Tropical Calabi-Yau manifolds and tropical line bundles

An affine manifold is a real manifold with coordinate charts whose transition maps are in $\text{Aff}(\mathbb{R}^n)$.

We will call a tropical Calabi-Yau manifold a real manifold $B$ with a dense open subset $B_0\subseteq B$ which has an affine structure with transition maps in $\mathbb{R}^n\rtimes \text{GL}_n(\mathbb{Z})$, and such that $B\setminus B_0=:\Delta$ is a locally finite union of locally closed submanifolds of $B$.

It makes sense to call $B_0$ a tropical variety. Certainly $B_0$ locally looks like tropical affine space, and maps in $\mathbb{R}^n\rtimes \text{GL}_n(\mathbb{Z})$ look like maps defined by tropical monomials, so this seems natural. One can additionally talk about the sheaf of piecewise linear functions on $B_0$ with integral slope, or the sheaf of continuous functions on $B$ which restrict to piecewise linear functions on $B_0$ with integral slope. This should play the role of the structure sheaf.

Question (Sturmfels): Is it natural to call $B$ a tropical Calabi-Yau variety? In other words, do these singularities make sense in the tropical context? This is related to Zharkov's question of cutting tentacles.

Let $\text{Aff}(B,\mathbb{R})$ denote the sheaf of functions on $B$ which are continuous and restrict to affine linear functions with integral slope on $B_0$. We define a tropical line bundle to be an element of $H^1(B,\text{Aff}(B,\mathbb{R}))$. Representing an element by a Cech 1-cocycle

$(\alpha_{ij})$ for an open cover $\{U_i\}$, a section of this tropical line bundle is a collection of tropical functions $s_i$ on $U_i$ such that $s_i-s_j=\alpha_{ij}$. (Here this is ordinary subtraction).

We saw how sections of tropical line bundles over tori are tropical theta functions.

Question (Eisenbud, see also the question on Riemann-Roch): What is tropical Riemann-Roch?

The above discussion should go over to tropical varieties in general, if we have the right definitions. The same question applies.

Questions: What is the notion of an ample line bundle? Is it interesting to study embeddings into tropical projective space?

Exercise: Consider a tropical plane cubic, say

$$-6x^3-4x^2y-3xy^2-6y^3-4y^2z-3yz^2-0xyz-3x^2z-1xz^2-3z^3.$$

Draw a picture of this curve. Cut off the infinite rays, to get a polygon. The affine length of each edge is defined as follows. For the vertices of an edge, $v$ and $w$, write $v-w=l d$, where $d$ is a primitive integral vector and $l$ is a real number. Then the affine length is $\vert l\vert$. Check that the sum of the affine lengths of the edges is 13. Show this polygon can be obtained as an embedding $\mathbb{R}/13 \mathbb{Z} \rightarrow T \mathbb{P}^2$, using three tropical sections of a tropical line bundle of degree five.

Question: This seems a bit strange, doesn't it?

Observation: If one uses a line bundle of degree 3 to try to map to $T\mathbb{P}^2$, certain line segments in the circle will be contracted! Does this mean that the line bundle of degree 3 isn't very ample?

Given such a $B$, we can form two manifolds of twice the dimension, both torus bundles over $B_0$. Let $\Lambda\subseteq \mathcal{T}_{B_0}$ be a family of lattices in the tangent bundle generated locally by $\partial/\partial y_1,\ldots,\partial/\partial y_n$ where $y_1,\ldots,y_n$ are local affine coordinates on $B_0$. Because of the $\text{GL}_n(\mathbb{Z})$ restriction on transition functions, this is well-defined. Let $X(B_0)=\mathcal{T}_{B_0}/\Lambda$. This carries a complex structure which interchanges horizontal and vertical directions in the tangent bundle. Similarly, let $\check\Lambda\subseteq \mathcal{T}_{B_0}^*$ be the dual family of lattices generated by $dy_1,\ldots, dy_n$. Then we set $\check X(B_0)=\mathcal{T}_{B_0}^*/\check\Lambda$. This is canonically a symplectic manifold.

One particularly important question relevant for the Strominger-Yau-Zaslow conjecture is the following. We would like to find classical sections of tropical line bundles (i.e. smooth functions $(U_i,s_i)$ with $s_i-s_j=\alpha_{ij}$) satisfying the Monge-Ampère equation

$$\det(\partial^2 s_i/\partial y_j\partial y_k)=constant.$$

If one does this, then pulling back the functions $s_i$ to $X(B_0)$ will give Kähler potentials for Ricci-flat metrics.

Question (Gross, Siebert, Zharkov): Is there a tropical Monge-Ampère equation? Fixing the line bundle $\mathcal{L}$, can one find a sequence of sections $s_i\in\Gamma(B,\mathcal{L}^n)$ (hopefully satisfying this tropical form) such that $\hbar s_n$ converges to a classical solution of the equation. (Here $\hbar=1/n$). Write a computer program to produce numerical solutions in this way, and draw a picture of a genuine Ricci-flat metric!

Remark: There was some discussion of phases for complex patching during the conference. This can be interpreted as the $B$-field. See Gross' book with Joyce and Huybrechts for details, but the basic idea is that one can twist the standard complex structure on $X(B_0)$ with an element of $H^1(B_0, \Lambda\otimes\mathbb{R}/\Lambda)$. Under mirror symmetry, this element corresponds to what physicists call the $B$-field.

References: Thinking about Calabi-Yau manifolds in a tropical sort of way first arose in Kontsevich's and Soibelman's paper from 2000. For examples of tropical Calabi-Yau manifolds, see Gross' book with Joyce and Huybrechts, and the preprints of Haase and Zharkov. For more types of tropical varieties of this flavor, see Symington's work. For a general construction of tropical Calabi-Yau manifolds arising from degenerations of genuine Calabi-Yau manifolds, see Gross' recent paper with Siebert. This latter paper includes quite a bit on tropical Calabi-Yau manifolds (section 1) and gives applications to mirror symmetry.

(contributed by Mark Gross)

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