1. Log $K_\circ$ -theory

We amalgamate [ Reference HerrHer19 ], [ Reference QuQu18 ], [ Reference ManolacheMan08 ] to develop the basic properties of log virtual fundamental classes in $K_\circ$ -theory. We owe the reader reminders on K theory on algebraic stacks, (log) normal cones in intersection theory, log structures and Artin fans, etc. but this section should not be taken as an introduction to these topics.

A locally noetherian algebraic stack X has a lis-ét site $X_{lis-\acute{e}t}$ consisting of smooth X-schemes $T \to X$ with T affine, and étale X-maps $T' \to T$ between them. We define the categories $\textrm{QCoh}(X), \textrm{Coh}(X)$ of quasi-coherent and coherent sheaves to be cartesian sections of the stack of sheaves of modules, with or without finite generation hypotheses [ Reference OlssonOls07 ]. These are ordinary quasicoherent or coherent sheaves on the étale site if X is a scheme.

Define $K_\circ(X) \;:\!=\; K(\textrm{Coh}(X))$ to be the group with:

• generators: the classes [F] for F a coherent sheaf on the lis-ét site of X,

• relations: an equality $[G] = [F] + [H]$ for any exact sequence

  \begin{align*} 0 \to F \to G \to H \to 0\end{align*}
  of coherent sheaves on X.

See also [ Reference FultonFul98 , section 15·1], [ Reference QuQu18 , 1·3·2], [ Reference Hoyois and KrishnaHK19 , 3A]. This group was denoted $G^{naive}_0(X)$ and shown to coincide with the Thomason-Trobaugh definition of G-theory under the assumption $D_{qc}(X)$ is compactly generated in [ Reference Hoyois and KrishnaHK19 , lemma 5·5].

To use excision sequences, we need our stacks to be quasicompact and quasiseparated in the sense of [ Sta23 ]. The morphism $X \to {\mathcal{L}} Y$ on which all our definitions depend has non-quasicompact target ${\mathcal{L}} Y$ , but we can fix this:

Recall the intrinsic normal cone $C_{M/N}$ of a DM type map $M \to N$ defined by Behrend-Fantechi [ Reference Behrend and FantechiBF96 ] or in [ Reference ManolacheMan08 ]. If $M \to N$ is a closed immersion with ideal I, the normal cone is

\begin{align*} C_{M/N} = {\textrm{Spec}\:} \bigoplus_n I^n/I^{n+1}.\end{align*}

This is the affine cone over the exceptional divisor of the blowup of N at M. If $M \to N$ is smooth, the normal cone is the classifying stack of the tangent bundle:

\begin{align*} C_{M/N} \;:\!=\; BT_{M/N}.\end{align*}

If $M \to N$ factors through a closed immersion into affine space ${\mathbb{A}}^k_N$ , the normal cone is the stack quotient

\begin{align*} C_{M/N} \;:\!=\; \left[C_{M/{\mathbb{A}}^k_N}/T_{{\mathbb{A}}^k}|_{M}\right].\end{align*}

This is a hybrid of the smooth case $BT_{{\mathbb{A}}^k_N/N}$ and the closed immersion $C_{M/{\mathbb{A}}^k_N}$ part. After a smooth cover of N and étale cover of M, any DM type morphism of algebraic stacks locally of finite type over $\mathbb{C}$ can be written as a closed immersion into affine space:

\begin{align*} M \subseteq {\mathbb{A}}^k_N \to N,\end{align*}

and the normal cone is defined by gluing together the above definitions.

Write ${\widetilde{M}_{X/Y}}$ for the deformation to the normal cone for a DM type morphism of stacks $f\;:\; X \to Y$ as in [ Reference QuQu18 , section 1·2], where it is denoted by $M^{\circ}_f$ . This stack is a flat family over ${\mathbb{A}}^1$ with general fiber Y and fiber $C_{X/Y}$ over the origin $0 \in {\mathbb{A}}^1$ . As far as Chow groups are concerned, the classes of Y and $C_{X/Y}$ are equivalent.

Definition 1·2. Assume X, Y are quasiseparated and X is quasicompact. The log (intrinsic) normal cone of a DM type map $f\;:\;X \to Y$ is defined by

\begin{align*} {C_{X/Y}^\ell} \;:\!=\; C_{X/{\mathcal{L}} Y}. \end{align*}

A log perfect obstruction theory for f is an embedding ${C_{X/Y}^\ell} \subseteq E$ into a vector bundle over X. One similarly has a log deformation to the normal cone ${\widetilde{M}_{X/Y}^\ell} = {\widetilde{M}_{X/{\mathcal{L}} Y}}$ [ Reference QuQu18 ]. We think of each as X-stacks, writing ${C_{f}^\ell}|_T$ for the pullback ${C_{f}^\ell} \times_X T$ along a map $T \to X$ .

Pick a factorization $X \to U \to Y$ with $X \to U$ strict, $U \to Y$ quasiseparated and log étale as in Remark 1·1. Given a log perfect obstruction theory for f, define the log Gysin map or log virtual pullback as

\begin{align*} f^\dagger\;:\;K_\circ({\mathcal{L}} Y) \to K_\circ(U) \to K_\circ(X),\end{align*}

the composite of restriction to U and the Gysin map with no log structure for $X \to U$ [ Reference ManolacheMan08 ] using the obstruction theory $C_{X/U} \simeq {C_{X/Y}^\ell} \subseteq E$ . The log virtual fundamental class ${[X/Y]^{\ell vir}}$ is $f^\dagger[\mathcal{O}_{{\mathcal{L}} Y}]$ , if $[\mathcal{O}_{{\mathcal{L}} Y}]$ is defined. This operation and notation may similarly be extended to the $K_\circ$ -theory of log stacks over Y.

Intersection theory is the study of the cones $C_{M/N}$ and their classes in Chow groups. Now that we have log versions ${C_{M/N}^\ell}$ , we can define and work with log intersection theory analogously. We remind the reader how the normal cone fits into usual intersection theory:

Remark 1·3. The map $f^\dagger\;:\;K_\circ({\mathcal{L}} Y) \to K_\circ(X)$ does not depend on the choice of $U \subseteq {\mathcal{L}} Y$ . Any pair of such choices U, U ^′ intersect in a third U” and the normal cones do not change under étale modifications of the target.

The map ${\mathcal{L}} Y \to Y$ is locally of finite presentation [ Reference OlssonOls03 , theorem 1·1], so ${\mathcal{L}} Y$ is locally noetherian [ Sta23 , 01T6] and the map $i\;:\;U \subseteq {\mathcal{L}} Y$ is quasicompact [ Sta23 , 01OX]. The restriction $K_\circ({\mathcal{L}} Y) \to K_\circ(U)$ is consequently well-defined because pullback preserves coherence [ Reference OlssonOls07 , lemma 6·5].

The Gysin map $g^!$ without log structures for $g\;:\;X \to U$ is defined by the usual commutative diagram [ Reference QuQu18 , (0·2)]

Then $g^! = i^! \circ ``{(j^\ast)^{-1}}\hbox{''} \circ \pi^\ast$ . One composes with the inclusion and intersection with the zero section $K_\circ(C_{X/U}) \to K_\circ(E) \to K_\circ(X)$ to get a class in the $K_\circ$ -theory of X.

The above serves as a proxy for the analogous diagram with ${\mathcal{L}} Y$ in place of U because we aren’t aware of a proof of excision for non-quasicompact stacks. Remark that $C_{X/U} \simeq {C_{X/Y}^\ell} \simeq {C_{X/Y}^\ell} \times_Y U$ , ${\widetilde{M}_{X/U}} \simeq {\widetilde{M}_{X/Y}^\ell}|_U$ because $U \subseteq {\mathcal{L}} Y$ is open.

Examine the case of $f^\dagger[\mathcal{O}_V]$ , for V a log smooth U-stack. Take the fs pullback

An inclusion ${\widetilde{M}_{W/V}} \subseteq {\widetilde{M}_{X/Y}}|_W$ results, and witnesses $j^\ast(\mathcal{O}_{{\widetilde{M}_{W/V}}}) = \mathcal{O}_{V \times {\mathbb{A}}^1}$ . Thus $g^![\mathcal{O}_V] = i^![\mathcal{O}_{{\widetilde{M}_{W/V}}}] = [\mathcal{O}_{{C_{W/V}^\ell}}]$ , and virtual fundamental classes are again fundamental classes of log normal cones.

If Y is log smooth, ${[X/Y]^{\ell vir}}$ is independent of Y but not of the obstruction theory E. By this, we mean that a composite $X \to Y' \to Y$ with obstruction theories E for $X \to Y$ and E ^′ for $X \to Y'$ coming from two-term perfect complexes $\mathcal{E}^\bullet, \mathcal{E}'^\bullet$ with a compatibility datum

\begin{align*} \mathbb{L}^{\ell}_{Y'/Y}|_X \to \mathcal{E}^\bullet \to \mathcal{E}'^\bullet \to \end{align*}

will produce the same virtual fundamental class ${[X/Y]^{\ell vir}} = {[X/Y']^{\ell vir}}$ . The proof is as in [ Reference HerrHer19 , theorem 3·12]. We omit the proof but use the notation ${[X]^{\ell vir}}$ for ${[X/Y]^{\ell vir}}$ if Y is log smooth.

The log structure of a map $X \to Y$ is encoded in the map $X \to {\mathcal{L}} Y$ . The all-purpose log stacks ${\mathcal{L}}, {\mathcal{L}} Y$ encode any log structure whatsoever. Artin fans are more particular, individually tailored to the log structures of the stacks X, Y involved. By ranging over all possible Artin fans, we get étale covers of ${\mathcal{L}}, {\mathcal{L}} Y$ .

Let P be an fs sharp monoid and ${\mathbb{A}}_P = {\textrm{Spec}\:} \mathbb{C}[P]$ the corresponding affine toric variety. An Artin cone is the stack quotient of a toric variety by its dense torus:

\begin{align*} \mathscr{A}_{P} \;:\!=\; \left[{\mathbb{A}}_P/{\mathbb{A}}_{{P^{gp}}}\right].\end{align*}

The fs log stack $\mathscr{A}_{P}$ is like $``{\textrm{Spec}\:} P\hbox{''}$ for monoids:

\begin{align*} \textrm{Hom}_{fs}(X, \mathscr{A}_{P}) = \textrm{Hom}_{mon}(P, \Gamma(\overline{M}_X)).\end{align*}

Surprisingly, the stack $\mathscr{A}_{P}$ represents a strict-étale sheaf among fs log schemes. Put another way, $\mathscr{A}_{P} \to {\mathcal{L}}$ represents an étale sheaf on ${\mathcal{L}}$ -schemes.

Any fine, finite type log scheme X has a locally closed stratification $X = \bigsqcup X_i$ such that $\overline{M}_X|_{X_i}$ is locally constant. Suppose $\overline{M}_X|_{X_i}$ were actually constant, and write $\overline{M}_{X_i} = \Gamma(X_i, \overline{M}_X|_{X_i})$ . There are generization maps if $X_i \subseteq \overline{X}_j$ :

\begin{align*} \overline{M}_{X_i} \to \overline{M}_{X_j}.\end{align*}

If I is the category of strata with generizations $X_i \subseteq \overline{X_j}$ as morphisms, this defines a functor

\begin{align*} I \to (Mon)^{fine} \to {\mathcal{L}}_{\rm{\acute{e}t}}; \quad \quad \quad i \mapsto \overline{M}_{X_i} \mapsto \mathscr{A}_{\overline{M}_{X_i}}.\end{align*}

Taking the colimit as étale sheaves on ${\mathcal{L}}$ gives the Artin fan of X [ Reference Abramovich, Chen, Wise and MarcusACWM17 ]:

\begin{align*} \textrm{colim}_I \mathscr{A}_{\overline{M}_{X_i}} \to {\mathcal{L}}.\end{align*}

Artin fans generalize the “skeleton” (cone over the dualizing complex) of an s.n.c. divisor $D \subseteq X$ .

If $\overline{M}_{X_i}$ are locally constant but not constant, they entail monodromy representations of the strata $\pi_1(X_i)$ . Cover X instead by strict étale maps from schemes with no monodromy. Define $\mathscr{A}_{X}$ to be the colimit of the Artin fans of this cover, again as sheaves over ${\mathcal{L}}$ .

The same procedure was done with $\textrm{Hom}(P, \mathbb{R}_{\geq 0})$ or $\textrm{Hom}(P, {\mathbb{N}})$ in place of $\mathscr{A}_{P}$ in [ Reference Kempf, Knudsen, Mumford and Saint-DonatKKMSD73 ] and [ Reference Gross and SiebertGS11 ]. These constructions and their variants have gone by “generalized cone complexes,” “Kato fans,” “monoschemes,” “tropicalization,” etc. in the literature.

The Artin fan need not be functorial in X. See [ Reference Abramovich, Chen, Marcus, Ulirsch and WiseACM⁺15 , 5·4·1] for an example.

They define a relative Artin fan as a workaround. For a morphism $X \to Y$ , define $\mathscr{A}_{X/Y}$ similarly, using ${\mathcal{L}} Y$ in place of ${\mathcal{L}}$ above. The map $X \to \mathscr{A}_{X/Y}$ is strict and $\mathscr{A}_{X/Y} \to \mathscr{A}_{Y}$ is log étale. See [ Reference Herr, Molcho, Pandharipande and WiseHMPW23 ] for more on functoriality of the Artin fan.

All log schemes have a map $X \to \mathscr{A}_{X}$ to an object analogous to the fan of a toric variety. Fans are unions of cones that can be subdivided into further cones. These subdivisions induced normalized blowups of toric varieties, which can be pulled back to X. We define subdivisions of Artin fans analogously before pulling them back to get “log blowups” of our scheme or stack X. These toric operations are central to log geometry, and the results in this paper revolve around log blowups.

The representability condition merely excludes changing the lattice structure by root stacks. Connectedness precludes taking disjoint unions $\mathscr{C} \bigsqcup \mathscr{C} \to \mathscr{C}$ .

A map $\mathscr{B} \to \mathscr{C}$ is proper and representable if and only if points over $\mathscr{A}$ lift uniquely [ Reference Abramovich and WiseAW18 , theorem 2·4·1]

For toric varieties, points of the fans correspond to morphisms $\mathscr{A} \to \mathscr{B}$ . Scilicet, maps between toric varieties are proper when the map on fans is surjective.

Definition 1·5. A morphism $Y \to X$ is a log blowup if, étale locally on X, there is a morphism $X \to \mathscr{A}_{P}$ to an Artin fan and a projective subdivision $\mathscr{B} \to \mathscr{A}_{P}$ fitting into an fs pullback diagram

Since the condition is étale local on X, we could instead take pullbacks along maps to (affine) toric varieties $X \to {\mathbb{A}}_P$ of normalized blowups $B \to {\mathbb{A}}_P$ corresponding to subdivisions of the fan. One can equivalently ask that $X \to {\mathbb{A}}_P$ be strict or X be an atomic neighborhood by localizing further.

Log blowups are stable under fs pullback, since they are defined by fs pullback of a toric subdivision. Their fibers are the fibers of equivariant normalized blowups of toric varieties, but they need not be defined by any scheme-theoretic blowup of the base scheme X.

Log blowups are clearly projective, but more surprisingly log étale and monomorphic (in the category of fs log schemes). One can see $B \to {\mathbb{A}}^2$ from Example 1·6 is a monomorphism because the map on fans is an inclusion. Another way of saying this is that the fs fiber product $B \times^\ell_{{\mathbb{A}}^2} B$ is B, unlike the scheme-theoretic fiber product. The same is true of $E \to 0$ , that the fs fiber product is $E \times_0^\ell E \simeq E$ , while the scheme theoretic fiber product is $\mathbb{P}^1 \times \mathbb{P}^1$ .

Figure 1. The map on toric fans corresponding to the diagonal map ${\mathbb{A}}^1 \to {\mathbb{A}}^2$ . Subdividing along the image, the diagonal line in ${\mathbb{N}}^2$ , corresponds to blowing up ${\mathbb{A}}^2$ at the origin. The resulting map ${\mathbb{A}}^1 \to Bl_0 {\mathbb{A}}^2$ is strict.

Such a monomorphism would not be possible if it were not for the rank-two log structure on the origin 0. The log structure remembers the coordinates x, y from the embedding $0 \in {\mathbb{A}}^2$ even though the functions x, y both vanish at 0.

Monomorphisms $E \to 0$ are automatically log unramified, but log blowups are even log étale. The underlying map of schemes $\mathbb{P}^1 \to {\textrm{Spec}\:} \mathbb{C}$ is far from unramified, and the blowup $B \to {\mathbb{A}}^2$ is far from flat.

Log blowups can be used to make log smooth schemes and stacks smooth on the underlying space. As a first application, we see that the hypotheses “log smooth and equidimensional” for log virtual classes in [ Reference HerrHer19 ] were redundant.

For Chow groups on Artin stacks X, one needs to impose that X admits a stratification by quotient stacks. This guarantees that a vector bundle stack $E \to X$ has the same Chow groups as X

\begin{align*} A_*(X) \overset{\sim}{\longrightarrow} A_*(E).\end{align*}

Without this isomorphism, there can be no virtual fundamental classes and no intersection theory.

The analogous isomorphism in $K_\circ$ -theory is automatic:

Remark 1·8. Consider a short exact sequence

\begin{align*} E \to C \to D\end{align*}

of cone stacks over noetherian X. The pullback is an isomorphism:

\begin{align*} f^\ast\;:\;K_\circ(D) \simeq K_\circ(C).\end{align*}

The proof of [ Reference Hoyois and KrishnaHK19 , theorem 5·7] shows $G^{naive}(D) \simeq G^{naive}(C)$ for any vector bundle-torsor, and $K_\circ(\cdot)=G^{naive}_0(\cdot)$ . For example, the intersection with the zero section used to define $f^\dagger$ above is an isomorphism.

In view of [ Reference KhanKha20 , proposition 3·3, corollary 3·4], this is an application of [ Reference KhanKha20 , theorem 3·16] for perfect stacks X.

The proof of the product formula has two ingredients – Costello’s pushforward theorem and the compatibility of Gysin maps around a pullback square. The latter is simple in $K_\circ$ -theory.

Remark 1·10. Consider an fs pullback square of quasicompact and quasiseparated stacks

where f and q are endowed with log perfect obstruction theories ${C_{f}^\ell} \subseteq E$ , ${C_{q}^\ell} \subseteq F$ . The two composite Gysin maps are equal

\begin{align*} f'^\dagger q^\dagger = p^\dagger f^\dagger\;:\;K_\circ({\mathcal{L}} Y) \to K_\circ(X').\end{align*}

This is seen by replacing ${\mathcal{L}} Y, {\mathcal{L}} Y', {\mathcal{L}} X$ with quasicompact open substacks $U_Y, U_{Y'}, U_X$ containing the images of X ^′ and fitting into a commutative diagram

applying [ Reference Abramovich, Chen, Marcus, Ulirsch and WiseQu18 , proposition 2·5] to the pullback square, and [ Reference QuQu18 , proposition 2·11] to $X' \to U_X \times_{U_{Y}} U_{Y'} \to U_X$ , $X' \to U_X \times_{U_{Y}} U_{Y'} \to U_{Y'}$ .

2. Pushforward Theorems of Hironaka and Costello

Unlike the previous section, these pushforward theorems are not simple applications of ordinary intersection theory [ Reference QuQu18 ] to maps $X \to {\mathcal{L}} Y$ . Even for a proper birational morphism $q\;:\;Y' \to Y$ between log smooth stacks, one might not have $Rq_\ast \mathcal{O}_{{\mathcal{L}} Y'} = \mathcal{O}_{{\mathcal{L}} Y}$ . Indeed, a log blowup $Y' \to Y$ results in an open embedding ${\mathcal{L}} Y' \subseteq {\mathcal{L}} Y$ . Nevertheless, analogues of the pushforward theorems of Hironaka and Costello remain true. We also verify the “birational invariance” property of [ Reference Abramovich and WiseAW18 ] generalized in [ Reference HerrHer19 , theorem 3·10].

A DM type map $f\;:\;X \to Y$ of algebraic stacks is birational [ Reference Hassett and HyeonHH09 , definition A·1] if there is a dense open substack $V \subseteq Y$ whose preimage $f^{-1}V \subseteq X$ is dense and on which f restricts to an isomorphism. This definition coincides with the stacks project if f is locally of finite presentation [ Sta23 , 0BAC]. Birationality is smooth-local on the target and satisfies the “3 for 2” property:

If $X \overset{f}{\to} Y \overset{g}{\to} Z$ have composite $h = g \circ f$ and two of f, g, h are birational, so is the third.

Recall Hironaka’s pushforward theorem: a proper birational map $p\;:\;X \to Y$ of locally finite presentation with X, Y smooth or rational singularities satisfies $Rp_\ast \mathcal{O}_X = \mathcal{O}_Y$ [ Reference ChatzistamatiouCR15 , theorem 1·1], [ Reference Kollár and MoriKM98 , theorem 5·10], [ Reference HironakaHir64 , corollary 2 p. 153]. The reduction to rational singularities merely resolves the singularities of each and then resolves the map between them in such a way that $Rp_\ast \mathcal{O}_X = \mathcal{O}_Y$ . We remind the reader we work over $\mathbb{C}$ to use Hironaka’s pushforward theorem and the usual constructions of Gromov–Witten theory.

Lemma 2·1. Let $p\;:\;\widetilde{X} \to X$ be a log blowup of an fs log smooth algebraic stack X. Then the structure sheaf pushes forward to the structure sheaf

\begin{align*} Rp_\ast \mathcal{O}_{\widetilde{X}} = \mathcal{O}_X.\end{align*}

Proposition 2·3 (“Hironaka’s Pushforward Theorem”).Consider a proper birational morphism $p\;:\;X \to Y$ of DM type and locally finite presentation between log smooth algebraic stacks over $\mathbb{C}$ . Then

\begin{align*} Rp_\ast \mathcal{O}_X = \mathcal{O}_Y.\end{align*}

Corollary 2·4. In the setting of Proposition 2·3,pushforward identifies fundamental classes:

\begin{align*} p_\ast [\mathcal{O}_X] = [\mathcal{O}_Y] \quad \in K_\circ(Y).\end{align*}

Hironaka’s pushforward theorem details when fundamental classes are identified under pushforward. What about log virtual fundamental classes?

Proposition 2·5. Let $X \to F$ be a strict morphism of DM type to an Artin fan and $\widetilde{F} \to F$ a proper, birational, DM-type morphism. Write $p\;:\;\widetilde{X} \to X$ for the pullback

This setup allows $\widetilde{X} \to X$ to be a log blowup, a root stack, or composites of such.

Suppose $f\;:\;X \to Y$ is a morphism to a log smooth algebraic stack equipped with a log perfect obstruction theory ${C_{X/Y}^\ell} \subseteq E$ and equip $\widetilde{X} \to Y$ with the induced log perfect obstruction theory [ Reference HerrHer19 , remark 2·14,definition 3·1].Giving $\widetilde{X}$ the induced log perfect obstruction theory, we have

\begin{align*} p_\ast {[\widetilde{X}/Y]^{\ell vir}} = {[X/Y]^{\ell vir}} \quad \in K_\circ(X) . \end{align*}

Proof. Apply compatibility of pushforward and the Gysin map [ Reference QuQu18 ,proposition 2·4]and Corollary 2·4to the strict pullback square on Artin fans:

\begin{align*} p_\ast [\mathcal{O}_{C_{\widetilde{X}/\widetilde{F}}}] = [\mathcal{O}_{C_{X/F}}] \quad \in K_\circ(X).\end{align*}

Note that $C_{\widetilde{X}/\widetilde{F}} = {C_{\widetilde{X}}^\ell}$ , and the same for $X/F$ . Consider the map of short exact sequences of cone stacks

from [ Reference HerrHer19 , proposition 2·5]. Remark that ${C_{\widetilde{X}/Y \times \widetilde{F}}^\ell} \simeq {C_{\widetilde{X}/Y}^\ell}$ , etc. After pulling back the bottom row to $\widetilde{X}$ , the leftmost vertical map ${T^{\ell}_{Y}}|_{\widetilde{X}} \to {T^{\ell}_{Y}}|_X$ becomes an isomorphism. In any morphism of short exact sequences of cone stacks with an isomorphism on the leftmost arrow, the right square must be a pullback.

Remark 1·8 attests $r^\ast, \widehat{r}^\ast$ are isomorphisms, so the commutative square

coming from compatibility of pullback and pushforward and the equalities $r^\ast[\mathcal{O}_{{C_{X}^\ell}}] = [\mathcal{O}_{{C_{X/Y}^\ell}}]$ etc. imply

\begin{align*} \widehat{t}_\ast [\mathcal{O}_{{C_{\widetilde{X}/Y}^\ell}}] = [\mathcal{O}_{{C_{X/Y}^\ell}}]. \end{align*}

Composing with the inclusions into the obstruction theories and the Gysin map of the zero section of the obstruction theories, we get our result.

The rest of this section concerns Costello’s formula, which will be half of our proof of the log product formula. Recall that ${\mathcal{L}}^1$ parameterizes a pair of log structures with a map $M^1 \to M^2$ [ Reference OlssonOls05 , example 2·2].

Construction 3·6. Suppose $f\;:\;X \to Y$ is a DM type map between quasicompact log algebraic stacks and Y is log smooth. Quasicompact log algebraic stacks have smooth-locally connected log strata, so [ACWM17, Proposition 3.2.1, Proposition 3.3.2] produces an Artin fan $Y \to \mathscr{A}_{Y}$ and a relative Artin fan $\mathscr{A}_{X/Y}$ for the pair, both quasicompact:

The map $\mathscr{A}_{X/Y} \to \mathscr{A}_{Y}$ is log étale, W is log smooth, and $X \to W$ is strict. All the maps are DM type because Olsson showed ${\mathcal{L}}^1 \to {\mathcal{L}}$ is and the maps $\mathscr{A}_{Y} \to {\mathcal{L}}$ , $\mathscr{A}_{X/Y} \to {\mathcal{L}}^1$ are representable by construction. The Artin fan of a fine log algebraic stack is locally noetherian. The map $Y \to \mathscr{A}_{Y}$ is smooth and thus noetherian; the same argument shows W is noetherian.

Costello’s original pushfoward formula [ Reference CostelloCos06 , theorem 5·0·1] is incorrect as stated; see [ Reference Herr and WiseHW21 ]. We prove a $K_\circ$ -theoretic version of Costello’s corrected pushforward formula in pure degree one:

Theorem 2·7 (“Costello’s pushforward formula in $K_\circ$ -theory”).Consider an fs pullback square of DM type maps between locally noetherian, locally finite type log algebraic stacks over $\mathbb{C}$ :

Suppose f has a log perfect obstruction theory ${C_{X/Y}^\ell} \subseteq E$ and endow f ^′ with the induced log perfect obstruction theory ${C_{X'/Y'}^\ell} \subseteq {C_{X/Y}^\ell}|_{X'} \subseteq E|_{X'}$ . Assume Y ^′, Y are log smooth, X is quasicompact, and q is proper birational. Then

\begin{align*} p_\ast {[X'/Y']^{\ell vir}} = {[X/Y]^{\ell vir}} \quad \in K_\circ(X).\end{align*}

The schematic fiber product $W \times^{sch}_Y Y' \to W$ is proper and $W' \to W \times^{sch}_Y Y'$ is finite [ Reference OgusOgu18 , proposition III·2·1·5].

Birationality of s is smooth-local in W and Y, so assume both are affine schemes. Find a commutative square

with horizontal maps log blowups such that ${\widetilde{W}} \to {\widetilde{Y}}$ is integral [ Reference KatoKat21 , theorem]. Take fs pullbacks ${\widetilde{W}}' \;:\!=\; W' \times^\ell_W {\widetilde{W}}, {\widetilde{Y}}' \;:\!=\; Y' \times^\ell_Y {\widetilde{Y}}$ to obtain

The 3 for 2 property of birationality and [ Reference NiziołNiz06 , proposition 4·3] reduce us to showing ${\widetilde{s}}$ is birational and ensure that ${\widetilde{Y}}' \to {\widetilde{Y}}$ is. We conclude by observing the log étale and integral ${\widetilde{W}} \to {\widetilde{Y}}$ is flat [ Reference OgusOgu18 , theorem IV·4·3·5(1)].

Corollary 2·4 shows $s_\ast [\mathcal{O}_{W'}] = [\mathcal{O}_W]$ . Remark ${C_{X/Y}^\ell} \simeq C_{X/W}$ , so we have an ordinary obstruction theory for $X \to W$ . Commutativity of Gysin maps and pushforward [ Reference QuQu18 , proposition 2·4] gives

\begin{align*} p_\ast {[X'/W']^{vir}} = {[X/W]^{vir}}\end{align*}

and this translates to the statement above via the identifications ${C_{X/Y}^\ell} \simeq C_{X/W} \subseteq E$ , etc.

Remark 2·8 (Due to G.Martin).One might wonder whether the analogous equality

\begin{align*} p_\ast {[X'/Y']^{\ell vir}} = d \cdot {[X/Y]^{\ell vir}} \quad \in K_\circ(X)\end{align*}

holds for proper maps $q\;:\;Y' \to Y$ that are of pure degree d instead of birational. This is false even for $f = id_Y$ in ordinary K Theory.

Consider the k-algebra

\begin{align*} A = k[x_1, x_2, x_3]/(x_1^3 + x_2^3 + x_3^3)\end{align*}

and its spectrum $Z = {\textrm{Spec}\:} A$ . Let X be the minimal resolution of Z given by blowing up the ideal $(x_1,x_2,x_3)$ :

\begin{align*} X \;:\!=\; \textrm{Proj}\Big(A[y_1,y_2,y_3]/(x_1y_2-x_2y_1,x_1y_3-x_3y_1,x_2y_3-x_3y_2) \Big).\end{align*}

The composite $q\;:\;X \to Y = {\mathbb{A}}^2 = {\textrm{Spec}\:} k[x_1, x_2]$ of the blowup and the natural projection is of pure degree 3. Now compute $R^iq_\ast(\mathcal{O}_X) = H^i(\mathcal{O}_X)$ using the Čech complex of the open cover $U_i=(y_i \neq 0) \subset X, i = 1, 2, 3$ :

\begin{align*} 0 \rightarrow C^0(\mathcal{U},\mathcal{O}_X) \rightarrow C^1(\mathcal{U},\mathcal{O}_X) \rightarrow C^2(\mathcal{U},\mathcal{O}_X) \rightarrow 0,\end{align*}

with cohomology

\begin{align*} \begin{split}H^0(X,\mathcal{O}_X) &= k[x_1, x_2] \cdot (1 \oplus 1 \oplus 1), \\H^1(X,\mathcal{O}_X) &= k[x_1, x_2] \cdot \left( \frac{y_1^2}{y_2y_3}\oplus \frac{-y_2^2}{y_1y_3} \oplus \frac{y_3^2}{y_1y_2} \right), \\ H^2(X,\mathcal{O}_X) &=0.\end{split}\end{align*}

Conclude that

\begin{align*} Rq_\ast \mathcal{O}_X = [k[x_1, x_2]] - [k[x_1, x_2]] + 0 = 0 \neq 3 \cdot [\mathcal{O}_Y].\end{align*}

The structure sheaf $\mathcal{O}_Y$ generates the $K_\circ$ -theory of $Y = {\mathbb{A}}^2$ and cannot be zero.

3. Log $K_\dagger$ -theory

This section emerged from conversations with Sam Molcho and Jonathan Wise. We analyze the category of log blowups (Definition 1·5) of a base log algebraic stack X for the sake of intersection theory.

Write ${{{\textrm{Sub}_{X}}}}$ for the category of log blowups ${\widetilde{X}} \to X$ of a log algebraic stack X, for example if X is an Artin fan. Log blowups are monomorphisms among fs log stacks and fs fiber products preserve log blowups, so ${{{\textrm{Sub}_{X}}}}$ is a cofiltered preorder. All X-morphisms ${\widetilde{X}}_1 \to {\widetilde{X}}_2$ between log blowups of X are themselves log blowups.

Given a functor $p\;:\;I \to J$ and an element $j \in J$ , form the comma category $(p/j)$ :

Objects: objects $i \in I$ together with a morphism $p(i) \to j$

Morphisms: maps $i \to i' \in I$ such that the composite $p(i) \to p(i') \to j$ is the same as the fixed morphism $p(i) \to j$ .

The functor p is initial if the comma category is nonempty and connected for each $j \in J$ [ nLa20b ]. This is the dual to concepts variously called “final” and “cofinal” in the literature and has nothing to do with initial objects in functor categories. If p is initial and $f\;:\;J \to C$ any functor, the natural map

\begin{align*} \lim_J f \to \lim_{I} f \circ p\end{align*}

is an isomorphism. A subcategory is an initial system if the inclusion functor is initial.

A map $f\;:\;X \to Y$ of log algebraic stacks induces a functor $f^\ast\;:\;{{{\textrm{Sub}_{Y}}}} \to {{{\textrm{Sub}_{X}}}}$ sending a log blowup ${\widetilde{Y}} \to Y$ to ${\widetilde{Y}} \times^\ell_Y X \to X$ . If f is itself a log blowup, the map $f_!\;:\;{{{\textrm{Sub}_{X}}}} \to {{{\textrm{Sub}_{Y}}}}$ sending ${\widetilde{X}} \to X$ to the composite ${\widetilde{X}} \to Y$ is a section of $f^\ast$ and each is initial.

The key technical observation of this section is that $f^\ast$ is initial for f strict. This allows us to enrich log Gysin maps and proper pushforwards to a version of “log $K_\circ$ theory” we now introduce.

Definition 3·1. Define log $K_\dagger$ -theory of a log algebraic stack X

\begin{align*} K_\dagger(X) \;:\!=\; \lim \limits_{\widetilde{X} \to X} K_\circ(\widetilde{X})\end{align*}

as the inverse limit under pushforwards along log blowups of X. It has natural maps $K_\dagger(X) \to K_\circ(X)$ and $K_\dagger(X) \to K_\circ(\widetilde{X})$ for any log blowup $\widetilde{X} \to X$ .

Remark 3·2. Given a proper morphism $p\;:\;X \to Y$ , compose the natural map on limits with the levelwise pushforward

\begin{align*} \lim_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} K_\circ({\widetilde{X}}) \to \lim_{{\widetilde{Y}} \in {{{\textrm{Sub}_{Y}}}}} K_\circ({\widetilde{Y}} \times^\ell_Y X) \to \lim_{{\widetilde{Y}} \in {{{\textrm{Sub}_{Y}}}}} K_\circ({\widetilde{Y}})\end{align*}

to get a morphism $p_\ast\;:\;K_\dagger(X) \to K_\dagger(Y)$ . If ${{{\textrm{Sub}_{Y}}}} \to {{{\textrm{Sub}_{X}}}}$ is initial, f has a log perfect obstruction theory, and levelwise Gysin maps are compatible with pushforwards, we will also have log Gysin maps $f^\dagger\;:\;K_\dagger(Y) \to K_\dagger(X)$ . A similar definition $\lim_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} A_\ast({\widetilde{X}})$ for Chow groups was made in [ Reference Holmes, Pixton and SchmittHPS19 ].

Log $K^\dagger$ -theory can likewise be defined as the colimit under pullbacks of log blowups as in [ Reference Ito, Kato, Nakayama and UsuiIKNU20 ], [ Reference BarrottBar18 ] using $K^\circ$ -theory. One could instead take Gysin maps as the transition morphisms, provided you require the total spaces to be smooth and use the canonical obstruction theory of an l.c.i. There are natural variants taking (co)limits over log blowups as well as root stacks.

The idea of log cohomology theories is always to invert log étale morphisms somehow, either through these limits and colimits or through the topology. There are analytic spaces (the valuativization and the Kato-Nakayama space) defined by taking the limit of all possible log blowups or root stacks of a scheme X, and the log Betti and log de Rham cohomology are defined through these spaces. Taking these limits is meant to define the log $K_\circ$ -theory or log Chow groups for X the same way.

Definition 3·4. Suppose $f\;:\;X \to Y$ is equipped with a log perfect obstruction theory and Y is log smooth. Define the log virtual fundamental class in $K_\dagger$ -theory to be the sequence

\begin{align*} ({[{\widetilde{X}}/Y]^{\ell vir}})_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} \quad \in K_\dagger(X).\end{align*}

Proposition 2·5 verifies this sequence lies in the limit $K_\dagger(X)$ . One can similarly define $f^\dagger[\mathcal{O}_V]$ for log smooth stacks $V \to Y$ but we don’t define this operation in full generality on $K_\dagger$ .

Lemma 3·5. Suppose $E \to F$ is a strict map of quasicompact Artin fans and ${\widetilde{E}} \to E$ a subdivision. There exists a subdivision ${\widetilde{F}} \to F$ such that the pullback refines ${\widetilde{E}}$ :

The construction of [ Reference Scherotzke, Sibilla and TalpoSST18 , proposition 4·6] could also be done by taking star subdivisions at all possible rational rays of the subcone $\mathscr{A}_{\sigma}$ . Star subdivision along one ray and then another differs from the opposite order, but there is a mutual refinement.

Proposition 3·8. Suppose a quasicompact DM type morphism $f\;:\;X \to Y$ of log algebraic stacks with Y quasicompact induces epimorphisms

\begin{align*} \overline{M}_{Y, f(\overline{x})} \to \overline{M}_{X, \overline{x}}\end{align*}

on stalks at geometric points $\overline{x} \to X$ . There is a log blowup ${\widetilde{Y}} \to Y$ such that the fs pullback

has ${\widetilde{f}}$ strict.

Proof. Case: first suppose X, Y are schemes.

Locally on Y, one chooses a log blowup making $X \to Y$ ${\mathbb{Q}}$ -integral [ Reference OgusOgu18 , theorem III·2·6·7]. We can refine these local log blowups by a global one using quasicompactness of Y and assume $f\;:\;X \to Y$ is ${\mathbb{Q}}$ -integral, in particular exact [ Reference Illusie, Kato and NakayamaIKN05 , remark pg. 58]. The stalks of the characteristic monoids for a log blowup are epimorphisms, so the property of epimorphic stalks is preserved.

Exactness entails a pullback

with horizontal arrows both epimorphisms. Epimorphisms of groups are surjections [ nLa20a ], so the top horizontal arrow is a surjection. Exactness ensures it is also an injection, hence an isomorphism.

General Case: We now reduce the general statement for log stacks to the case of schemes. Cover Y by a strict smooth map $V \to Y$ from a quasicompact scheme and choose a blowup ${\widetilde{V}} \to V$ pulled back from $\mathscr{A}_{V}$ such that is strict. Find a log blowup ${\widetilde{Y}} \to Y$ refining ${\widetilde{V}} \to V$ using Lemma 3·7 to conclude.

The hypotheses of Proposition 3·8 include strict maps, log blowups, root stacks, and generally proper monomorphisms of log schemes. The diagonal $\Delta\;:\;X \to X \times X$ of any log algebraic stack is eventually strict.

Definition 3·10. If $f\;:\;X \to Y$ satisfies the hypotheses of Proposition 3·8 and furthermore is equipped with a log perfect obstruction theory ${C_{f}^\ell} \subseteq E$ , we can define $f^\dagger$ on $K_\dagger$ -theory as the composite:

\begin{align*} K_\dagger(Y) \;:\!=\; \lim_{{\widetilde{Y}} \in {{{\textrm{Sub}_{Y}}}}}K_\circ({\widetilde{Y}}) \overset{{\widetilde{f}}^\dagger}{\to} \lim_{{\widetilde{Y}} \in {{{\textrm{Sub}_{Y}}}}} K_\circ({\widetilde{Y}} \times^\ell_Y X) \simeq \lim_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} K_\circ({\widetilde{X}}) \;=\!:\; K_\dagger(X).\end{align*}

The first map is induced from the levelwise Gysin maps on ${\widetilde{f}}\;:\;{\widetilde{Y}} \times^\ell_X Y \to {\widetilde{Y}}$ , while the isomorphism comes from Corollary 3·9 showing $f^\ast\;:\;{{{\textrm{Sub}_{Y}}}} \to {{{\textrm{Sub}_{X}}}}$ is initial.

We now enrich the log Costello Theorem 2·7 and compatibility of log virtual classes with log Gysin maps to our log $K_\circ$ -theory $K_\dagger$ .

Proposition 3·11. Consider an fs pullback square

of DM type maps between log algebraic stacks with Y ^′, Y log smooth. Endow f with a log perfect obstruction theory ${C_{f}^\ell} \subseteq E$ and equip f ^′ with the induced log perfect obstruction theory ${C_{f'}^\ell} \subseteq {C_{f}^\ell}|_{X'} \subseteq E|_{X'}$ .

1. (1) If q is proper birational and X is quasicompact, then

   \begin{align*} p_\ast {[X'/Y']^{\ell vir}} = {[X/Y]^{\ell vir}} \quad \in K_\dagger(X).\end{align*}

2. (2) Suppose $q\;:\;Y' \to Y$ satisfies the hypotheses of Proposition 3·8.Endow q with a log perfect obstruction theory ${C_{q}^\ell} \subseteq F$ . Then

   \begin{align*} q^\dagger {[X/Y]^{\ell vir}} = {[X'/Y']^{\ell vir}} \quad \in K_\dagger(X').\end{align*}

Proof. The map

\begin{align*} t\;:\;K_\dagger(X') \to \lim_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} K_\circ({\widetilde{X}} \times^\ell_X X')\end{align*}

sends the sequence ${[X'/Y']^{\ell vir}} \;:\!=\; ({[{\widetilde{X}}'/Y']^{\ell vir}})_{{\widetilde{X}}' \in {{{\textrm{Sub}_{X'}}}}}$ to the sequence $({[{\widetilde{X}} \times^\ell_X X'/Y']^{\ell vir}})_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}}$ . Costello’s pushforward Theorem 2·7 shows the levelwise pushforward sends this sequence to the $({[{\widetilde{X}}/Y]^{\ell vir}})_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}} \;=\!:\; {[X/Y]^{\ell vir}} \in K_\dagger(X)$ .

Similarly, one argues the levelwise Gysin maps send $({[{\widetilde{X}}/Y]^{\ell vir}})_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}}$ to $({[{\widetilde{X}} \times^\ell_X X'/Y']^{\ell vir}})_{{\widetilde{X}} \in {{{\textrm{Sub}_{X}}}}}$ by applying Remark 1·10 and then $t^{-1}$ sends this to $({[{\widetilde{X}}'/Y']^{\ell vir}})_{{\widetilde{X}}' \in {{{\textrm{Sub}_{X'}}}}} \;=\!:\; {[X'/Y']^{\ell vir}}$ .

Remark 3·12. Write $s\;:\;K_\dagger(X) \to K_\circ(X)$ , etc. for the natural projections. Under the assumptions of Proposition 3·11,

\begin{align*} s q^\dagger {[X/Y]^{\ell vir}} = {[X'/Y']^{\ell vir}} = q^\dagger s {[X/Y]^{\ell vir}},\end{align*}

\begin{align*} s p_\ast {[X'/Y']^{\ell vir}} = {[X/Y]^{\ell vir}} = p_\ast s {[X'/Y']^{\ell vir}}.\end{align*}

In this sense, the operations $q^\dagger, p_\ast$ defined variously on $K_\dagger$ -theory or $K_\circ$ -theory are compatible.

4. The Log Product Formula

Let V, W be log smooth quasiprojective schemes throughout this section. Write ${\overline{M}_{g, n}} \subseteq {\mathfrak{M}_{g, n}}$ for the open substack of stable curves in the moduli space of all genus-g, n-marked nodal curves. Equip each with the divisorial log structure from the singular locus, an s.n.c. divisor. The stack $\overline{M}_{g, n}^\dagger(V)$ of log stable maps sends an fs log scheme T to diagrams of fs log schemes

where C is an fs log smooth curve over T with genus g and n-markings such that the underlying diagram of schemes is a stable map of curves. We emphasize that $\overline{M}_{g, n}^\dagger(V)$ denotes the stack “ $\mathscr{M}(V)$ ” of Gross-Siebert [ Reference Gross and SiebertGS11 ] parametrizing log stable maps and not the ordinary space of stable maps. Ordinary Gromov–Witten theory would require V, W smooth and not merely log smooth.

The stack ${\mathfrak{D}}$ consists of diagrams $(C' \leftarrow C \rightarrow C'')$ of genus g, n-pointed prestable curves over T such that $C \to C' \times C''$ is stable. Equivalently, no unstable rational component of C is contracted under both projections.

We have an fs pullback square:

(1)

One lemma and the study of this diagram in [ Reference HerrHer19 ] suffice to achieve the log product formula.

the natural inclusion ${C_{f'}^\ell} \subseteq {C_{f}^\ell}|_{X'}$ is an isomorphism.

Write

as in Construction 2.6 and take the fs pullback of the factorization:

Then $X \to W$ is strict, $W \to Y$ is log étale, and we have a map of short exact sequences of cone stacks

Log flatness of q ensures the leftmost arrow becomes an isomorphism after pullback to X ^′ [ Reference OlssonOls05 , 1·1 (iv)], so the right square of cones is cartesian. It suffices to show the map of ordinary normal cones is an isomorphism. Note $W' \to W$ is log flat and integral, hence it is flat [ Reference OgusOgu18 , theorem IV·4·3·5(1)] and the result is standard.

Theorem 4·3 (“The log product formula”).The log product formula holds in $K_\circ$ -theory as well as $K_\dagger$ -theory: the classes

\begin{align*} h_\ast {[\overline{M}_{g, n}^\dagger(V \times W)]^{\ell vir}} = \Delta^\dagger {[\overline{M}_{g, n}^\dagger(V) \times \overline{M}_{g, n}^\dagger(W)]^{\ell vir}}\end{align*}

are equal in $K_\circ(Q)$ as well as $K_\dagger(Q)$ .

Proof. We claim both sides of the equality compute ${[Q/Q']^{\ell vir}}$ as in [ Reference BehrendBeh99 ]. Remark 3·12 shows it suffices to prove the equality in $K_\dagger(Q)$ .

Note that $\phi$ induces surjective maps

\begin{align*} \overline{M}_{{\mathfrak{M}_{g, n}} \times {\mathfrak{M}_{g, n}}, \phi(\overline{x})} \to \overline{M}_{Q', \overline{x}}\end{align*}

at geometric points $\overline{x} \to Q'$ , so $\phi^\dagger$ is well-defined on $K_\dagger$ -theory as in Definition 3·10. Proposition 3·11 refines Costello’s Formula 2.7 and the compatibility we’ll need of Remark 1·10 to $K_\dagger$ -theory. In particular, $\phi^\dagger$ and $\nu_\ast$ both preserve log virtual fundamental classes:

\begin{align*} \phi^\dagger {[\overline{M}_{g, n}^\dagger(V) \times \overline{M}_{g, n}^\dagger(W)]^{\ell vir}} = {[Q/Q']^{\ell vir}}\end{align*}

\begin{align*} \nu_\ast {[\overline{M}_{g, n}^\dagger(V \times W)]^{\ell vir}} = {[Q/Q']^{\ell vir}}.\end{align*}

Combining these with the equalities ${[\overline{M}_{g, n}^\dagger(V \times W)]^{\ell vir}} = {[\overline{M}_{g, n}^\dagger(V \times W)]^{\ell vir}}$ and $\phi^\dagger = \Delta^\dagger$ from Remark 4·2 gives the result.

5. Relative variants and a Counterexample

We offer a relative variant of Theorem 4·3 as in [ Reference Lee and QuLQ18 , section 2·4]. This variant makes it easier to see the necessity of $\Delta^\dagger$ instead of $\Delta^!$ in Theorem 4·3. We end with a counterexample to the version of the relative variant where $\Delta^!$ replaces $\Delta^\dagger$ , justifying our technology.

We do not mean “relative” as in stable maps relative to a divisor, but to families of targets. The target V will be a log smooth quasiprojective map $V \to S$ to a base. Relative log stable maps to $V \to S$ over a base T are commutative squares

where $C \to T$ is a family of prestable log curves and the square has finitely many automorphisms restricting to the identity on V, S, and T. Log stable maps to $V/S$ are parameterized by a stack $\overline{M}_{g, n}^\dagger(V/S)$ of DM type over S. These include rubber log stable maps, which will give our counterexample in Section 5·2.

Example 5·1. Let $V \to S$ be the map ${[}\mathbb{P}^1/\mathbb{G}_m{]} \to B\mathbb{G}_m$ obtained from $\mathbb{P}^1 \to {\textrm{Spec}\:} \mathbb{C}$ by modding out by $\mathbb{G}_m$ . We claim that maps to $V/S$ are the same as rubber stable maps without expansions.

Rubber stable maps are equivalence classes of maps $C \to \mathbb{P}^1$ up to an isomorphism which need not fix $\mathbb{P}^1$

The automorphism of $\mathbb{P}^1$ must be action by a unit. Stackifying this equivalence relation in the base T of the family of curves means that the unit $\mathbb{P}^1 \overset{\sim}{\longrightarrow} \mathbb{P}^1$ must form a cocycle. Thus T comes equipped with a line bundle $T \to B\mathbb{G}_m$ and the curve C maps to its associated projective bundle $T \times_{B\mathbb{G}_m} {[}\mathbb{P}^1/\mathbb{G}_m{]}$ . This coincides with stable maps to ${[}\mathbb{P}^1/\mathbb{G}_m{]} \to B\mathbb{G}_m$ .

5·1. Relative variants.

Let $V \to S$ , $W \to T$ be log smooth, quasiprojective maps. We allow V, W, S, T to be algebraic stacks as long as $V \to S$ , $W \to T$ are representable by schemes; the ensuing fibers of $\overline{M}_{g, n}^\dagger(V/S) \to S$ are stable maps to schemes. Our counterexample uses relative stable maps to ${[}\mathbb{P}^1/\mathbb{G}_m{]} \to B\mathbb{G}_m$ .

Form the analogue of Diagram (1):

(2)

Remark 5·2. Log stable maps $C \to V$ have more discrete data than simply genus g and marked points n. They also have contact orders at the marked points and the curve class $\beta$ of their image.

Let $\Gamma$ contain the genus g, the number n of marked points, and the contact orders of a stable map $C \to V \times W$ over $S \times T$ . This induces discrete data $\Gamma', \Gamma''$ describing the maps $C' \to V$ , $C'' \to W$ obtained by stabilization.

One way to bundle this data together is in the Artin fan $\mathscr{A}_{\overline{M}_{g, n}^\dagger(V/S)}$ . The map from a log stack X to its Artin fan is geometrically connected by construction, but not necessarily surjective if X is not log smooth. This induces an injection from the connected components of log stable maps to those of its Artin fan, which is more closely related to tropical curves.

Given $\Gamma', \Gamma''$ discrete data for maps to V, W, there is at most one $\Gamma$ for maps to $V \times W$ that induces $\Gamma'$ and $\Gamma''$ . If g, n differ in $\Gamma'$ , $\Gamma''$ , there are none.

Theorem 5·3 (=4.3^′).Fix discrete data $\Gamma$ for a stable map $C \to V \times W$ over $S \times T$ as in Remark 5·2.The log product formula holds in $K_\dagger$ -theory and $K_\circ$ -theory:

\begin{align*} h_\ast {[\overline{M}_{\Gamma}^\dagger(V \times W/S \times T)]^{\ell vir}} = \Delta^\dagger {[\overline{M}_{\Gamma'}^\dagger(V/S) \times \overline{M}_{\Gamma''}^\dagger(W/T)]^{\ell vir}}\end{align*}

in $K_\dagger(Q)$ and $K_\circ(Q)$ .

Proof. Omitted. For the discrete data, the normal cones themselves ${C_{X/Y}^\ell}$ respect disjoint unions in the source.

5·2. Counterexample to the ordinary log product formula

We claim the “ordinary product formula” with $\Delta^!$ in the place of $\Delta^\dagger$ is false:

\begin{align*} h'_\ast {[\overline{M}_{g, n}^\dagger(V \times W)]^{\ell vir}} \neq \Delta^! {[\overline{M}_{g, n}^\dagger(V) \times \overline{M}_{g, n}^\dagger(W)]^{\ell vir}},\end{align*}

where h ^′ is the map

\begin{align*} \overline{M}_{g, n}^\dagger(V \times W) \overset{h}{\to} Q \overset{\pi}\to \overline{M}_{g, n}^\dagger(V) \times_{{\overline{M}_{g, n}}}^{sch} \overline{M}_{g, n}^\dagger(W)\end{align*}

to the ordinary scheme-theoretic fiber product. This necessitates the introduction of $\Delta^\dagger$ in Section 1.

We outline a counterexample due to Dhruv Ranganathan [ Reference HerrRan20 ] and David Holmes, which gives the following:

\begin{align*} \begin{split}&\Delta^! {[\overline{M}_{1, 2}^{\dagger}(\mathbb{P}^1) \times \overline{M}_{1, 2}^{\dagger}(\mathbb{P}^1)]^{\ell vir}}\\&\hspace{50pt}\neq \pi_\ast \Delta^\dagger {[\overline{M}_{1, 2}^{\dagger}(\mathbb{P}^1) \times \overline{M}_{1, 2}^{\dagger}(\mathbb{P}^1)]^{\ell vir}}.\end{split}\end{align*}

Example 5·1 equates relative (log) stable maps to ${[}\mathbb{P}^1/\mathbb{G}_m{]} \to B\mathbb{G}_m$ with rubber (log) stable maps to $\mathbb{P}^1$ without expansions. W For notational simplicity, write $\textrm{DR}_\Gamma$ for this space of rubber log stable maps $\overline{M}_{\Gamma}^\dagger([\mathbb{P}^1/\mathbb{G}_m] / B\mathbb{G}_m)$ . The $\Gamma$ signifies fixed discrete data. Here we equip $\mathbb{P}^1$ with divisorial log structure from 0 and $\infty$ and the moduli space consists of relative stable maps

of degree 2 from genus-one, 2-marked curves with contact order 2 at both 0 and $\infty$ .

If no confusion may arise, we denote also $\textrm{DR}_\Gamma$ as its image on $\overline{M}_{1, 2}$ , which has $\mathbb{Z}/2\mathbb{Z}$ -action at all points as their stack structures. $\textrm{DR}_\Gamma$ can be understood as a degree-three multisection from $\overline{M}_{1,1}$ to $\overline{M}_{1, 2}$ as in Figure 2.

Figure 2. The map $\overline{M}_{1, 2}^\dagger \to \overline{M}_{1, 2}$ blowing up the self-intersection point of one of the n.c. divisors. The log structure of each comes from the bolded n.c. divisors.

We forbid the two marked points from coming together in $\textrm{DR}_\Gamma$ , a component that is sometimes included in double ramification cycles. This convention simply excludes that component for simplicity. One can include the component where they meet instead as an equally valid convention.

Counterexample in Chow. We first give the computation in Chow theory. By definition, we have

\begin{align*} \Delta^! (\textrm{DR}_\Gamma \times \textrm{DR}_\Gamma) = \textrm{DR}_\Gamma \cdot \textrm{DR}_\Gamma.\end{align*}

To compute $\Delta^{\dagger}(\textrm{DR}_\Gamma \times \textrm{DR}_\Gamma)$ explicitly, we introduce the following procedure using tropical geometry.

Let $\textrm{DR}_\Gamma^{trop}$ and $\overline{M}_{1, 2}^{trop}$ be the Artin fans of $\textrm{DR}_\Gamma$ and $\overline{M}_{1, 2}$ respectively. Think of them as stacky cone complexes with strict maps $\textrm{DR}_\Gamma \rightarrow \textrm{DR}_\Gamma^{trop}$ and $\overline{M}_{1, 2} \rightarrow \overline{M}_{1, 2}^{trop}$ . Both $\textrm{DR}_\Gamma^{trop}$ and $\overline{M}_{1, 2}^{trop}$ have $\mathbb{Z}/2$ stack structure from interchanging the edges $e_1$ and $e_2$ .

Figure 3. Above left: $\textrm{DR}_\Gamma^{trop}$ . Above right: $\overline{M}_{1, 2}^{trop}$ . Below: The subdivision $\overline{M}_{1, 2}^{\dagger,trop}$ . Typical members of top-dimensional cones are encircled. Quotient by the $\mathbb{Z}/2$ action interchanging $e_1$ and $e_2$ .

These tropical edges $e_1, e_2$ correspond to self-nodes in a genus-one curve and the tropical edge lengths keep track of logarithmic data. The edge lengths encode logarithmic data roughly equivalent to the power q of a parameter t in a local equation

\begin{align*} xy = t^q\end{align*}

for the node. The locus where both edge lengths $e_1, e_2$ are positive corresponds to the algebraic curve which is two $\mathbb{P}^1$ ’s meeting at a pair of distinct nodes. The $\mathbb{Z}/2$ action arises because these two nodes are not consistently ordered in families.

The map $\textrm{DR}_\Gamma^{trop} \to \overline{M}_{1, 2}^{trop}$ is the inclusion of the diagonal in the quadrant spanned by $e_1$ and $e_2$ . It does not map cones surjectively onto cones, which corresponds to a non-flat map of toric stacks. Take a subdivision $\overline{M}_{1, 2}^{\dagger, trop}$ of $\overline{M}_{1, 2}^{trop}$ by adding the diagonal to the $e_1$ - $e_2$ quadrant as in Figure 3. This subdivision induces a log blowup

\begin{align*} \pi\;:\;\overline{M}_{1, 2}^\dagger \;:\!=\; \overline{M}_{1, 2} \times_{\overline{M}_{1, 2}^{trop}} \overline{M}_{1, 2}^{\dagger,trop} \to \overline{M}_{1, 2}\end{align*}

such that the factorization $\textrm{DR}_\Gamma^{\dagger} \to \overline{M}_{1, 2}^{\dagger}$ is strict, where $\textrm{DR}_\Gamma^{\dagger}$ is the strict transformation of $\textrm{DR}_\Gamma$ . Let $E = \pi^{-1}(p)$ be the exceptional divisor with $\mathbb{Z}/2\mathbb{Z}$ -action at all points as stack structure.

Now we have

\begin{align*} \Delta^{\dagger} (\textrm{DR}_\Gamma \times \textrm{DR}_\Gamma) = \textrm{DR}_\Gamma^{\dagger} \cdot \textrm{DR}_\Gamma^{\dagger}\end{align*}

and hence

\begin{align*} \begin{split}\pi_\ast \Delta^{\dagger} (\textrm{DR}_\Gamma\times \textrm{DR}_\Gamma) &= \pi_\ast (\textrm{DR}_\Gamma^{\dagger} \cdot \textrm{DR}_\Gamma^{\dagger})\\[5pt] &= \pi_\ast \Big((\pi^\ast\textrm{DR}_\Gamma - E)\cdot(\pi^\ast\textrm{DR}_\Gamma - E)\Big)\\[5pt] &= \textrm{DR}_\Gamma \cdot \textrm{DR}_\Gamma -\frac{1}{4}\neq \Delta^! (\textrm{DR}_\Gamma \times \textrm{DR}_\Gamma).\end{split}\end{align*}

Note that $\pi^\ast\textrm{DR}_\Gamma \cdot E = 0$ by projection formula and $E\cdot E = -\frac{1}{4}$ . This gives the inequality for Chow theory.

Counterexample in $\boldsymbol{{K}}_{\boldsymbol\circ}$ - theory. For K-theory, note first that

\begin{align*} \begin{split}\Delta^![\mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma}] &\;:\!=\; \phi \Big(\left[ \mathcal{O}_{\Delta(\overline{M}_{1, 2})}\otimes^L \left( \mathcal{O}_{\textrm{DR}_\Gamma}\times \mathcal{O}_{\textrm{DR}_\Gamma} \right) \right] \Big)\\& = \lambda_{-1} (N^\ast_{\textrm{DR}_\Gamma / \overline{M}_{1, 2}})\cdot \mathcal{O}_{\textrm{DR}_\Gamma} ,\end{split}\end{align*}

where $\lambda_{-1}(F) \;:\!=\; \sum_i (-1)^i \Lambda^i F$ is the alternating sum of exterior power of F. We view both $\mathcal{O}_{\Delta(\overline{M}_{1, 2})}$ and $\mathcal{O}_{\textrm{DR}_\Gamma}\times \mathcal{O}_{\textrm{DR}_\Gamma}$ as sheaves on $\overline{M}_{1, 2} \times \overline{M}_{1, 2}$ and $\phi\;:\; K_{\circ}^{\Delta(\overline{M}_{1, 2})} (\overline{M}_{1, 2}\times \overline{M}_{1, 2}) \simeq K_{\circ}(\overline{M}_{1, 2})$ is a canonical isomorphism, where $K^Z_{\circ} (X)$ denote the K-theory of X supported on Z. The last equality is the excess intersection formula in K-theory [ Reference Fulton and LangFL85 , VI·theorem 1·3]. One can also compute the derived tensor product $\otimes^L$ by taking locally free resolution of $\mathcal{O}_{\Delta(\overline{M}_{1, 2})}$ and $\mathcal{O}_{\textrm{DR}_\Gamma}$ .

The computation of $\Delta^{\dagger}[\mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma}]$ is similar.

\begin{align*} \Delta^{\dagger}[\mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma}] \;:\!=\; \phi \Big(\left[\mathcal{O}_{\Delta(\overline{M}_{1, 2}^{\dagger})}\otimes^L ( \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}}\times \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} ) \right] \Big),\end{align*}

Therefore, assuming the third equality below, we have

(3) \begin{equation}\begin{split}& \pi_\ast \Delta^{\dagger} [\mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma}]\\= \, &\pi_\ast \phi \Big(\left[ \mathcal{O}_{\Delta(\overline{M}_{1, 2}^{\dagger})} \otimes^L \left(\mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} \times \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} \right) \right] \Big) \\ = \, & \pi_\ast \Big(\lambda_{-1}(N^\ast_{\textrm{DR}_\Gamma^{\dagger}/\overline{M}_{1, 2}^{\dagger}}) \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}}\Big)\\= &- 2\mathcal{O}_p + \lambda_{-1} (N^\ast_{\textrm{DR}_\Gamma / \overline{M}_{1, 2}})\cdot \mathcal{O}_{\textrm{DR}_\Gamma} \\= &- 2\mathcal{O}_p + \Delta^! (\mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma} )\\ \neq& \, \Delta^! [ \mathcal{O}_{\textrm{DR}_\Gamma} \times \mathcal{O}_{\textrm{DR}_\Gamma} ],\end{split}\end{equation}

exactly as claimed. We are left to show the third equality in (3). Write

\begin{align*} \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} = \pi^\ast \mathcal{O}_{\textrm{DR}_\Gamma} -\mathcal{O}_{E} + \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E}.\end{align*}

The desired equality follows easily from the following equations

\begin{align*} \begin{split} \pi_\ast( \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} \cdot \mathcal{O}_E) &= \pi_\ast\mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E} = \mathcal{O}_p; \\ \pi_\ast (\mathcal{O}_{\textrm{DR}_\Gamma^{\dagger}} \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E}) &= 0; \\ \pi_\ast(\mathcal{O}_E \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E}) &= \pi_\ast ( \pi^\ast \mathcal{O}_p \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E} ) = \mathcal{O}_p \cdot \mathcal{O}_p = 0; \\ \pi_\ast(\mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E} \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E}) &= 0.\end{split}\end{align*}

The last three equations use the following simple observation. For any locally free sheaf V,

\begin{align*} V \cdot \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E} = \mathcal{O}_{\textrm{DR}_\Gamma^{\dagger} \cap E}^{\oplus \textrm{rk}(V)},\end{align*}

since $\textrm{DR}_\Gamma^{\dagger} \cap E$ is a point.