Derived Relative Virtual Localization

Definitions.

• \(X_0\) a toric variety with toric log structure over the standard log point \(b_0\)
• \(M = \mathcal M(X_0/b_0, \beta) \) be the moduli space of basic stable log maps with class \(\beta = (A, g, u_{p_1}, ..., u_{p_n})\).
• \(\mathfrak M = \mathfrak M(\mathcal X, \beta') \) the moduli space of pre-stable log maps to the relative Artin fan \(\mathcal X_0 = b_0\times_B \mathcal X\) of \(X_0\) over \(b_0\). Curve classes don’t make sense in \(\mathfrak M\) so \(\beta' = (g, u_{p_1},...,u_{p_k})\) is \(\beta\) with the curve class removed.
• \(M^T\) is the torus fixed locus of \(M\), right now it is left ambiguous which version we are using but it will likely need to be the reparameterized homotopy fixed locus \(M^{rhT}\).
• We have the diagram

where \(T\) acts trivially on \(M^T\), all maps are \(T\)-equivariant and all stacks are over \(S\) (which need only be a \(1\)-Artin stack, but here is a field). Need to verify that all stacks in sight are \(1\)-Artin and that thefixed stack \(M^T\) is quasi-DM. Then we get a canonical homotopy \[(\pi\circ \iota)^!_T\simeq \iota^!\circ \pi^!\] of maps \(C_\bullet^{BM, T}(\mathfrak M/S)_{loc} \to C^{BM,T}_\bullet(M^T/S)\). Here we’re using Theorem B.5 in “Virtual Localization Revisted”.

1. We take the identity class \(1 \in C^{BM,T}_\bullet(\mathfrak M/S)_{\text{loc}}\)
2. We take the Gysin pullback to obtain the relative virtual fundamental class \(\pi^!(1) = [M/\mathfrak M]^{\text{vir}}\) (this is the definition of the relative virtual fundamental class, see “Virtual fundamental class for derived stacks I, Construction 3.6”).
3. We then take the composition with \(\iota^!\) and apply (1) above to get

\[\iota^!_T([M/\mathfrak M]^{\text{vir}}) = (\pi\circ \iota)^!(1) = [M^T/\mathfrak M]^{\text{vir}}.\]

1. When \(X^T\) is the homotopy fixed point scheme (or consequently a reparameterization) then pushforward along \(\iota\) induces an isomorphism

\[\iota_*:C^T_\bullet(M^{hT}_{ / \mathfrak M})_{\text{loc}}\to C_\bullet^T(M_{ / \mathfrak M})_{\text{loc}}\] hence there is some class \(\beta\) such that \[\iota_*(\beta) = [M/\mathfrak M]^{\text{vir}}\]

1. Applying the Gysin pullback operator \(\iota^!_T\) to both sides of this give

\[\iota^!_T(\iota_*(\beta)) = \iota^!_T\big([M/\mathfrak M]^{\text{vir}}\big) = [M^T/\mathfrak M]^{\text{vir}}\] by what we have already shown.

1. The left-hand side of the above can be written using the intersection formula \(\iota^!_T(\iota_*(\beta)) = e_T(N^{\text{vir}}) \cap \beta,\) so we get

\[e_T(N^{\text{vir}}) \cap \beta = [M^T/\mathfrak M]^{\text{vir}}\]

1. Inverting \(e_T(N^{\text{vir}})\) gives us

\[\beta =[M^T/\mathfrak M]^{\text{vir}} \cap e_T(N^{\text{vir}})^{-1}\] and then taking \(\iota_*(-)\) of both sides and using the definition of \(\beta\) yields \[[M/\mathfrak M]^{\text{vir}} =\iota_*\big([M^T/\mathfrak M]^{\text{vir}} \cap e_T(N^{\text{vir}})^{-1}\big)\]

We should use the decomposition formula from [Decomposition of Degenerate Logarithmic Gromov-Witten Invariants, Theorem 1.2], so we’ll work with the moduli space of basic stable log maps marked by a decorated type. See the disambiguation of moduli spaces of basic stable log maps