv:0706.391
Gluing formula for Gromov-Witten invariants in a triple product
We present a gluing formula for Gromov-Witten invariants in the case of a
triple product. This gluing formula is a simple case of a much more general
gluing formula proved and stated using exploded manifolds. We present this
simple case because it is relatively easy to explain without any knowledge of
exploded manifolds or log schemes.Comment: 15 pages, 8 pictures. v2: minor improvements to exposition, updated
reference
Holomorphic curves in exploded manifolds: virtual fundamental class
We define Gromov--Witten invariants of exploded manifolds. The technical
heart of this paper is a construction of a virtual fundamental class of any Kuranishi category (which is a simplified, more general
version of an embedded Kuranishi structure.) We also show how to integrate
differential forms over to obtain numerical invariants, and push
forward differential forms from over suitable evaluation maps. We
show that such invariants are independent of any choices, and are compatible
with pullbacks, products, and tropical completion of Kuranishi categories.
In the case of a compact symplectic manifold, this gives an alternative
construction of Gromov--Witten invariants, including gravitational descendants.Comment: 60 pages. Final version to appear in Geometry and Topolog
Notes on exploded manifolds and a tropical gluing formula for Gromov-WItten invariants
Notes for a short lecture series, covering exploded manifolds, the moduli
stack of curves in exploded manifolds, and a tropical gluing formula for
Gromov-Witten invariants: a gluing formula providing a degeneration formula for
Gromov-Witten invariants in normal-crossing degenerations. I gave the original
lecture series in April 2016 at the Simons Center for Geometry and Physics at
Stonybrook. Video of the lectures is available on the SCGP website,
http://scgp.stonybrook.edu/video_portal/video.php?id=2595Comment: 17 page
Exploded Fibrations
Initiated by Gromov, the study of holomorphic curves in symplectic manifolds
has been a powerfull tool in symplectic topology, however the moduli space of
holomorphic curves is often very difficult to find. A common technique is to
study the limiting behavior of holomorphic curves in a degenerating family of
complex structures which corresponds to a kind of adiabatic limit. The category
of exploded fibrations is an extension of the smooth category in which some of
these degenerations can be described as smooth families.
The first part of this paper is devoted to defining exploded fibrations and a
slightly more specialized category of exploded torus fibrations. Later sections
contain the transverse interesction theory for exploded fibrations and some
examples of holomorphic curves in exploded torus fibrations, including a brief
discussion of the relationship between tropical geometry and exploded torus
fibrations. In the final section, the perturbation theory of holomorphic curves
in exploded torus fibrations is sketched.Comment: 39 pages, 16 figures, to appear in proceedings of the 13th Gokova
Geometry and Topology conferenc
Tropical gluing formulae for Gromov-Witten invariants
We prove two tropical gluing formulae for Gromov-Witten invariants of
exploded manifolds, useful for calculating Gromov-Witten invariants of a
symplectic manifold using a normal-crossing degeneration. The first formula
generalizes the symplectic-sum formula for Gromov-Witten invariants. The second
formula is stronger, and also generalizes Kontsevich and Manin's splitting and
genus-reduction axioms. Both tropical gluing formulae have versions
incorporating gravitational descendants.Comment: 41 page
De Rham theory of exploded manifolds
This paper extends de Rham theory of smooth manifolds to exploded manifolds.
Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality,
and integration along the fiber. The resulting cohomology theory is used to
define Gromov Witten invariants of exploded manifolds in a separate paper.Comment: 57 pages. v4: Post-publication update fixing an error in the
computation of compactly supported cohomology of a coordinate chart. An
additional assumption that a polytope was simplicial at infinity was
required. As a consequence, this additional assumption is also required for
the integration pairing to be a perfect pairing. See Appendix for further
detail
On the value of thinking tropically to understand Ionel's GW invariants relative normal crossing divisors
Ionel's GW invariants relative normal-crossing divisors appear different from
Gromov-Witten invariants defined using log schemes or exploded manifolds.
Appearances are, in this case, deceiving. I sketch the relationship between
Ionel's invariants and their exploded cousins using the example of the moduli
space of lines in the complex projective plane relative two coordinate lines.
Even in this simplest of examples, 13 different types of curves appear in
Ionel's compactified moduli space, but these different types of curves can be
understood in a unified and intuitive fashion using tropical curves.Comment: 16 pages, mostly picture
Gromov Witten invariants of exploded manifolds
This paper describes the structure of the moduli space of holomorphic curves
and constructs Gromov Witten invariants in the category of exploded manifolds.
This includes defining Gromov Witten invariants relative to normal crossing
divisors and proving the associated gluing theorem which involves summing
relative invariants over a count of tropical curves.Comment: 102 page
Tropical enumeration of curves in blowups of the projective plane
We describe a method for recursively calculating Gromov-Witten invariants of
all blowups of the projective plane. This recursive formula is different from
the recursive formulas due to G\"ottsche and Pandharipande in the zero genus
case, and Caporaso and Harris in the case of no blowups. We use tropical curves
and a recursive computation of Gromov-Witten invariants relative a normal
crossing divisor.Comment: 25 pages, 22 pictures. A talk with many more pictures, and a
Mathematica program computing these invariants is available on my website:
http://maths-people.anu.edu.au/~parkerb/publications.html. v2: minor
improvements in exposition, and updated reference
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