Tom Graber (2021) Localization for relative obstruction theories and stable log maps

These are notes following a lecture of Tom Graber on virtual localization. See the link here: recorded lecture.

Original Bott Residue Theorem (1950s)

Here's the original paper: The Moment Map and Equivariant Cohomology, Atiayh and Bott.

This is the algebraic geometry version of Bott's original paper; he would've taken a smooth compact manifold rather than a smooth projective variety and a $S^1$-action rather than a $\mathbb G_m = \mathbb C^*$-action.

Let $X$ be a smooth projective variety $X$ with a $T$-action. Here, $T$ is either $\mathbb G_m^r$ or $\mathbb G_m$; most of the time it will be the latter, and it is here. The differences are not drastic. Further let $E$ be a vector bundle over $X$ with a lift of the $T$-action (meaning the bundle map $E\to X$ is $T$-equivariant), $p$ a polynomial in the chern classes of $E$ (for instance $p = c_1^2 \cdot c_7$) with $\deg(p) = \operatorname{dim} X$. Then

where $\bigstar$ is some explicit formula depending on $E|_{X^T}$. Here $X^T$ is the fixed locus of the $T$ action on $X$. The representation theory of $\mathbb G_m$ is quite nice: it says that $E|_{X^T}$ splits as a direct sum of eigen bundles upon which $T$-acts by weight $i\in \mathbb Z$:

Let's explain this a bit more. The torus isn't acting on $X^T$, but it's still acting on $E$ over $X^T$. Because the torus action fixes $X^T$, it must act fiberwise on $E$; in other words, it preserves the fibers of $E$ over each point in $X^T$. The fibers of $E\to X$ are a vector space, so the action of $T$ on $E$ actually gives us a representation of $T$.

Every representation of $T = \mathbb G_m$ splits as a direct sum of eigen bundles $E_i$. Here, $E_i$ denotes the eigenbundle of $E|_{X^T}$ upon which $T$-acts by weight $i\in \mathbb Z$.

To compute $\int_X p$ by the Bott residue formula, I need to know $X^T$ and this direct sum decomposition of $E|_{X^T}$; in particular, the weights $i\in \mathbb Z$.

Case of the tangent bundle

The above discussion applies generally to any random vector bundle, and its not particularlly helpful for computing on a random vector bundles over $X$ since it may not have a $T$-equivariant structure. However, the tangent bundle $TX$does have a natural $T$-equivariant structure ($T$ is used both for the tangent space and the torus, pick out which one is meant from context). So we have a decomposition

Since $T$-acts trivially on $X^T$, it acts trivially on the tangent space to $X^T$:

Here $TX_0$ is the weight $0$ piece of $TX|_{X^T}$, sometimes denotes $TX^f$ for "fixed". The above statement thus means that the only directions in the tangent bundle which are fixed are no other directions in $TX|_{X^T}$ which are fixed besides those in $TX$. Note that we need smooth for this to be true.

This discussion means that all other directions not in $T_{X^T}$ are not fixed, hence the normal bundle $N_{X^T}$ of $X^T$ in $TX|_{X^T}$ consists of the sum of all weights other than $0$:

where "$m$" stands for "moving".

So $\bigstar$ depends on $c(E_i)$ and $c(N_{X^T/X})$.

The most useful place to use this is when $X^T$ consists of finitely many isolated fixed points, in which case $\int_{X^T}\bigstar$ just becomes a finite sum, and you've reduced a potentially nasty integral to a combinatorial expression in the chern classes $c(E_i)$ and the normal bundle $N_{X^T/X}$.

This formula can be proven and understood in terms of equivariant cohomology, or in terms of equivariant chow.

Equivariant Chow

Edin-Graham construct equivariant chow groups $A^T_X(X)$ which are formally similar to $A_*(X)$, in that they have

• functoriality (for $T$-equivariant morphisms)

• Chern classes (for $T$-equivariant bundles)

• intersection product for smooth $X$ (you get an equivariant Chow ring A^*_T(X))

• specialization map $A^T_*(X)\to A_*(X)$.

A novel feature of the theory, however, is that $A^*_{\mathbb G_m}(pt) = \mathbb Q[\lambda]$, where $A^T_{-i}(pt) = \mathbb Q\cdot \lambda^i$. This turns out to be a massive boon to this area of study. It implies that for any $X$ with a $T$-action, $A^T_*(X)$ has the natural structure of a $\mathbb Q[\lambda]$-module.

Localization Theorem and Bott-residue

To recover the Bott residue theorem, observe that

in $A^*_T(X^T) = A^*(X^T) \otimes \mathbb Q[\lambda].$ Because torus weights on $N$ are all non-zero, we get that $e(N)$ (the Euler class of $N = N_{X^T/X}$) is invertible in $A^*_T(X^T)$. Therefore, if we set $\alpha = \frac{1}{e(N)}$, then

the fundamental class in $X^T$. If I pull back the fundamental class I get back the fundamental class, so $i_*\alpha = [X]$. Localization tells us that this is the unique class in $A^*_T(X^T)$ which pulls back to the fundamental class of $[X]$. Thus,

where the last equality follows from the projection formula from intersection theory. Note that $i^*p$ is a polynomial in the equivariant Chern classes of $i^*E$.

Conclusion: $\int_Xp(E) = \int_{X^T}\frac{p(i^*E)}{e(N)}$.

Key point

If we can identify $\alpha\in A^*_T(X^T)$ such that $i_*\alpha = [X]$ then we can reduce integrals over $X$ to integrals over $X^T$. Proving a Bott-like residue theorem essentially boils down, therefore, to finding this class $\alpha$. It's not known how to do this in general for smooth $X$. However, here's a note:

We still have localization when $X$ is singular, but it's no longer obvious how to find this class $\alpha$. We can still say something about this case, however.

Deformation to the normal cone gives us a map $X^T\hookrightarrow C_{X^T/X}$. We can therefore factor the residue map $\operatorname{Res}:A^T_*(X)\to A^T_*(X^T)$ as

The residue map $\operatorname{Res}:A^T_*(C_{X^T/X})\to A^T_*(X^T)$ can be made sense of; the torus acts on $C_{X^T/X}$ and the fixed locus is still just $X^T$ and the "moving" part is the normal cone.

This doesn't necessarily improve anything, we just changed which residue map we have to calculate. However, if we have an embedding $C\hookrightarrow V$ of $C$ into some equivariant vector bundle (this often happens, it's like taking a resolution by locally free sheaves) then you can calculate the residue of $[C]$:

where $s_i(C)$ are the Segre classes at $C$. If we take $T=\mathbb G_m$ with the standard scaling action, then

Cone stacks

This construction also makes sense for cone stacks which admit $T$-embeddings in $T$-vector bundle stacks via the same formula. The reason this Segre class perspective is useful is because it still makes sense for cone stacks – even in situations where localization doesn't apply, the formula

still makes sense. In fact, we can simply define

to be some sort of "generalized Segre class associated to a cone" and in this way still have a means to associate a class on the base to the cone.

Comment from Ravi: to de-mystify cone stacks, one should view them as some sort of cone living in the quotient of vector bundles. That is, given a map of vector bundles $E_0 \to E_1$, you can mod out by the additive action of $E_0$ to get a stack $[E_1/E_0]$, and if you additionally have a cone in $E_1$ invariant under the action of $E_0$ then you obtain a "cone stack" inside of $[E_1/E_0]$.

We'll just think of a cone stack as a cone in some "vector bundle quotient" as Ravi suggests.

Fixed stacks

The fixed scheme makes sense, but the correct notion of "fixed locus of a group acting on a stack" is more subtle.

For a scheme: if you have $T$ actings on $X$, then you can think of the fixed locus $X^T$ in two different ways.

1. As the intersection $X^T = \bigcap_{t\in T}X^t$. Here we view $t\in T$ as a map $X\xrightarrow{t\cdot} X$, and then $X^t$ is the locus on $X$ where $t\cdot$ is the identity map. From this point of view, one might worry that if $X$ isn't separated then $X^T$ won't be closed, but because $X$ is connected it still is.

2. From the point of view of the functor of points: a map $S\to X^T$ corresponds to a $T$-equivariant map $S\to X$ where $S$ is equipped with the trivial $T$-action. This is the definition that works better for the fixed stack.

Danger: The natural map $X^T\to X$ is not a monomorphism for $T$ a torus, $X$ a DM stack and $X^T$ a closed substack.

Example: The stack of nodal curves

Take $\mathcal M_g$ to be the stack of nodal curves and let $T$ act trivially. Then $\mathcal M_g(S)$ is flat families of nodal curves over $S$. Surprisingly, using this definition of the fixed stack, the fixed family $\mathcal M_g(S)^T$ is the flat family of nodal curves over $S$ with a *fiberwise $T$-action, that is, you can still have non-trivial $T$-actions in the family.

Example: Quotient stack with a $T$-action

Take $X = [M/G]$ with a $T$-action induced by

1. The extension of a group $1 \to G\to \tilde{G} \to T \to 1$ and

2. an action of $\tilde{G}$ on $M$.

Then the fixed stack looks like the collection of conjugacy classes of splittings,

where $C(\varphi)$ denotes the centralizer. Note that this formula only works for tori!

Subexample

If $T$ acts trivially on $BG = [pt/G]$, then

Take $G = \mathbb{P}\operatorname{GL}_2$ then $BG\subset \mathcal M_0$ and we see that we have a copy of $B\mathbb G_m \subset BG^T$.

WARNING

$X^T$ need not be an algebraic stack. It already fails to be one for $\mathcal M_1$ and $T = \mathbb G_m$.

Main Theorem

Setup

We take

• $M$ to be a Deligne-Mumford stack

• $\mathcal Y$ an Artin stack (why do we take the base to be an Artin stack?)

• $M\xrightarrow{\pi} \mathcal Y$ a $T$-equivariant morphism (what does this mean for stacks?)

• $E \to L_{M/\mathcal Y}$ a perfect obstruction theory (what is a perfect obstruction theory?)

• Given this data, we get a normal cone to the map $\pi$: $C_{M/\mathcal Y} \hookrightarrow \mathbb E$ which embeds in the normal bundle stack $\mathbb E = [E_{-1} \to E_0]$.

All this data gives rise to the virtual pullback

This is important. You need to think about this virtual machinery in the way Christina Manolache describes (see Manolache, "Virtual pull-backs"). What this gives you isn't only a virtual class on $M$, it gives you a virtual pull-back – a way to pullback cycles from $\mathcal Y$ to $M$. Usually you pullback only the fundamental class of $\mathcal Y$, which you then call the virtual fundamental class, but we need to use the whole machinery to pullback other classes as well using the data of the normal cone stack $C_{M/\mathcal Y}\hookrightarrow \mathbb E$.

Case of a point (non-relative case)

When $\mathcal Y = pt$, then the virtual localization theorem (Graber, Rahul) says the fixed locus $M^T$ has a perfect obstruction theory given by $E^f$ and

General case (relative case)

In the relative case you don't get an "absolute" perfect obstruction theory, you only get a relative perfect obstruction theory given by $E^f$:

and you get a natural class $\alpha \in \widetilde{A}^T_*(\mathcal Y ^T)_\lambda$ by taking $s(C_{\mathcal Y^T/\mathcal Y})$.

We obtain the following formula:

It looks almost the same as before, the only difference is the class $\alpha$ on the base which plays a role. Tom Graber thinks of $\alpha$ as being the class in $\mathcal Y^T$ which "pushes forward to the fundamental class of $\mathcal Y$".

Some notes

• Because $M$ is a DM-stack and $T$ is a torus, $M^T$ is fine – it's a closed substack of $M$.

• With additional technical assumptions we can assume that $\mathcal Y^T$ is an algebraic stack, or at least that the "image" of $M^T\to \mathcal Y^T$ is an algebraic stack. This happens in all natural examples, claims Tom Graber, but certainly need not be the case always without the unspecified additional technical assumptions.

• $\mathcal Y^T$ is hideous, but we assume $M^T$ only hits nice stuff.

• The $s(C_{\mathcal Y^T/\mathcal Y})$ thing is the "generalized Segre class" we had earlier.

• The map $\pi^T_{!}$ is the virtual pullback

• Tom hints a little bit at some of the technical hypotheses:

  • you want $M^T\xrightarrow{\pi^T} \mathcal Y^T$ to factor through an open inclusion $U\hookrightarrow \mathcal Y^T$ so that $U$ is algebraic

  • to define the Segre classes $s(C_{\mathcal Y^T/\mathcal Y})$ Tom used some embedding of the cone in a vector bundle. You need this to exist, or at least for it to exist after some base change (he calls this hypoethesis "existence of resolutions").

• Before you start, you KNOW there is an answer – $M$ is Deligne-Mumford, the localization theorem works there and so you know there is something that pulls back to the virtual fundamental class, it's simply a questsion of identifying this class.

• In specific examples the main formula can be made sense of in more straightforward ways, you don't necessarily need to use deformation to the normal cone to make sense of it (says Graber).

Examples

Log stable maps

Take $X$ to be a projective log smooth scheme. You get a moduli space $M = \overline{M}_{g,n}(X,\beta)$ of log stable maps; if $S$ is a log scheme then

It is represented by a DM-stack with a log structure. Note that the functor of points for $M$ is hard to describe in generals schemes, but its easy for log schemes. $M$ comes with a perfect obstruction theory relative to a singular base $\mathcal Y$.

Etale locally, $\mathcal Y$ is of the form $[U/T_U]$, i.e. a toric variety modulo the action of its torus. Near a point, $\mathcal Y^T \to \mathcal Y$ looks like $T\to T_U$ and $U^T\subset U$.

General question: Given a toric variety $V$ and a subtorus $T\subset T_V$, can we compute $\operatorname{Res}([U])\in A^T_*(V^T)_\lambda$?