# Axiomatic characterization of virtual fundamental classes?

There are several objects called "virtual fundamental classes." For example, certain Deligne-Mumford stacks, quasi-smooth derived schemes, etc. will admit a "perfect obstruction theory" as defined by Behrend-Fantechi and then the mojo of intrinsic normal cones produces a "virtual fundamental class" $[X] \in A_d(X)$ of virtual dimension $d$ in the Chow ring of the stack (or whatever) $X$.

Already here, there could be different perfect obstruction theories e.g. different morphisms $\varphi : E^\cdot \to \mathbb{L}_X$, $\varphi' : E'^\cdot \to \mathbb{L}_X$. How can we compare the virtual classes $[X]_{\varphi}, [X]_{\varphi'}$ coming from the two different perfect obstruction theories? Are they always the same?

Further, there are many approaches to defining virtual classes for moduli problems in symplectic geometry. There, one does not use perfect obstruction theories. Rather, one looks for "Polyfold structures" or "Kuranishi structures" which can be used to produce a homology class which is then called "the virtual fundamental class." I suppose one would like to compare the classes gotten from Polyfolds with the ones gotten from Kuranishi structure. And there are many appraoches to Kuranishi structure, so I suppose one would like to compare the virtual classes gotten from those approaches.

I apologise if there is a big well-known theorem stating asserting they are all equal. But even in that event, I think comparing the virtual classes in algebraic geometry with those in symplectic geometry is interesting.

And so, I ask: What makes these classes "virtual fundamental." Is there an axiomatic list of properties which either determines the thing or allows us to say general "model independent" things?

• Do you mean $\varphi$ and $\varphi'$ to have the same domain $E^\bullet$, or can they have different $E^\bullet$ and $E^{\prime\bullet}$? – John Pardon Dec 12 '17 at 21:07
• @JohnPardon Thank you for catching this. I do not see any reason to assume that $\varphi$ and $\varphi'$ have the same domain. – user100272 Dec 16 '17 at 18:46

Question 1 (compare virtual fundamental cycles of different perfect obstruction theories on space underlying space): There is essentially no relation between $[X]_\varphi$ and $[X]_{\varphi'}$ for different perfect obstruction theories $\varphi:E^\bullet\to\mathbb L_X$ and $\varphi':E^{\prime\bullet}\to\mathbb L_X$. The "derived" structure on $X$ encoded in $\varphi$ is essential for defining the vfc; knowing $X$ as a topological space (or variety, stack, etc.) determines essentially nothing about vfc (except trivial things like the fact that the vfc vanishes if the virtual dimension is larger than the classical dimension of $X$).

A very special case of your question is "Does the euler class $e(E)\in H^\bullet(M)$ of a vector bundle $E$ over a manifold $M$ depend only on $M$?" whose answer should be clear.

Question 2 (axiomatic characterization of virtual fundamental classes): I've thought extensively about this problem, and as far as I know, no axiomatic characterization of virtual fundamental classes/cycles/chains has been formulated and proved in the literature. There is definitely no satisfying general result which allows one to compare all reasonable approaches to defining virtual fundamental cycles in symplectic geometry. The philosophical reason why this seems like a difficult problem is that it's much easier to work "infinitesimally" in algebraic geometry than in differential geometry (or, at least, the sort of differential geometry relevant to moduli spaces of pseudo-holomorphic curves). Thus, all existing methods for defining the VFC in symplectic geometry "remember" much more of the ambient geometry of the entire space of smooth (as opposed to pseudo-holomorphic) maps than should be necessary for defining the VFC. The comparison between them is very technical because, although morally all approaches give rise to exactly the same VFC, we don't currently have a good language for recording the minimal amount of "derived" information that the moduli spaces carry (and which should be sufficient for defining the VFC).

Ideally, one would like to define some (derived?) moduli problem in the smooth or topological category for pseudo-holomorphic curves. Then one would like to show that this moduli problem is representable by a reasonable "derived topological manifold" (or orbifold) whose underlying topological space is the usual moduli space and whose derived structure is the analogue of a perfect obstruction theory. The last step (and probably the easiest, actually) is defining the VFC from this derived structure.

In my view, an axiomatic characterization of virtual fundamental classes is unlikely to be helpful with the question of comparing different constructions in symplectic geometry, unfortunately. This is simply because the "problem" is more than just having various ways of extracting the VFC, rather it's that we don't even know what the right canonical extra "derived" structure on the moduli space is from which we should extract the VFC. I'd be thrilled if I'm wrong, though!

I'll stop here, although it's possible to write endlessly on this topic. If you have other questions, I'm happy to expand this answer or answer a subsequent question you ask.

• Thank you for this detailed reply. And excuse my ignorance, but I thought the condition that the derived manifold has cotangent complex concentrated in degrees $-1$ and $0$ was the correct amount of derived structure for obtaining virtual fundamental classes. – user100272 Dec 16 '17 at 18:49
• @JacobGross: It is true that given a derived moduli problem, you can ask whether it admits a cotangent complex and whether it is of tor-amplitude $[-1,0]$, in which case it would admit a virtual fundamental class. However you need first of all to extend your classical moduli problem to a derived moduli problem in a reasonable way, which may not always be trivial at all. – AAK Dec 16 '17 at 19:26

Joyce has some work in this direction: in particular, he defines a functor from Deligne-Mumford $\mathbb{C}$-stacks with perfect obstruction theories to what he names 'd-orbifolds'. These should be thought of, essentially, as 'derived' $\mathbb{R}$-stacks. Since we are in characteristic 0, this means these spaces come with a homotopy sheaf of simplicial $\mathbb{R}$-algebras, or equivalently, commutative differential graded algebras over $\mathbb{R}$. I strongly suspect that from this perspective, one can define an actual quasi-smooth derived geometric stack from which one can obtain a virtual fundamental cycle. Indeed, one would need to work over the appropriate site, and the Joyce's work gives some indications on what one should do.