Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost toric fibrations on symplectic blow ups

Pranav Chakravarthy, Yoel Groman

Abstract

Given a symplectic 4-manifold with an almost toric fibration and a symplectic ball embedding whose image under the moment map is contained in an affine convex set R, we produce a symplectomorphism between the almost toric blow-up and the symplectic blow-up which is the identity on the pre-image of the complement of R. Furthermore, under a compatibility condition of the ball embedding with the boundary divisor, we show that the symplectomorphism can be chosen to preserve the induced symplectic log canonical divisors.

Almost toric fibrations on symplectic blow ups

Abstract

Given a symplectic 4-manifold with an almost toric fibration and a symplectic ball embedding whose image under the moment map is contained in an affine convex set R, we produce a symplectomorphism between the almost toric blow-up and the symplectic blow-up which is the identity on the pre-image of the complement of R. Furthermore, under a compatibility condition of the ball embedding with the boundary divisor, we show that the symplectomorphism can be chosen to preserve the induced symplectic log canonical divisors.

Paper Structure

This paper contains 13 sections, 9 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Symplectic and Almost-Toric Blow-up
  3. Definitions
  4. Symplectic blow-up
  5. Almost toric blow up
  6. Blow-ups of symplectic divisors
  7. Symplectic log Calabi Yau pairs
  8. Symplectic reduction along boundary components
  9. Global isomorphism for blow up toric varieties
  10. The $D$-compatible case
  11. The non-$D$-compatible case
  12. Adjustment near the boundary
  13. Proof of Theorem \ref{['tmMain']}

Key Result

Theorem 1.1

Let $R\subset B$ be an affine convex subset with an edge $e$ contained in $\partial_i B$, the $i$th edge of the boundary. Let $T(c)\subset R$ have $e$ as one of its edges. Let $R^o$ be the relative interior of $R$ and let $\iota:B^4(c)\to\mu^{-1}(R^o)$ be a symplectic ball embedding contained in $\m

Figures (1)

  • Figure 1: Almost toric blow-up

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1: Almost toric blow-up
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['globalCase']}
  • Proposition 3.3
  • Theorem 3.4
  • proof
  • ...and 8 more