Tropical Calabi-Yau structures

An example of a tropical Calabi-Yau is the base of a Lagrangian fibered K3 surface. This is a sphere with, generically, an affine structure $\mathcal{A}$ on the complement of $24$ points where the singularity at each point has a structure specified by two features:

  1. The monodromy in the affine structure $\mathcal{A}$ along a simple loop around a singular point is conjugate to

    \begin{displaymath}\left(\begin{array}{cc}1&1\\ 0&1\end{array}\right)\end{displaymath}

    and
  2. there is an injective map $\Phi:(U-R,\mathcal{A})\rightarrow(\mathbb{R}^2,\mathcal{A}_0)$ where $U$ is a neighborhood of the singularity and $R$ is a ray based at the singular point. (Here the map $\Phi$ is assumed to be a local isomorphism of the affine structures $\mathcal{A}$ and $\mathcal{A}_0$.) The injectivity follows from an argument involving three-dimensional contact geometry.

A natural question is what closed surfaces admit such a singular affine structure, and how many singular points there can be on such a surface. In fact, the possibilities are: a torus or Klein bottle with no singular points, a sphere with $24$ singular points, or an $\mathbb{R} P^2$ with $12$ singular points. Each one can be realized as the base of a (singular) Lagrangian fibration. The singular fibers in each are diffeomorphic to the singular fibers in a genus one Lefschetz fibration, i.e. they are spheres with one positive self-intersection.


Question: What can one say about the geometry or topology of the set of tropical Calabi-Yau structures on $S^2$?


Remark: If one is willing to give up the second condition on the singular points, retaining only the monodromy constraint, then one can construct affine structures on $S^2$ with $12k$ singularities for any $k\ge2$.


Motivated by the moment map images of Kähler toric varieties, one can consider tropical manifolds that are not necessarily Calabi-Yau. Such a manifold would be built out of strata that are tropical Calabi-Yau manifolds with boundary that satisfy appropriate compatibility conditions. A simple example would be a cylinder equipped with an affine structure such that the boundary of the cylinder is an affine submanifold.


Question: Zharkov asked whether one can perform tropical Gromov-Witten calculations on a Calabi-Yau. Continuing on this line of thought, can one make such calculations on manifolds that have Lagrangian fibrations over these more general tropical manifolds? In particular, on $S^2\times T^2$ fibering over the cylinder?


(contributed by Margaret Symington)



Back to the main index for Amoebas and tropical geometry.