2. Definition of kappa classes

A KSBA-stable pair $(X, D=\sum _i a_iD_i)$ consists of an equidimensional variety $X$ and integral Weil divisors $D_i$ taken with rational coefficients $0\lt a_i\le 1$ such that $X$ is deminormal (and, in particular, has only double crossings in codimension $1$ ), $D_i$ are Mumford divisors (so do not contain components of the double locus of $X$ ), the pair $(X,D)$ has semi-log-canonical singularities and the divisor $K_X+D$ is ample.

We refer to Kollár [Reference KollárKol23, Definition 8.13] for the definition of the moduli functor, which is quite delicate and involves an important notion of a K-flat family of Mumford divisors. The main result of [Reference KollárKol23] is that after fixing the basic invariants, dimension $d=\dim X$ , the coefficient set $\mathbb {M}{a} = (a_i)$ and the volume $\nu = (K_{X}+D)^d$ , this moduli functor admits a projective coarse moduli space $\operatorname {SP}(\mathbb {M}{a}, d, \nu )$ . The moduli stack $\operatorname {\mathcal {S}\mathcal {P}}(\mathbb {M}{a}, d, \nu )$ is a proper Deligne–Mumford stack.

Mumford [Reference MumfordMum83] defined the kappa classes $\kappa _i$ on $\overline {{\mathcal {M}}}_g$ as the pushforwards of the cycles $K_{\mathcal {X}/\overline {{\mathcal {M}}}_g}^{i+1}$ in the universal family $f\colon \mathcal {X}\to \overline {{\mathcal {M}}}_g$ . Similarly, Arbarello and Cornalba [Reference Arbarello and CornalbaAC96, Reference Arbarello and CornalbaAC98] defined $\kappa _i$ on $\overline {{\mathcal {M}}}_{g,n}$ as the pushforwards of $(K_{\mathcal {X}/\overline {{\mathcal {M}}}_{g,n}}+\,\mathcal {D})^{i+1}$ in the universal family $f\colon (\mathcal {X},\mathcal {D}=\sum _{k=1}^n\mathcal {D}_k)\to \overline {{\mathcal {M}}}_{g,n}$ . In both cases we are greatly helped by the fact that $\overline {{\mathcal {M}}}_g$ and $\overline {{\mathcal {M}}}_{g,n}$ are smooth Deligne–Mumford stacks, so $\kappa _i$ can be considered to be cocycles in $A^i(\overline {{\mathcal {M}}}_{g,n})_{\mathbb {Q}}$ and $H^{2i}(\overline {{\mathcal {M}}}_{g,n},\mathbb {Q})$ .

We would like to define the kappa classes on KSBA spaces similarly, as the pushforwards of $(K_{\mathcal {X}/\operatorname {\mathcal {S}\mathcal {P}}}+\mathcal {D})^{i+d}$ . The question is: in what generality does this definition make sense and which properties does it have? We propose several versions.

Definition 2.4 (Cycle version). For any family $f\colon (X,D)\to S$ with equidimensional base $S$ , define the cycles $\kappa _i(S)\in A_{\dim S-i}(S)$ as proper pushforwards

\begin{align*} \kappa _i(S) = f_*\left ( (K_{X/S} + D)^{i+d} \cap [X] \right ) \quad \text {under}\quad \quad f_*\colon A_{m}(X) \to A_{m}(S), \end{align*}

where $m = \dim X-i-d = \dim S-i$ . Consider the following commutative square.

Here $g\colon S'\to S$ is a proper generically finite morphism of degree $e$ , with reduced $S,S'$ . Then $g'$ is also a proper generically finite morphism of degree $e$ . We have that $g'_*[X'] = e[X]$ , and by the projection formula

\begin{eqnarray*} e \kappa _i(S) &=& f_*\left ( (K_{X/S} + D)^{i+d} \cap g'_* [X']\right )\\ &=& f_* g'_* \left((K_{X'/S'} + D')^{i+d} \cap [X'] \right ) \\ &=& g_* f'_* \left ( (K_{X'/S'} + D')^{i+d} \cap [X'] \right ) = g_* \kappa _i(S'). \end{eqnarray*}

The same definition, and with the same functoriality, also works for the morphisms of DM stacks using the intersection theory on stacks [Reference VistoliVis89]. In particular, let $\operatorname {\mathcal {S}\mathcal {P}}'$ be an irreducible component of $\operatorname {\mathcal {S}\mathcal {P}}_{\textrm {red}}$ , or its normalization. Then we get cycles $\kappa _i(\operatorname {\mathcal {S}\mathcal {P}}')\in A_{\dim \operatorname {\mathcal {S}\mathcal {P}}'-i}(\operatorname {\mathcal {S}\mathcal {P}}')$ and $\kappa _i(\operatorname {SP}')\in A_{\dim \operatorname {SP}'-i}(\operatorname {SP}')$ on its coarse moduli space.

Definition 2.7 (Cohomological version). Without assuming that $f$ is lci, for any family of stable pairs $f\colon (X,D)\to S$ over a smooth, not necessarily proper $S$ , we can use Gysin pushforward $H^i(X,\mathbb {Q})\to H^i(S,\mathbb {Q})$ defined as the composition

\begin{align*} H^{2(i+d)}(X) \xrightarrow {\cap [X]} H^{\textrm {BM}}_{2\dim X-2i-2d}(X) \xrightarrow {f_*} H^{\textrm {BM}}_{2\dim S-2i} \xrightarrow {\sim } H^{2i}(S), \end{align*}

where $H^{\textrm {BM}}_*$ is a Borel–Moore homology (see e.g. [Reference FultonFul97, Appendix B]). I do not know if this Gysin pushforward has enough functorial properties to imply $\kappa _i(S') = g^*\kappa _i(S)$ .

At this point an educated reader will certainly think of many other ways to define kappa classes in cohomology, e.g. using operational Chow rings and bivariant theories, étale Borel–Moore homology. I think all of them deserve serious consideration.

In what follows, I assume that we have well-defined kappa classes in cohomology as in one of the above ways. In the three examples considered later in this paper the moduli stacks are smooth and we use Definition 2.5 to compute $\kappa _i$ .

The semipositivity results for families of stable pairs [Reference FujinoFuj18, Reference Kovács and PatakfalviKP17] imply that $K_{X/S} + D$ is nef, by a standard argument. It follows that whenever $\kappa _i$ are defined in cohomology, they are nef. In fact, $\kappa _1$ is the CM line bundle, known to be ample. This is true for $\overline {M}_g$ by [Reference MumfordMum77], for $\overline {M}_{g,n}$ by [Reference CornalbaCor93] and, in general, by [Reference Patakfalvi and XuPX17].

A special and very interesting case is the KSBA compactification of the moduli of log Calabi–Yau pairs $(X,\Delta +\epsilon B)$ such that, generically, $K_X+\Delta \sim _{\mathbb {Q}} 0$ and $B$ is $\mathbb {Q}$ -Cartier and ample. By [Reference Kollár and XuKX20, Reference BirkarBir23], after fixing basic numerical invariants there exists $\epsilon _0\gt 0$ such that for any $0\lt \epsilon \lt \epsilon _0$ the compactification for the stable pairs $(X,\Delta +\epsilon B)$ does not depend on $\epsilon$ . Some concrete cases are the compactified moduli spaces of toric and abelian varieties [Ale02] and of K3 surfaces [Reference Alexeev and EngelAE23].

Obviously, $\kappa _i(\epsilon )$ are polynomials in $\epsilon$ , $\lambda$ and $f_* (B^{i+d})$ . For example,

\begin{align*} \kappa _1(\epsilon ) / \epsilon ^d = (n+1)\nu (B) \lambda + \epsilon f_*B^{d+1}, \end{align*}

where $\nu (B) = B_s^d$ is the volume of a general fiber. Nefness of $\kappa _1(\epsilon )$ for $0\lt \epsilon \ll 1$ implies that $\lambda$ is nef as well. For toric pairs $(X,\Delta + \epsilon B)$ with toric boundary $\Delta$ , one has $\lambda=0$ . For abelian and K3 pairs $(X,\epsilon B)$ , $\lambda$ is the pullback of the ample Hodge bundle on the Satake–Baily–Borel compactification. So it is true in these cases. This was also proved for degenerations of pairs $(\mathbb {P}^2,D)$ in [ABB $^{+}$ 23]. One may therefore ask if $\lambda$ is always semiample.

3. Products of curves

Consider surfaces of the form $X=C_g\times C_h$ which are products of smooth curves of genus $g$ and $h$ . Obviously, the stable limits of one-parameter degenerations of such surfaces are products of two stable curves. By van Opstall [Reference van OpstallvO05], there is an irreducible component of the moduli of stable surfaces isomorphic to $\overline {M}_g \times \overline {M}_h$ if $g\ne h$ , or a quotient of it by an involution if $g=h$ . This construction is further extended by [Reference van OpstallvO06] to finite quotients of $C_g\times C_h$ .

For a universal family $\mathcal {X}$ over the stack $\overline {{\mathcal {M}}}_g\times \overline {{\mathcal {M}}}_h$ , the line bundle $\omega _{\mathcal {X}/\overline {{\mathcal {M}}}_g\times \overline {{\mathcal {M}}}_h}$ is simply $p_1^*(\omega _{\mathcal {C}_g/\overline {{\mathcal {M}}}_g}) + p_2^*(\omega _{\mathcal {C}_h/\overline {{\mathcal {M}}}_h})$ , and the kappa classes on $\overline {M}_g\times \overline {M}_h$ are merely appropriate sums of monomials in the pullbacks of kappa classes from $\overline {M}_g$ and $\overline {M}_h$ . So this case is reduced to $\overline {M}_g$ . The cases of quotients of $C_g\times C_h$ can be treated similarly; at the stack level there is not much difference.

4. Kappa classes on moduli of $\mathbb {Z}_2^k$ -covers

In §§ 4 and 5, we treat the cases of Burniat and Campedelli surfaces described in [Reference Alexeev and PardiniAP23] and [Reference Alexeev and HuAH23]. These are surfaces of general type that are certain branched $\mathbb {Z}_2^k$ -covers $(k=2,3)$ of pairs $(Y, \frac 12 D)$ , $D=\sum _i D_i$ . The stable surfaces on the boundary are $\mathbb {Z}_2^k$ -covers of stable pairs $(Y,\frac 12 D)$ . In each case, the compactified coarse moduli space of surfaces $X$ is a finite quotient of the compactified fine moduli space for the pairs $(Y,\frac 12 D)$ by a symmetry group permuting the labels of the $D_i$ .

Any family $f\colon X\to S$ after a finite base change $S'\to S$ can be written as a $\mathbb {Z}_2^k$ -cover $X'\to (Y',\frac 12 D')$ , where $X' = X\times _S S'$ , $(Y',D') = (Y,D)\times _S S'$ and the $\mathbb {Q}$ -line bundle $\omega _{X'/S'}$ is the pullback of $\omega _{Y'/S'}(\frac 12 D')$ . Thus, the kappa classes for the covers $X$ are proportional to the kappa classes for the pairs $(Y,\frac 12 D)$ by some multiples that are powers of $2$ . So it is enough to study the kappa classes for the pairs $(Y,\frac 12 D)$ .

5. Burniat surfaces

Burniat surfaces are certain surfaces of general type of degree $3\le d=K_X^2\le 6$ with $p_g=q=0$ which can be obtained as $\mathbb {Z}_2^2$ -covers of degree- $d$ del Pezzo surfaces ramified in a set of $12$ curves coming from a particular configuration of lines in $\mathbb {P}^2$ .

Primary Burniat surfaces are those of degree $6$ ; they are covers of Cremona surface $\Sigma = \textrm {Bl}_3\mathbb {P}^2$ . Secondary Burniat surfaces have degrees $5$ and $4$ , they are $\mathbb {Z}_2^2$ -covers of del Pezzo surfaces of degrees $5$ and $4$ obtained by further blowups of $\Sigma$ at the points where some three of the $12$ curves pass through the same point.

An explicit KSBA compactification of the moduli space of primary Burniat surfaces was described in [Reference Alexeev and PardiniAP23]. Using it, explicit KSBA compactifications for the moduli of secondary Burniat surfaces were described in [Reference Alexeev and HuAH23].

For Burniat surfaces of degrees $6$ and $5$ and for the non-nodal Burniat surfaces of degree $4$ , the above papers give compactifications of the entire irreducible components in the moduli space of surfaces of general type. In the nodal degree $4$ and $3$ cases they are closed subsets of irreducible components or larger dimensions. We do not discuss the nodal cases here.

As was pointed out to me by Yunfeng Jiang, for numerical applications the most interesting degrees are $4$ and $5$ . Indeed, for degree $d$ the dimension of the compactification $\overline {M}^{\mathrm { Bur}}_d$ is $d-2$ . On the other hand, by [Reference DonaldsonDon20, Reference DevelopersJia22] the dimension of the virtual fundamental class is $10\chi (\mathcal {O}_X)-2K_X^2=10-2d$ . Thus, for $d=6$ the virtual fundamental class is zero, and for $d=5$ it is a multiple of a point.

In §§ 5.1 and 5.2, we consider the degree $6$ and $4$ non-nodal cases, respectively. Here, the virtual fundamental case has dimension $2$ and coincides with $[\overline {M}_4^{\mathrm { Bur}}]$ . In § 5.3, we compute the kappa classes on $\overline {M}_4^{\mathrm { Bur}}$ .

5.1 Degree $6$

Burniat surfaces $X_6$ of degree $6$ are $\mathbb {Z}_2^2$ -covers of a Cremona surface $Y_6^{\mathrm { tor}} = \Sigma =\textrm {Bl}_3\mathbb {P}^2$ ramified in a configuration of $12$ curves shown in the left panel of Figure 1. A $\mathbb {Z}_2^2$ -cover is determined by three divisors $R,G,B$ satisfying certain conditions (see [Reference Alexeev and PardiniAP23]). We use the primary colors red, green and blue to draw them. In this case, $R=\sum _{i=0}^3 R_i$ , $G=\sum _{i=0}^3 G_i$ , $B=\sum _{i=0}^3 B_i$ . The curves with $i=0,3$ form the toric boundary $D_{6,\textrm {bry}}$ of $\Sigma$ . The curves with $i=1,2$ form the interior divisor $D_{6,{\mathrm { int}}}$ . The total branch divisor of $X_6\to Y_6$ is $D_6=D_{6,\textrm {bry}}+D_{6,{\mathrm { int}}} = R+G+B$ .

Figure 1. Burniat configurations of degree 6 and 4 non-nodal cases.

In [Reference Alexeev and PardiniAP23] is a construction of a compactified moduli space for the pairs $(Y_6,\frac 12 D_6)$ , which we will denote by $\overline {M}_6$ here. It comes with a universal family $(\mathcal {Y}_6^{\mathrm { tor}},\frac 12 \mathcal {D}_6) \to \overline {M}_6$ . Then the compactified moduli space $\overline {M}_6^{\mathrm { Bur}}$ of degree $6$ Burniat surfaces is the quotient of $\overline {M}_6$ by a finite group $S_3\ltimes S_2^4$ shuffling the labels of $R_i,G_i,B_i$ .

In more detail, $(\mathcal {Y}_6,\frac 12 \mathcal {D}_6)\to \overline {M}_6$ is obtained from an explicit morphism of toric varieties $\mathcal {Y}_6^{\mathrm { tor}}\to \overline {M}_6^{\mathrm { tor}}$ by a series of smooth blowups, followed by a contraction to the relative canonical model. The morphism $\overline {M}_6\to \overline {M}_6^{\mathrm { tor}}$ is a composition of a blowup $\rho _1$ at the central point $1\in \mathbb {C}^*{}^4\subset \overline {M}_6^{\mathrm { tor}}$ , followed by a blowup $\rho _2$ along six disjoint $\mathbb {P}^1$ . A family $\mathcal {Y}'_6\to \overline {M}_6$ is obtained from $\mathcal {Y}_6^{\mathrm { tor}}$ by doing the base changes under $\rho _1,\rho _2$ and additional smooth blowups in the fibers. On $\mathcal {Y}_6'$ , the divisor $K_{\mathcal {Y}_6'/\overline {M}_6} + \frac 12 D_6$ is relatively big and nef over $\overline {M}_6$ . The universal family $\mathcal {Y}_6\to \overline {M}_6$ is its relative canonical model.

The boundary divisor $\mathcal {D}_{6,\textrm {bry}}^{\mathrm { tor}}$ is the union of the boundary curves on the fibers; it is the horizontal part of the toric boundary of $\mathcal {Y}_6^{\mathrm { tor}}$ . The interior divisor $\mathcal {D}_6^{\mathrm { tor}}$ on $\mathcal {Y}_6^{\mathrm { tor}}$ is constructed in [Reference Alexeev and PardiniAP23, Sec. 4] as follows. In addition to the map $p_1\colon \mathcal {Y}_6^{\mathrm { tor}}\to \overline {M}_6^{\mathrm { tor}}$ , there a second projection, a birational morphism $p_2\colon \mathcal {Y}_6^{\mathrm { tor}}\to V_{P_6}$ to a projective toric variety $V_{P_6}$ defined by a lattice polytope $P_6$ that is the convex hull of a $46$ -point set $A_6$ . Under this projection, the fibers $Y_6$ become closed subvarieties of $V_{P_6}$ . Then $\mathcal {D}_{\mathrm { int}}$ is a section of $p_2^*\mathcal {O}_{V_{P_6}}(2) \otimes p_1^*\mathcal {O}_{\overline {M}_6}(-F)$ for a certain effective divisor $F$ on $\overline {M}_6$ that is defined in the proof of [Reference Alexeev and PardiniAP23, Proposition 4.20].

5.2 Degree $4$

Non-nodal Burniat surfaces of degree $4$ are defined as follows. One considers the special configurations for which the triples of the curves $(R_1,G_1,B_1)$ and $(R_2,G_2,B_2)$ pass through common points, as in the right panel of Figure 1. Let $Y\to \Sigma$ be the blowup at these points. The strict preimages of $R,G,B$ give a $\mathbb {Z}_2^2$ -cover $\pi \colon X\to Y$ that is a Burniat surface of degree $4$ . Note that the exceptional divisors are not included in the branch divisor of $\pi$ .

The compactified moduli space $\overline {M}_4^{\mathrm { Bur}}$ of degree $4$ Burniat surfaces was constructed in [Reference Alexeev and HuAH23] as an $S_3\ltimes S_2^2$ -quotient of the compactified moduli space $\overline {M}_4$ of pairs $(Y,\frac 12\mathcal {D})$ , as follows.

Remark 5.1. As we only consider the non-nodal degree $4$ case in this paper, in order to simplify the notation, in this section we write simply $D$ , $Z$ , $Y$ … instead of $D_{4a}$ , $Z_{4a}$ , $Y_{4a}$ …, as in [Reference Alexeev and HuAH23]. To distinguish the parent degree $6$ case, we keep the subscripts there: $D_6$ , $Z_6$ , $Y_6$ ….

There exists a closed subvariety $Z\subset \overline {M}_6$ , a complete intersection of two divisors, over which the curves $(R_1,G_1,B_1)$ and $(R_2,G_2,B_2)$ are incident. The degenerate pairs appearing in this family are shown in the upper row of Figure 2. The restricted family $\mathcal {Y}_6|_Z$ comes with two disjoint sections $s_1,s_2$ . Let $\mathcal {Y}'\to Z$ be the blowup of $\mathcal {Y}_6|_Z$ along $s_1,s_2$ .

The variety $Z$ is the strict preimage of a toric variety $Z^{\mathrm { tor}} \subset \mathcal {Y}^{\mathrm { tor}}$ under the blowup $\rho _1$ . It turns out that $Z^{\mathrm { tor}}\simeq \Sigma$ and $Z=\textrm {Bl}_1\Sigma$ . The divisor $K_{\mathcal {Y}'/Z}+\frac 12\mathcal {D}'$ is relatively nef over $Z$ ; let $\mathcal {Y}''$ be its relative canonical model. The degenerate fibers appearing in $\mathcal {Y}''\to Z$ are shown in the lower row of Figure 2. They are a union of two $\mathbb {P}^1\times \mathbb {P}^1$ glued along the diagonal, a union of four $\mathbb {P}^2$ and another union of two $\mathbb {P}^1\times \mathbb {P}^1$ with a different configuration of branch divisors.

Figure 2. Surfaces over $Z\subset \overline {M}_6$ of degrees $6$ and $4$ .

Over the exceptional divisor of $\textrm {Bl}_1\Sigma \to \Sigma$ (the divisor of type E) all the fibers are isomorphic, so the family $(\mathcal {Y}'',\frac 12\mathcal {D}'') \to Z$ descends to a family $(\mathcal {Y},\frac 12\mathcal {D})\to \overline {M}_4=Z^{\mathrm { tor}}=\Sigma$ . Thus, the final family $\mathcal {Y}\to \Sigma$ is obtained from the toric family $\mathcal {Y}^{\mathrm { tor}}\to \Sigma$ as follows. There is a sequence of smooth blowups

(1) \begin{align} \mathcal {Y}^{\mathrm { tor}} = \mathcal {Y}_0 \xleftarrow {\beta _1} \mathcal {Y}_1 \xleftarrow {\beta _2} \mathcal {Y}_2 \xleftarrow {\beta _3} \mathcal {Y}_3 \xleftarrow {\beta _4} \mathcal {Y}_4 = \mathcal {Y}', \end{align}

in which

1. (i) $\beta _1$ is the blowup of the fiber $(\mathcal {Y}_6^{\mathrm { tor}})_1\simeq \Sigma$ of $\mathcal {Y}_6^{\mathrm { tor}}$ over $1\in \Sigma$ (this is the base change $\mathcal {Y}_1 = \mathcal {Y}_0 \times _\Sigma \textrm {Bl}_1\Sigma \to \mathcal {Y}_0$ );

2. (ii) $\beta _2$ is the blowup of $\mathbb {P}^1=\beta ^{-1}(1)$ , preimage of the central point $1\in (\mathcal {Y}_6^{\mathrm { tor}})_1$ ;

3. (iii) $\beta _3$ and $\beta _4$ are the blowups of the sections $s_i = R_i\cap G_i\cap B_i$ , $i=1,2$ .

This sequence is followed by a contraction $\mathcal {Y}'\to \mathcal {Y}''$ followed by a contraction $\mathcal {Y}''\to \mathcal {Y}$ covering the contraction $\textrm {Bl}_1\Sigma \to \Sigma$ .

5.3 Kappa classes

Theorem 5.2. In $A^*(\Sigma )$ one has $\kappa _0=1$ , $\kappa _1 = \mathcal {O}_\Sigma (1)=\mathcal {O}(-K_\Sigma )$ , $\kappa _2 = \tfrac {47}{4}\cdot [\textrm {pt}]$ .

Proof. The class $\kappa _0$ is the degree of the divisor $K_Y+\frac 12 D$ on a general fiber, so $\kappa _0 = (K_Y + \frac 12 D)^2 = (-\frac 12 K_Y)^2=1$ .

For the rest, we begin by computing the divisor $K_{\mathcal {Y}^{\mathrm { tor}}/\Sigma } + \frac 12 \mathcal {D}^{\mathrm { tor}}$ . Let us denote the toric boundary of the toric variety $\mathcal {Y}^{\mathrm { tor}}$ by $\Delta$ . Denote the part of $\Delta$ that maps to the toric boundary of $\Sigma$ by $\Delta ^{\mathrm { ver}}$ and the remaining part by $\Delta ^{\mathrm { hor}}$ . Obviously, one has $\Delta ^{\mathrm { ver}} = p_1^*(\Delta _\Sigma ) = p_1^*\mathcal {O}_\Sigma (1)$ for the projection $\mathcal {Y}^{\mathrm { tor}}\to Z^{\mathrm { tor}} =\Sigma$ .

As explained above, the family $\mathcal {Y}_6^{\mathrm { tor}}\to \overline {M}_6^{\mathrm { tor}}$ comes with a second projection $p_2\colon \mathcal {Y}^{\mathrm { tor}}_6\to V_{P_6}$ and the fibers $Y_6^{\mathrm { tor}}$ are closed subvarieties sweeping out $V_{P_6}$ . Restricting this family to $Z$ gives a family that sweeps out a smaller toric variety $V$ for the lattice polytope $P$ obtained by an appropriate projection of $P_6$ . By Lemma 5.4, the projection $\mathcal {Y}^{\mathrm { tor}}_6\to V$ is small, since both varieties have $18$ toric boundary divisors. By Lemma 5.3, $P$ is reflexive with a unique interior point. This implies that $\Delta = p_2^*\Delta _{V}$ and $-K_{V} = \Delta _{V} = \mathcal {O}_{V}(1)$ . Restricting the divisor $\mathcal {D}_{6,}{}_{\mathrm { int}}$ on $\overline {M}_6^{\mathrm { tor}}$ to the family over $Z^{\mathrm { tor}}$ gives $\mathcal {O}(\mathcal {D}_{\mathrm { int}}) = p_2^* \mathcal {O}_{V}(2) \otimes p_1^* \mathcal {O}_\Sigma (-1)$ . Putting this together, writing additively, and for convenience mixing up sheaves and divisors, we get

\begin{eqnarray*} &&K_{\mathcal {Y}^{\mathrm { tor}}/\Sigma } = -\Delta ^{\mathrm { hor}}, \quad \Delta = p_2^*\mathcal {O}_V(1),\quad \mathcal {D}_{\mathrm { bry}} = \Delta ^{\mathrm { hor}},\\ &&\mathcal {D}_{\mathrm { int}} = p_2^*\mathcal {O}_V(2) - p_1^*\mathcal {O}_\Sigma (1)=2\Delta - \Delta ^{\mathrm { ver}},\\ &&K_{\mathcal {Y}^{\mathrm { tor}}/\Sigma } + \tfrac 12 \mathcal {D}^{\mathrm { tor}} = -\Delta ^{\mathrm { hor}} + \tfrac 12(\Delta ^{\mathrm { hor}} + 2\Delta -\Delta ^{\mathrm { ver}}) = \tfrac 12 \Delta = p_2^*\mathcal {O}_V(\tfrac 12). \end{eqnarray*}

By symmetry, $\kappa _1$ is a multiple of $\mathcal {O}_\Sigma (1)$ . To find this multiple it is enough to find its intersection with a boundary $(-1)$ -curve $C$ on $\Sigma$ , which equals $(K_{\mathcal {Y}}+\tfrac 12 \mathcal {D})^3$ on the divisor $F = f^{-1}(C) \subset \mathcal {Y}$ . To compute it, we can ignore the blowup $\rho _1$ since it does not touch $F$ . We can also compute on the family $\mathcal {Y}'$ since the contraction $\mathcal {Y}'\to \mathcal {Y}''$ from a relative minimal model to a relative canonical model is crepant.

The restriction of $\mathcal {Y}^{\mathrm { tor}}$ to $F$ has two irreducible components corresponding to the two surfaces $\textrm {Bl}_1\mathbb {F}_1$ in case D of Figure 2. Each component maps birationally to a boundary divisor of $V$ . Thus, the degree of $p_2^*\mathcal {O}_V(1)$ on $F$ is twice the degree of $\mathcal {O}_V(1)$ on a boundary divisor of $V$ . The latter degree is the lattice volume of the corresponding facet of $P$ , which by Lemma 5.3 is $7$ . So the degree of $K_{\mathcal {Y}^{\mathrm { tor}}/\Sigma } + \frac 12\mathcal {D}$ on $F$ is $2\cdot \frac 78$ .

Restricting $\mathcal {Y}^{\mathrm { tor}}$ to $C$ gives a family $\mathcal {Y}^{\mathrm { tor}}_C$ with two disjoint sections corresponding to the two special points. The family $\mathcal {Y}'_C$ is obtained from it by blowups at the two special sections, one in each irreducible component of $\mathcal {Y}^{\mathrm { tor}}_C$ . One has

\begin{align*} K_{\mathcal {Y}'_C/C} + \tfrac 12\mathcal {D}|_F = \beta ^*\big ( K_{\mathcal {Y}_C ^{\mathrm { tor}}/ C} + \tfrac 12\mathcal {D}^{\mathrm { tor}}_C \big ) - \tfrac 12 E_1 -\tfrac 12 E_2, \end{align*}

where $E_1$ , $E_2$ are the exceptional divisors. Using the blowup formula [Reference FultonFul84, 3.3.4] and Lemma 5.5, we get that the degree of $K_{\mathcal {Y}/\Sigma } + \frac 12\mathcal {D}$ on $F$ is $2\cdot \frac {7-3}{8} = 1$ . So, $\kappa _1 = \mathcal {O}_\Sigma (1)$ .

To compute $\kappa _2 = (K_{\mathcal {Y}'/\textrm {Bl}_1\Sigma } + \frac 12\mathcal {D}')^4$ , we can compute on $\mathcal {Y}'\to \textrm {Bl}_1\Sigma$ using the functorial property of the diagram (1). On $\mathcal {Y}^{\mathrm { tor}}$ one has $\left ( K_{\mathcal {Y}^{\mathrm { tor}}/\Sigma } + \frac 12\mathcal {D}^{\mathrm { tor}} \right )^4 = \mathcal {O}_V\left (\frac 12\right )^4 = \frac {18\cdot 7}{2^4} = \frac {63}{8}$ by Lemma 5.3. Then we trace how this number changes under the four blowups in (1) using [Reference FultonFul84, 3.3.4].

The next two lemmas are proved by direct computations with polytopes.

Lemma 5.3. The polytope $P$ is a reflexive four-dimensional polytope with the $f$ -vector $(1, 30, 84, 72, 18, 1)$ and a unique interior point. Its $18$ facets are isomorphic three $3$ -dimensional polytopes with $8$ vertices and lattice volume $7$ . One has $\textrm {vol}(P)=18\cdot 7$ .

Lemma 5.4. The toric family $\mathcal {Y}^{\mathrm { tor}}$ is a projective toric variety for a four $4$ -dimensional lattice polytope $P+14 P_\Sigma$ with the $f$ -vector $(1, 42, 96, 72, 18, 1)$ .

Here, $P_\Sigma$ is the hexagon corresponding to the toric variety $(\Sigma, \mathcal {O}_\Sigma (1))$ , and $14 P_\Sigma$ is the fiber polytope coming from the construction of the toric family in [Reference Alexeev and PardiniAP23].

Lemma 5.5. For each of the special sections $s_1=R^{\mathrm { tor}}_1\cap G^{\mathrm { tor}}_1\cap B^{\mathrm { tor}}_1$ and $s_2=R^{\mathrm { tor}}_2\cap G^{\mathrm { tor}}_2\cap B^{\mathrm { tor}}_2$ of $\mathcal {Y}^{\mathrm { tor}}\to \Sigma$ , the normal bundle is trivial, i.e. equal to $\mathcal {O}_\Sigma ^{\oplus 2}$ .

Proof. Consider the union $U$ of torus orbits in $\mathcal {Y}^{\mathrm { tor}}$ containing $s_1$ . Since $U$ comes with a section and a free action of the vertical torus $\mathbb {C}^*{}^2 = \ker (\mathbb {C}^*{}^4 \to \mathbb {C}^*{}^2)$ , one has $U = \Sigma \times \mathbb {C}^*{}^2$ . So $N_{s_1/U} = \mathcal {O}_{s_1}^{\oplus 2}$ , and the same works for $s_2$ .

6. Campedelli surfaces

The Campedelli surfaces considered in [Reference Alexeev and PardiniAP23] are surfaces of general type with $K^2=\,2$ and $p_g=0$ which are $\mathbb {Z}_2^3$ -covers of $\mathbb {P}^2$ ramified in seven lines in general position. The branch data for this cover consists of these seven lines $D_g$ , $g\in \mathbb {Z}_2^3\setminus 0$ . The moduli space has dimension $6$ , which coincides with the dimension of the virtual fundamental class, equal to $10\chi -2K_X^2$ . So the virtual fundamental class in this case is $[\overline {{\mathcal {M}}}_{\mathrm { Cam}}]$ .

By [Reference Alexeev and PardiniAP23], the moduli stack $\overline {{\mathcal {M}}}_{\mathrm { Cam}}$ is a global quotient $[\overline {M}:\Gamma ]$ , where $\overline {M}$ is the compactified moduli space of labeled log canonical pairs $(\mathbb {P}^2,\sum _{g\in \mathbb {F}_2^3\setminus 0} \frac 12 D_g)$ and $\Gamma =\textrm {GL}(3,\mathbb {F}_2^3)$ . Further, $\overline {M}$ in this case is the GIT quotient $(\mathbb {P}^2)^7//\textrm {SL}(3)$ for the symmetric polarization $(1,\dotsc, 1)$ . The sets of stable and semistable points in this case coincide and the $\textrm {SL}(3)$ -action on it is free. Therefore, $(\mathbb {P}^2)^7//\textrm {SL}(3)$ is smooth.

The universal family $(\mathcal {Y}, \sum \frac 12 D_g)\to \overline {M}$ is itself a GIT quotient of a family of hyperplane arrangements in $(\mathbb {P}^2)^7\times (\mathbb {P}^2)^\vee$ by the action of the group $\textrm {SL}(3)$ for the polarization $(1,\dotsc, 1, \epsilon )$ , $0\lt \epsilon \ll 1$ .

As a first step, I compute the Chow ring of $\overline {M} = (\mathbb {P}^2)^7//\textrm {SL}(3)$ . Then the rational Chow ring of the stack $\overline {{\mathcal {M}}}_{\mathrm { Cam}}$ is identified with its $\Gamma$ -invariant subring.

6.1 Chow ring

Let $X=(\mathbb {P}^2)^7$ with the diagonal action of $G=\textrm {SL}(3)$ . Consider the GIT quotient $X//G$ for the symmetric polarization $(1,\dotsc, 1)$ . In this case, the stable and semistable loci coincide and the $G$ -action on $X^{\mathrm { s}}$ is free, so the cohomology ring of $X//G$ can be identified with the equivariant cohomology ring $H_G(X^{\mathrm { ss}},\mathbb {Q})$ and the Chow ring of $X//G$ with the equivariant Chow ring $A_G(X^{\mathrm { ss}})_{\mathbb {Q}}$ .

Remark 6.1. As Michel Brion explained to me, for any semisimple group $G$ with a maximal torus $T$ and a $G$ -variety $V$ that admits a $T$ -invariant cell decomposition, the $G$ -equivariant Chow ring $A^*_G(V)_{\mathbb {Q}}$ and the $G$ -equivariant cohomology ring $H^*_G(V,\mathbb {Q})$ coincide. Indeed, for the $T$ -equivariant versions the cycle map $A^*_T(V)_{\mathbb {Q}} \to H^*_T(V,\mathbb {Q})$ is an isomorphism by [Reference BrionBri97], and the $G$ -equivariant versions $A^*_G$ and $H^*_G$ are the Weyl group invariant subrings of these.

Theorem 6.2. One has

\begin{align*} A^*(X//G)_{\mathbb {Q}}= H^*(X//G,\mathbb {Q}) = \mathbb {Q}[z_1,\dotsc, z_7, c_2, c_3] / J, \end{align*}

with the generators of degree $(1,\dotsc, 1,2,3)$ and the ideal $J$ generated by the relations:

1. (i) $z_1^3 + c_2 z_1 + c_3$ ;

2. (ii) $\sigma _3(z_1,z_2,z_3,z_4,z_5) - \sigma _1(z_1,z_2,z_3,z_4,z_5)c_2 + c_3$ ;

3. (iii) $\sigma _4(z_1,z_2,z_3,z_4,z_5) - \sigma _2(z_1,z_2,z_3,z_4,z_5)c_2 + c_2^2 + \sigma _1(z_1,z_2,z_3,z_4,z_5)c_3$ ;

4. (iv) $(z_1^2z_2^2+z_2^2z_3^2+z_3^2z_1^2) + (z_1+z_2+z_3)(z_1z_2z_3-c_3) + (z_1^2 + z_2^2+ z_3^3)c_2 + c_2^2$ ;

and the ones obtained from them by permuting the variables $z_1,\dotsc, z_7$ . Here, $\sigma _k$ are the elementary symmetric polynomials.

Moreover, the relations of types (i),(ii),(iv) and a single relation of type (iii), or the sum of all relations of type (iii), suffice. The relations of types (i),(ii) are independent.

The dimensions of $A^{i}$ for $i=0,\dotsc, 6$ are $(1,7,29,64,29,7,1)$ .

Proof. This is a direct application of Brion’s paper [Reference BrionBri91]. Let $T\subset G$ be the maximal torus. One has

\begin{align*} A_T(\cdot ) = S=\mathbb {Q}[\epsilon _0,\epsilon _1,\epsilon _2]/(\epsilon _0+\epsilon _1+\epsilon _2), \quad A_G(\cdot ) = S^W = \mathbb {Q}[c_2,c_3], \quad W=S_3, \end{align*}

where $c_2$ and $c_3$ are the Chern characters of the representation $V$ with $\mathbb {P}^2=\mathbb {P}(V)$ , the elementary symmetric polynomials in $\epsilon _i$ . Then the equivariant cohomology ring $A^*_T (X)$ equals $S[z_1,\dotsc, z_7]$ modulo the basic relations $(z_i+\epsilon _0)(z_i+\epsilon _1)(z_i+\epsilon _2)$ . Similarly, $A^*_G (X)$ equals $S^W[z_1,\dotsc, z_7]$ modulo the basic relations $z_i^3 + c_2z_i+c_3$ . Here, $z_i=c_1(\mathcal {O}_{\mathbb {P}^2}(1))$ for the $i$ th $\mathbb {P}^2$ , cf. Remark 6.3.

Let $X^{\mathrm { ss}}_T$ denote the set of semistable points for the action of $T$ . Then by [Reference BrionBri91, Theorem 2.1], $A^*_T(X^{\mathrm { ss}}_T) = A^*_T(X) / I$ , and the ideal $I$ is described with the help of the maximal unstable sets. Up to the permutation by $S_3\times S_7$ of indices, they are

\begin{align*} \{0\}^3,\ \{0,1,2\}^4 \quad \text {and}\quad \{1,2\}^5,\ \{0,1,2\}^2, \end{align*}

which in a transparent way correspond to the instability conditions of seven lines on $\mathbb {P}^2$ (see [Reference Alexeev and PardiniAP23]): when three lines coincide or five lines pass through a common point. Then $I$ up to permutation by $S_3\times S_7$ is generated by the expressions

\begin{eqnarray*} &(z_1+\epsilon _1)(z_1+\epsilon _2) (z_2+\epsilon _1)(z_2+\epsilon _2) (z_3+\epsilon _1)(z_3+\epsilon _2)\\ &\text {and} \quad (z_1+\epsilon _0)(z_2+\epsilon _0)(z_3+\epsilon _0) (z_4+\epsilon _0)(z_5+\epsilon _0). \end{eqnarray*}

Then, by [Reference BrionBri91, Section 1.2], one has $A^*_G(X^{\mathrm { ss}}) = A^*G(X) / p(I)$ , where $p$ is the anti-symmetrization operator

\begin{align*} p=D^{-1} \sum _{w\in W} (-1)^{\operatorname {sign}(w)} w, \quad D = (\epsilon _0-\epsilon _1)(\epsilon _1-\epsilon _2)(\epsilon _2-\epsilon _0). \end{align*}

Moreover, given generators $f_i$ of $I$ and an additive basis $\langle g_j\rangle$ for the harmonic module $\mathcal {H}$ , the ideal $p(I)$ is generated by $p(f_ig_j)$ . For $G=\textrm {SL}(3)$ the harmonic module is $\mathcal {H}=\langle 1, \epsilon _i, \epsilon _i^2-\epsilon _j^2, D\rangle$ . The rest is a computation in sagemath [Reference DevelopersSag22].

Remark 6.3. In [Reference BrionBri91] Grothendieck’s convention for a projective space $\mathbb {P} V$ as the space of one $1$ -dimensional quotient of $V$ is followed. We follow the convention that $\mathbb {P} V$ is the space of lines in $V$ , more common in the literature on equivariant cohomology. Then for us $z_i=c_1(\mathcal {O}_{\mathbb {P}^2}(1))$ , and in [Reference BrionBri91], $z_i=c_1(\mathcal {O}_{\mathbb {P}^2}(-1))$ .

6.2 $\textrm {GL}(3,2)$ -invariants

Denote by $s_k=\sigma _k(z_1,\dotsc, z_7)$ the elementary symmetric polynomials in $z_1,\dotsc, z_7$ . Also, denote by $s'_3$ , respectively $s''_3$ , the sums of $z_iz_jz_k$ with distinct $i,j,k$ such that the indices $i,j,k$ considered as points of the Fano plane $\mathbb {P}^2(\mathbb {F}_2)$ are incident, respectively are not incident. Obviously, $s'_3$ and $s''_3$ are $\textrm {GL}(3,\mathbb {F}_2)$ -invariant but not $S_7$ -invariant. One has $s_3 = s'_3 + s''_3$ and denotes $t=s''_3-4s'_3$ .

Theorem 6.4. For the ring of invariants $A^*(X//G)^{\textrm {GL}(3,2)}$ , the dimensions of the invariants of degree $0,\dotsc, 6$ are $(1,1,3,4,3,1,1)$ with an additive basis

\begin{align*} 1\quad s_1\quad s_1^2,c_2,s_2\quad s_1^3,c_2s_1,s_2s_1,t\quad c_2^2,c_2s_2,s_2^2\quad c_2^2s_1\quad c_2^3. \end{align*}

The algebra $A^*(X//G)^{\textrm {GL}(3,2)}$ is generated by $s_1$ , $c_2$ , $s_2$ , $t$ with relations

\begin{eqnarray*} &&ts_1,\quad tc_2,\quad ts_2,\quad t^2+126 c_2^3,\quad \\ &&45 s_1^4 - 1246 c_2^2+1090 c_2 s_2 - 240 s_2^2,\quad 15 c_2 s_1^2 - 28 c_2^2+35 c_2 s_2 - 10 s_2^2,\quad \\ &&3 s_2 s_1^2 - 49 c_2^2+46 c_2 s_2 - 11 s_2^2,\quad 5 c_2 s_2 s_1 - 16 c_2^2 s_1,\quad 5 s_2^2 s_1 - 59 c_2^2 s_1. \end{eqnarray*}

Proof. The irreducible characters of $\textrm {GL}(3,2)$ are $\chi _1$ , $\chi _3$ , $\chi _{\bar 3}$ , $\chi _6$ , $\chi _7$ , $\chi _8$ , where the subscript denotes the dimension and $\chi _1$ is the trivial representation. There are two basic permutation representations of $S_7$ on the $7$ variables $z_1,\dotsc, z_7$ and on the $21$ monomials $z_iz_j$ with $i\ne j$ . The induced $\textrm {GL}(3,2)$ -representations are

\begin{align*} \chi ^{\mathrm { p}}_7 = \chi _1 + \chi _6, \quad \chi _{21}^{\mathrm { p}} = \chi _1+2\chi _6+\chi _8. \end{align*}

From the relations of Theorem6.2, the representations on $A^{i}$ for $i=1,2,3$ are:

1. (i) $A^1=\chi _7^{\mathrm { p}} = \chi _1 + \chi _6$ , invariant: $s_1$ ;

2. (ii) $A^2=\operatorname {Sym}^2(\chi _7^{\mathrm { p}})+\chi _1\cdot c_2=3\chi _1+3\chi _6 + \chi _8$ , invariants $s_1^2$ , $s_2$ , $c_2$ ;

3. (iii) (from the generators and relations of type (i) and (ii))

   \begin{eqnarray*} A^3 &=& \operatorname {Sym}^3(\chi ) + \chi _7^{\mathrm { p}}\cdot c_2 + \chi _1\cdot c_3 - \chi _7^{\mathrm { p}} - \chi _{14}^{\mathrm { p}} \\ &=& (4\chi _1+7\chi _6+2\chi _7+3\chi _8) + \chi _1 - (\chi _1+2\chi _6+\chi _8)\\ &=& 4\chi _1+5\chi _6+2\chi _7+2\chi _8. \end{eqnarray*}

From this and Poincare duality, for $\dim (H^{2i})^{\textrm {GL}(3,2)}$ we get $1,1,3,4,3,1,1$ . By hard Lefschetz, $s_1^{2i}H^{6-2i} \simeq H^{6+2i}$ . This gives additive bases in the invariant subspaces of $H^8=A^4$ , $H^{10}=A^5$ , $H^{12}=A^6$ , and we check that our choices (leading to smaller formulas) also give bases. For $H^6$ , we get a subspace $s_1\cdot (H^4)^{\textrm {GL}(3,2)} = \langle s_1^3, c_2s_1, s_2s_1\rangle$ . Since these vectors are $S_7$ -invariant and $t$ is not, adding $t$ completes them to a basis of $(H^6)^{\textrm {GL}(3,2)}$ .

I checked the algebra relations and the fact that they suffice in sagemath.

Remark 6.5. The smaller subring of invariants $A^*(X//G)^{S_7}$ has $(1,1,3,3,3,1,1)$ for the dimensions of the graded pieces. It is generated by $s_1$ , $c_2$ , $s_2$ with the same relations, dropping those involving $t$ .

For § 6.3, I note the relation $c_3 = \frac 17 (2 s_1^3 - 6 s_2 s_1+17 c_2 s_1)$ .

6.3 Kappa classes

The moduli stack $\overline {{\mathcal {M}}}_{\mathrm { Cam}}$ is a global quotient $[\overline {M}:\textrm {GL}(3,2)]$ , where $\overline {M} = (\mathbb {P}^2)^7//\textrm {SL}(3,\mathbb {C})$ . Let $f\colon \mathcal {Y}\to \overline {M}$ be the universal family of stable pairs $(Y, \sum _{i=1}^7 \frac 12 B_i)$ (see [Reference Alexeev and PardiniAP23]). The kappa classes on $\overline {M}$ are

\begin{align*} \kappa _l = f_* L^{l+2} \quad \text {where}\quad L = \omega _{\mathcal {Y}/\overline {M}} \left ( \sum _{i=1}^7 \frac 12 B_i \right ). \end{align*}

I am grateful to William Graham for explaining to me how to do pushforward in equivariant cohomology, which is used in the proof of the next theorem.

Theorem 6.6. In the Chow ring $A^*(\overline {M}, \mathbb {Q})$ described above, one has

\begin{eqnarray*} 2^2\kappa _0 &=& 1,\\ 2^3\kappa _1 &=& 3s_1,\\ 2^4\kappa _2 &=& 6s_1^2 - c_2,\\ 2^5\kappa _3 &=& 10s_1^3 - 5s_1c_2 + c_3,\\ 2^6\kappa _4 &=& 15s_1^4 - 15s_1^2c_2 + c_2^2+6s_1c_3,\\ 2^7\kappa _5 &=& 21s_1^5 - 35s_1^3c_2+7s_1c_2^2+21s_1^2c_3 - 2c_2c_3,\\ 2^8\kappa _6 &=& 28s_1^6 - 70s_1^4c_2+28s_1^2c_2^2 - c_2^3+56s_1^3c_3 - 16s_1c_2c_3 + c_3^2. \end{eqnarray*}

Proof. Over $X = \mathbb {P} (V)^7$ we have the universal family $X\times \mathbb {P} (V^*)$ with seven divisors $\mathcal {B}_i$ , and the family over $\overline {M}$ is a quotient by a free action of $G=\textrm {SL}(V)$ . Therefore, it suffices to compute in $A^*_G(X)$ . Denote $h=\mathcal {O}_{\mathbb {P}(V^*)}(1)$ . Each $\mathcal {B}_i$ is the incidence divisor in $\mathbb {P}(V)\times \mathbb {P}(V^*)$ and is linearly equivalent to $z_i+h$ . Therefore,

\begin{align*} L = \omega _{X\times \mathbb {P}(V^*)/ X} \left ( \sum _{i=1}^7 \frac 12 B_i \right ) = -3h + \sum _{i=1}^7\frac 12(h+z_i) = \frac 12(h+s_1). \end{align*}

By projection formula, to compute $f_*L^{i+2}$ it suffices to know $f_* h^k$ under the homomorphism $A^*_G(\mathbb {P}(V^*))\to A^*_G(\cdot )=S$ induced by the morphism $\mathbb {P}(V^*)\to \textrm {pt}$ . For $s\in S^W$ one has $f_*(s)=f_*(hs)=0$ , $f_*(h^2s)=s$ , and the pushforwards of higher powers of $h$ follow by recursively using the basic relation $h^3 + c_2h - c_3=0$ . The rest is an easy computation.

Note that $s_1$ is the ample line bundle that comes with the GIT quotient construction, the $\mathcal {O}(1)$ on the $\textrm {Proj}$ of the graded algebra of invariants.