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 $[\mathcal K]$ of any Kuranishi category $\mathcal K$ (which is a simplified, more general version of an embedded Kuranishi structure.) We also show how to integrate differential forms over $[\mathcal K]$ to obtain numerical invariants, and push forward differential forms from $\mathcal K$ 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