Talk:Stable map

Semipositivity

Perhaps I missed it, but it seems to me that the semi-positive condition has not been emphasized here. The pseudocycle construction requires the compactified moduli space to have "boundary" of co-dimension 2. In general, this is not true. One must put conditions on the Chern numbers of spheres that can bubble off, hence the semi-positivity condition used in e.g. McDuff-Salamon. SammyBoy 05:07, 10 April 2006 (UTC)

You are right. The simplest fix is to assume semipositivity. Perhaps we should say "under suitable/reasonable assumptions" the boundary can be shown to have codimension at least 2. Taken literally, this is a vacuous statement, so perhaps we should at least define semipositivity and explain how it's used? I'm never sure how much precision is desirable in an encyclopedia article as difficult as this. Your improvements are welcome. Joshua Davis 19:30, 10 April 2006 (UTC)

Sorry I disappeared for so long. How about under "some simplifying assumptions on the geometry of the symplectic manifold" ? I think you are right that too much precision is undesirable. On the other hand, the huge amount of work that has been done in trying to explain and formalize the virtual moduli cycle would seem like a waste of time without some precision. (This basically is the construction of something like a fundamental class of the moduli space of pseudoholomorphic spheres, even though the boundary strata may be of too high a dimension. I don't understand it, alas.) SammyBoy 07:49, 16 September 2006 (UTC)

Requested move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Not moved. The proponent did not make the reason for the move very clear. At least there should be a proper reference saying that 'space of stable maps' is the usual name. EdJohnston (talk) 04:55, 14 January 2014 (UTC)

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Stable map → Space of stable mapsSpace of stable maps – As far as I understand, the object of study is a "space of stable maps" (in fact, stack), which is a generalization of the moduli space of stable curves. Taku (talk) 12:58, 17 December 2013 (UTC)

I don't think that this is a really pressing issue, but I also have no objection. It's fine by me. Mgnbar (talk) 22:40, 17 December 2013 (UTC)

• Comment how are we going to find out where exactly Jesus' manger was if there's no WP article on stable maps? Red Slash 05:54, 19 December 2013 (UTC)
• Agree with Red Slash. --SmokeyJoe (talk) 04:33, 30 December 2013 (UTC)

• I also agree with Red Slash. Indeed, as Taku pointed out, there is no description about the relationship to the concept of "stack" in this article. It should be written in the headline or the paragraph Deligne-Mumford stable mapping space, I think. For example, "the space of stable mapping inevitably reaches to the idea of 'stack' by compactificacation." But their explanations may be difficult.--Enyokoyama (talk) 05:44, 30 December 2013 (UTC)
• I have no idea what's going on here. What's a stack? --BDD (talk) 19:16, 8 January 2014 (UTC)

See Stack (mathematics). But the concept is not easy. Mgnbar (talk) 03:20, 9 January 2014 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Definition?

The article does not seem to contain a definition of the term stable map. BSpringborn (talk) 09:27, 22 September 2023 (UTC)

The definition is in the paragraph beginning "In order to make this precise, define a stable map to be...". Admittedly this is pretty late in the article, compared to most math articles. Its placement could probably be improved. Mgnbar (talk) 10:36, 22 September 2023 (UTC)

Thanks! I think it is no problem if the definition comes after introductory explanations. But it should have a separate section with the heading "Definition". This seems to be the usual style for Wikipedia articles about mathematical terms. BSpringborn (talk) 10:51, 22 September 2023 (UTC)

Algebraic geometry perspective?

This article seems to exclusively treat the symplectic point of view. As someone learning about GW invariants and stable maps from the point of view of algebraic geometry I feel this side should also be represented here. Additionally, the current description seems to go into great technical detail straight away which I feel is not useful for those looking to learn about the topic. At the risk of this being a very long post, here's what I'd write for such a page (most of my information comes from the book Mirror Symmetry by Hori et al):

Informally speaking, a stable map is a morphism from a marked curve into a smooth projective variety, subject to a "stability" condition requiring the map to have only a finite number of automorphisms. This constraint is necessary for the construction of a good moduli space of stable maps.

Fix a smooth projective variety $X$ over $\mathbb {C}$, and two nonnegative integers $g,n$. Consider a nodal curve $C$ over $\mathbb {C}$ of genus $g$ and with $n$ marked points $p_{1},\ldots ,p_{n}$.  A morphism $f:C\to X$ is a stable map if every component of $C$ which is contracted by $f$ is itself a stable curve. That is, contracted genus 0 components must have 3 or more markings, and contracted genus 1 components must have at least 1 marking. We typically write $(C;p_{1},\ldots ,p_{n};f)$ for such a map.

We say that two stable maps $(C;p_{1},\ldots ,p_{n};f)$ and $({\tilde {C}};q_{1},\ldots ,q_{n};{\tilde {f}})$ are isomorphic if there is an isomorphism of curves $\tau :C\to {\tilde {C}}$ taking each $p_{i}$ to $q_{i}$, such that ${\tilde {f}}\circ \tau =f$. The above stability condition is then equivalent to saying that the group of automorphisms of $f$ is finite.

With this in mind, we may construct moduli spaces of stable maps. Let $\beta \in H_{2}(X)$ be a curve class. Then the space ${\overline {\mathcal {M}}}_{g,n}(X,\beta )$ consists of stable maps $(C;p_{1}\ldots ,p_{n};f)$ such that $f_{*}[C]=\beta$.

And so on...

Of course some of this information is already present, and I don't feel qualified to shuffle things around as I don't understand what's already written. Feel free to pick and choose whatever (if anything) looks good from the above for inclusion in this page. Splodge5 (talk) 17:21, 7 November 2025 (UTC)

I've gone ahead and added some of this, as well as some examples (and moved the definition to the top!) I'm aware its a big edit, so I hope it doesn't annoy anyone - I'm Being Bold. Splodge5 (talk) 13:20, 10 November 2025 (UTC)

How much detail is needed?

I've just rewritten several parts of the article and cleaned up others (new wording, consistent notation etc). This has, in my opinion, the effect of making the article more readable, but it has also involved me removing some of the more technical parts. In particular, the section on how to compactify the moduli space has now been turned into a section about bubbling and stabilisation of limits, and the section on GW pseudocycles has been removed entirely.

I had several reasons for this:

• The GW pseudocycles section should really be in the GW invariants article; moreover it mainly seems to belong to the symplectic side of things - algebro-geometric approaches favour using a virtual fundamental class
• With regards to the section on compactification, it is not needed for the algebraic point of view, and so seems out of place in the new form of the article, which tries to balance the two approaches
• Both of these sections, and several other places in the old version of the article, half-explained some complex ideas whilst hand-waving others away, or used ideas that were not clearly defined (e.g. "iteratively rescaling" a sequence of maps, "energy captured", "under suitable conditions", "Fano in a certain sense", "after a significant analytical argument [long list of results]", etc). I understand the wish for the article to be technically correct, and that it's not possible to include enough detail for everything to make perfect sense, but I feel the best approach is to offload long technical details onto dedicated articles or external sources. I have tried to do this by name-dropping the Gromov-Uhlenbeck compactification and by linking to the wiki article on the virtual fundamental class instead of aiming to explain either one here.

That said, I'm not a symplectic geometer, so it's possible that I've removed something that is vital to the symplectic side of things. If this is the case, I urge you to add it to the article, but bear in mind that anything here should ideally be accessible to symplectic and algebraic geometers alike. Felixhens (talk) 18:02, 31 January 2026 (UTC)

It's great that you've tried to strike a balance between the algebro-geometric and symplectic treatments. The article was heavily tilted toward the symplectic side for years.

Some hand waving is okay, because the reader can still get the broad outline of the subject here and then consult a technical text for the technicalities. But you're right, that there was a lot of hand waving in this article until now.

I'll try to proofread some of the symplectic text sometime, although I'm not an expert. I might want to re-install the GW pseudocycle into the symplectic section, because it really is about the topology of the moduli space of stable maps, whether that moduli space can be compactified by adding strata of sufficient codimension, etc.

Thanks for your work. Mgnbar (talk) 21:14, 31 January 2026 (UTC)

By all means, feel free to add back the (sub)section on pseudocycles - the motivation you give here does seem reason enough to include it and is probably worth mentioning in the article! I think the symplectic construction I've gone for is the same as the one in the old form of the article; I was working from Cox and Katz's book (Mirror Symmetry and Algebraic Geometry, section 7.2) which doesn't delve into too much detail on these constructions so there might be some facts worth mentioning that I didn't come across. I'd particularly be interested in how close the two constructions are, e.g. whether the compactified spaces coincide (for some choice of $\nu$?), and how many of the bullets in the "properties" section apply to both spaces. Felixhens (talk) 00:22, 2 February 2026 (UTC)