Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Question

Consider the following question: Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let $\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic submanifolds (of the right dimensions) in $X$. What is $N_{g,\beta}(\mu_1, \ldots, \mu_k)$, the number of genus $g$ curves in $X$ representing the class $\beta$ and intersecting the submanifolds $\mu_i$?

If $X:= \mathbb{P}^2$, then there is a complete formula for $N_{g,d}(p_1, \ldots, p_{3d-1+g})$, the number of genus $g$ degree $d$ curves through $3d-1+g$ generic points, using Tropical Geometry (and also using the Caporaso Harris formula). In fact I think if $X$ is any toric two dimensional variety there is a formula for $N_{g, \beta}$ using Tropical Geometry.

$\textbf{Question:}$ Is there a $\textit{Tropical geometric}$ solution for computing $N_{g,d}$ for $\mathbb{P}^n$ where $n>2$? Kontsevich's derivation of $N_{0,d}$, the number of degree $d$ rational curves in $\mathbb{P}^2$ generalizes for any
$\mathbb{P}^n$. More generally, is there a tropical geometric solution (or actually any solution) for $N_{g, \beta}$ on complex manifolds $X$ other than $\mathbb{P}^n$ (my question here primarily being for $g>0$, although I am interested in knowing what is known for $g=0$ as well.)

$\textbf{Remark:}$ One of the reasons for asking this question is as follows (the reason/hope might be very naive): I am just wondering if using any of these enumerative results one can make some partial (but direct) verification of some predictions made by Mirror Symmetry (particularly higher genus mirror symmetry predictions). To take one example, it is predicted in this paper (page 34)

http://arxiv.org/pdf/hep-th/0612125v1.pdf

that the number of degree $4$, genus $2$ curves on the quintic three fold is $534750$. Is it conceivable that there could be a direct way to see that?

$\textbf{Added Later:}$ My last comment (about degree $4$ genus $2$ curves on the quintic threefold) is incorrect as has been pointed out. I was very naively concluding that the numbers on page $34$ (at least the initial ones) were enumerative, which they are not (the authors don't claim that either). My hope that using enumerative formulas for genus g curves one can explain some of these BPS numbers is perhaps very naive. I am nevertheless interested to know if there are enumerative formulas for genus g curves into manifolds with dimension greater than two (irrespective of whether they have any use in explaining the numbers obtained from Mirror Symmetry).

Answer 1

Here is an attempt at an overview of tropical curve counts by someone who has been involved in the story for a while but certainly hasn't followed everything that has happened. I look forward to being corrected by others.

Toric surfaces are done: That's Mikhalkin's work and has since been redone in from many perspectives.

Genus zero curves in any toric variety are done: That's Siebert and Nishinou.

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If you want to get beyond these cases, even for toric varieties, there are two major problems. I'll talk about them, and then move to the non-toric case.

First, I want to talk about expected dimension. Fix a toric variety $X$ of dimension $n$ and a class $\beta \in H_2(X)$. We can think of $\beta$ as the list (possibly with multiplicities) of directions for the unbounded rays of the tropical curve. Call the length of this list $x$; in terms of $\beta$ and $X$, this is $\beta \cup c_1(X)$. The ``expected dimension" of the moduli space of curves is $D:=x-(n-3)g+3(n-1)$.

Now, consider a corresponding tropical curve $\Gamma$ which could come from a limit of genus $g$ degree $\beta$ curves. So $\Gamma$ has $x$ unbounded rays, and $b_1(\Gamma) \leq g$. Let $e$ be the number of internal edges of $\Gamma$. Then $e \leq x+3(b_1(\Gamma)-1)$, with equality if and only if $\Gamma$ is trivalent, and the space of deformations of $\Gamma$ is of dimension $\geq e-n(b_1(\Gamma)-3)$. (See section 5.1 in my thesis.) In the nice cases which are done, we always wind up studying cases where $g=b_1(\Gamma)$, $\Gamma$ is trivalent and the space of deformations of $\Gamma$ has dimension $e-n(b_1(\Gamma)-3)$. When all these things hold, $\Gamma$ has deformations of dimension $e-n(b_1(\Gamma)-3)=x+3(b_1(\Gamma)-1)-n(b_1(\Gamma)-3) = x+3(g-1)-n(g-3)=D$.

In general, all of these things can go wrong:

Superabundant curves The space of classical curves doesn't have to be $D$ dimensional. When it isn't, we say that the curves are super-abundant, and we have to refer to virtual fundamental classes. But, even if the space of classical curves is $D$-dimensional, the space of tropical curves need not be. For example (see my thesis again; I learned this example from a talk of Mikhalkin) there are genus $1$ tropical cubic curves in $\mathbb{P}^3$ which vary in a $13$ dimensional family, whereas classical cubics only have $D=12$ dimensions of freedom.

Most of these extra tropical curves can not be limits of classical curves, so one must find additional conditions to exclude them before on can hope to do enumerative geometry, or even prove finiteness results. I did some of this for genus $1$ and Katz and Bogart-Katz did some more, but I think we are very far from a complete answer.

We cannot reduce to maximally degenerate curves A tropical curve encodes the dual graph of a nodal curve. I'll say that the tropical curve is maximally degenerate if the nodal curve has no moduli, so $\Gamma$ is trivalent and $b_1(\Gamma) = g$.

In the cases $g=0$ or $n=2$, we can force this to be true by generic conditions to impose on the curve. Roughly speaking, in order for the tropical curve to obey $D$ many conditions, it must be able to vary in a $D$-dimensional family. Taking our curve not to be trivalent reduces $e$, and hence reduces the dimensionality with which our curve can vary. Taking $b_1(\Gamma)<g$, when $n=2$, increases $(n-3) b_1(\Gamma)$ and thus likewise reduces the variation.

However, when $g \geq 1$ and $n=3$, we can take $\Gamma$ trivalent but with $b_1(\Gamma) < g$, and still have $x+3(b_1(\Gamma)-1)-n(b_1(\Gamma)-3)$ come out the same. So there are $D$-dimensional families of tropical curves, corresponding to limits where there are moduli hidden in the stable curve. Life is even worse when $n \geq 4$: Then the $(n-3) b_1(\Gamma)$ term REWARDS taking $b_1(\Gamma) < g$.

These issues are why I got a bit depressed about tropical curve counting.

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Moving beyond the toric case: This project is being driven almost entirely by Gross and Siebert. (Maybe I should be listing Konsevich and Soibelman as well? I'm not sure. And Hacking and Keel are also involved, but my ignorant impression is that they are not thinking about the actual question of lifting curves back out of the tropical realm.) I don't feel competent to summarize how far they have gotten. They are THE people to talk to about trying to do things beyond the toric case. (And beyond abelian varieties, which other people have also worked on.) But I don't see how leaving the toric case could possibly make the above issues easier.