Compact tropical varieties, Monge-Amp\`{e}re equation, Calabi conjecture and curve counting

Background: We adapt Gross's definition of tropical Calabi-Yau manifolds as well as notations (see his contribution on tropical Calabi-Yau manifolds and tropical line bundles).

Compact tropical varieties. The natural question is how to make sense of compact tropical manifolds, not necessarily Calabi-Yau. There has to be a procedure of deleting pseudo-pods and leaving as much of affine structure as possible. My guess is that this will require a choice of polarization (tropical Kähler class). But the affine structure should not depend on this choice and has to be of purely algebro-geometric nature.

Question: How to modify naturally the valuation map for compact tropical varieties?

Tropical Monge-Ampère equation. Let me also add to Gross's question on Monge-Ampère equation. The beauty and importance of the real (and complex) MA equation is that given a boundary conditions it has a unique solution. This is crucial in proving various Calabi conjectures.

In the differential-geometric picture a metric is determined locally in an affine chart $U\subset B_0$ by the graph of the differential of a potential $dK\subset \mathbb{R}^n\times (\mathbb{R}^n)^*$ (we can consistently identify tangent spaces at different points in $U$ with $\mathbb{R}^n$ and cotangent spaces - with $(\mathbb{R}^n)^*$). In the tropical world the $C^\infty$ graphs should be replaced by piece-wise linear ones, which will define distribution-like metrics (or measures) on $B$. The Monge-Ampère condition - the equality of euclidean measures on $\mathbb{R}^n$ and $(\mathbb{R}^n)^*$ provided by the graph $dK$ - has to be understood in this distributional sense as well.

Question: (Tropical Calabi conjecture) Is there a way to define a tropical Monge-Ampère operator whose solution gives a (unique?) Monge-Ampère measure on $B$ in a given polarization class in $H^1(B,\text{Aff}(B,\mathbb{R}))$?

The uniqueness seems to be false in an obvious assumption that the bendings of the potentials are regulated by the integral lattice. On a torus this corresponds to several possible Voronoi cell decompositions in dimension higher than one.

Curve counting. The symplectic area of a straight line interval is easily seen to be proportional to the scalar product of the primitive vector along the interval and the distance vector between the end points in the dual structure. The behavior of a curve near the singular locus is very restrictive. Namely, it can end on a singular point only coming from a unique (eigen) direction.

Question: Given this can we perform a tropical Gromov-Witten calculation on a Calabi-Yau?

As was shown by Mikhalkin the tropical invariants coincide with the genuine ones for curves in surfaces. In higher dimensions, however, there are tropical curves which are not limits of true holomorphic curves.

Question: Is there a simple recipe deciding which tropical curves are the limits of classical ones?

(contributed by Ilia Zharkov)

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