Exploded Manifolds

By Brett Parker. 2009

55 citations recorded by the source corpus.

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov-Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov-Witten invariants.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6c1b57a6f39aebe281a2bdce6edc7e84a3e7b031

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Linear series on metrized complexes of algebraic curves (2012, background): “As noted by Payne in [41], exploded tropicalizations can be thought of as an algebraic analogue of the “exploded torus fibrations” studied by Parker from a symplectic viewpoint [40].”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, methodology): “Gross and Siebert that have been outlined in [25, Appendix B]; their construction is motivated by applications to logarithmic Gromov–Witten theory in analogy with the tropical part of an exploded manifold in the sense of [40].”
Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations (2012, background): “The impact of the degeneration formula has been profound, but efforts to generalize the theory to other degenerations have until recently met with limited success (see [34, 33, 16, 20, 3] for recent progress).”
Stable logarithmic maps to Deligne-Faltings pairs I (2010, unknown): “s of deformations Gross and Siebert need in their mirror symmetry program [GS09, 2.2]. Adiﬀerent approachusingexploded manifoldstostudying holomorphiccurves was recently introduced by Brett Parker in [Par09a], [Par09b], and [Par09c]. It also aimed at deﬁning and computing relative and degenerated Gr”