Gaps in the support of canonical currents on projective K3 surfaces

Fix a set $S$, whose elements we regard as formal symbols which can be juxtaposed to form words. Assume that $S$ is equipped with a fixed-point-free involution $s\mapsto s^{-1}$, i.e. any element has a unique corresponding “inverse” in the set. Define the “derived series” of sets by

where $[A,B]$ denotes the set of commutators $[a,b]:=aba^{-1}b^{-1}$ with $a\in A,b\in B$, and we omit the trivial commutators $[a,a^{-1}]$. Denote the disjoint union by $S^{\bullet}:=\coprod_{n\geq 0}S^{(n)}$. We will use the same notation in the case of a pseudogroup. We also collect the next elementary result:

Later on, we will apply iteratively this proposition, starting with $k\geq 4$, an inequality which is preserved by $k\mapsto{\binom{k}{2}}$.

Let $B_{0}(\varepsilon)\subset\mathbb{C}^{d}$ denote the ball of radius $\varepsilon>0$ centered at the origin in $\mathbb{C}^{d}$. Let $\gamma_{1},\ldots,\gamma_{k}$ be injective holomorphic maps $\gamma_{i}\colon B_{0}(\varepsilon)\to\mathbb{C}^{d}$, which are thus biholomorphisms onto their ranges $\mathcal{R}_{\gamma_{i}}:=\gamma_{i}(B_{0}(\varepsilon))$.

Let $S$ denote the set with $2k$ symbols $\gamma_{1},\ldots,\gamma_{k},\gamma_{1}^{-1},\ldots,\gamma_{k}^{-1}$. With $S^{\bullet}$ as in § 2.1.1, assign to any element $\gamma\in S^{\bullet}$, whenever possible, the holomorphic map also denoted by $\gamma\colon\mathcal{D}_{\gamma}\to\mathcal{R}_{\gamma}$ with open sets $\mathcal{D}_{\gamma},\mathcal{R}_{\gamma}\subset\mathbb{C}^{d}$ by expressing $\gamma$ in reduced form in the letters from $S$, and shrinking the domains/ranges according to the word. For certain elements $\gamma$, these might well be empty sets.

Denote by ${\bm{1}}$ the identity transformation and by $\left\|f\right\|_{C^{0}(K)}$ the supremum norm of a function or map $f$ on a set $K$.

Note also that without further assumptions on the $\gamma_{i}$, it could happen that $S^{(n)}$ contains only identity mappings. In our intended applications, this will be excluded as the elements will act nontrivially in cohomology.

The next result, appearing in [RR21, Lemma 7.2], will be useful in exhibiting explicit examples satisfying the assumptions of Theorem 2.1.4. We will denote by ${\bm{1}}$ both the identity map and the identity matrix acting on $\mathbb{C}^{d}$, and by $\left\|-\right\|_{\rm Mat}$ the matrix norm on $n\times n$ matrices.

We work over the complex numbers. Consider the $3$-fold $(\mathbb{P}^{1})^{3}$, with its family of smooth anticanonical divisors given by degree $(2,2,2)$-surfaces, i.e. let $\mathcal{U}\subset\mathbb{C}^{27}$ denote the parameter space of coefficients of an equation

that yield smooth surfaces when compactified in $(\mathbb{P}^{1})^{3}$. We will call these $(2,2,2)$-surfaces. We consider for simplicity the full set of equations, without identifying surfaces equivalent under the action of $(\operatorname{\mathbf{PGL}}_{2})^{3}$.

Note that an at most countable dense union of codimension-one subsets in $\mathcal{U}$ consists of non-strict $(2,2,2)$-surfaces. For strict $(2,2,2)$-surfaces, the Néron–Severi group equipped with its intersection form is isometric to $\mathbb{R}^{1,2}$ (after extension of scalars to $\mathbb{R}$).

Recall that $F_{\sigma}$-sets are countable unions of closed sets, while $G_{\delta}$-sets are countable intersections of open ones. It follows from standard results in the moduli theory of K3 surfaces that strict $(2,2,2)$-surfaces form a dense $G_{\delta}$-set in $\mathcal{U}$, which in fact has full Lebesgue measure. Indeed, parameters giving strict $(2,2,2)$-surfaces are the complement of countably many divisors in the full parameter space, see e.g. [Ogu03].

For any $u\in\mathcal{U}$, denote the associated surface by $X_{u}\subset(\mathbb{P}^{1})^{3}$. The projection onto one of the coordinate planes $X_{u}\to(\mathbb{P}^{1})^{2}$ is two-to-one and so $X_{u}$ admits an involution exchanging the two sheets. Denote by $\sigma_{x},\sigma_{y},\sigma_{z}$ the three involutions obtained in this manner.

We can apply [FT23, Thm. 1] to any strict $(2,2,2)$-surface $X_{u}$ with $u\in\mathcal{U}$. In that theorem a certain space $\partial^{\circ}\operatorname{{Amp}}_{c}(X_{u})$ appears, which on strict $(2,2,2)$-surfaces reduces to the boundary of the ample cone $\partial\operatorname{{Amp}}(X_{u})$, so it consists of nef cohomology classes $[\eta]\in\operatorname{{NS}}_{\mathbb{R}}(X_{u})\subset H^{1,1}(X_{u})$ satisfying $[\eta]^{2}=0$. Since $\operatorname{{NS}}_{\mathbb{R}}(X_{u})$ equipped with the intersection pairing is isometric to $\mathbb{R}^{1,2}$, the space $\partial\operatorname{{Amp}}(X_{u})$ is isomorphic to one component of the null-cone in this Minkowski space. Note that in the general form of the result, one needs to replace the rational rays in $\partial\operatorname{{Amp}}(X_{u})$ by their blowups; since in the case of a rank $3$ Néron–Severi group it would mean blowing up rays on a surface, no extra points need to be added.

Next, [FT23, Thm. 1] shows that each cohomology class $[\eta]\in\partial\operatorname{{Amp}}(X_{u})$ has a canonical positive representative $\eta$, which additionally has $C^{0}$ potentials. The representative is unique when the class is irrational, and a preferred representative in the rational (also called parabolic) classes exists that makes the entire family of currents continuous in the $C^{0}$-topology of the potentials for the currents.

We will show in Theorem 2.3.5 below that some of the canonical representatives do not have full support in $X_{u}$. Specifically, we will show that there exists an open set $\mathcal{U}_{0}\subset\mathcal{U}$ and a dense $G_{\delta}$ set of $u\in\mathcal{U}_{0}$ for which some of the canonical currents $\eta$ do not have full support in $X_{u}$.

But first, we will show that the set of cohomology classes $[\eta]$ for which the gaps in the support are constructed contain, after projectivization, a closed uncountable set.

We will consider subgroups of automorphisms of $X_{u}$ freely generated by five elements. Specifically, $\sigma_{x},\sigma_{y},\sigma_{z}$ generate a group $\Gamma_{\sigma}\subseteq\operatorname{{Aut}}(X_{u})$ isomorphic to $\left(\mathbb{Z}/2\right)*\left(\mathbb{Z}/2\right)*\left(\mathbb{Z}/2\right)$, in other words there are no relations between them except that $\sigma_{i}^{2}={\bm{1}}$ for $i=x,y,z$. This can be verified by considering the action on the hyperbolic space inside the Néron–Severi group of $X_{u}$ (see for instance [Fil19, Prop.6.1] for the explicit matrices corresponding to the action in the upper half-space model).

The above homomorphism corresponds to evaluating the derivatives of the transformations at the common fixed point of the transformations described in § 2.3.4. § 2.2.6 will provide us with a free group with five generators, each of whose derivative at the fixed point is the identity.

To continue, we select $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}\subset K_{\sigma}$ to be five elements freely generating the group. Next, the construction of § 2.1.1 applies with $S:=\{\gamma_{1},\ldots,\gamma_{5},\gamma_{1}^{-1},\ldots\gamma_{5}^{-1}\}$ and yields a subset $S^{\bullet}\subset\operatorname{{Aut}}(X_{u})$ consisting of iterated commutators. Fix a Kähler metric $\omega_{0}$ on $X_{u}$, with volume normalized to $[\omega_{0}]^{2}=1$, and let $\mathbb{H}^{2}(X_{u})$ denote the hyperbolic plane of all nef cohomology classes satisfying $[\omega]^{2}=1$.

To show that the assumptions of § 2.1.3 are satisfied in practice, we start with an explicit equation:

Let us note that Eqn. (missing) 2.3.2 determines a singular $(2,2,2)$-surface, with a singularity at the origin $0\in\mathbb{C}^{3}$. We will construct an open set $\mathcal{U}_{0}$ of smooth $(2,2,2)$-surfaces by taking perturbations of the above equation.

Let $u_{0}\in\mathbb{C}^{27}$ denote the point corresponding to the choice of parameters as in Eqn. (missing) 2.3.2, it lies outside $\mathcal{U}$ but any analytic neighborhood of $u_{0}$ in $\mathbb{C}^{27}$ intersects $\mathcal{U}$ in a nonempty open set. We have three explicit involutions $\sigma_{u_{0},x},\sigma_{u_{0},y},\sigma_{u_{0},z}$:

and similarly for $\sigma_{u_{0},y},\sigma_{u_{0},z}$, which we view as holomorphic maps defined in a neighborhood of $0\in\mathbb{C}^{3}$.

It is immediate from the explicit formulas that all three involutions preserve the point $0\in\mathbb{C}^{3}$. Furthermore, their derivatives at that point are matrices of order two:

We now consider $\sigma_{u,x},\sigma_{u,y},\sigma_{u,z}$ for $u\in\mathbb{C}^{27}$ in a sufficiently small neighborhood of $u_{0}$. Then we can regard the $\sigma$’s as holomorphic maps defined in a neighborhood of $0\in\mathbb{C}^{3}$, preserving the intersection of $X_{u}$ with the fixed neighborhood.

We can now use these observations to establish:

Let $X_{0}$ denote the (singular) surface

compactified in $(\mathbb{P}^{1})^{3}$. Its real locus $X_{0}(\mathbb{R})$ is a real $2$-dimensional sphere.

As before let $\mathcal{U}\subset\mathbb{R}^{27}$ be the subset of smooth $(2,2,2)$-surfaces, parametrized by the possible coefficients, and normalized such that the parameter $0\in\mathbb{R}^{27}$ corresponds to $X_{0}$. Note that $0\notin\mathcal{U}$. Let next $\mathcal{U}^{\prime}\subset\mathcal{U}$ denote the subset of strict $(2,2,2)$-surfaces. By the discussion in § 2.2.3 the set $\mathcal{U}^{\prime}$ is the complement of countably many divisors in $\mathcal{U}$, and thus forms a dense $G_{\delta}$ set.

Let us first observe that the involution $\sigma_{x}$ acting on the surface $X_{0}$ in § 2.4.1 is given by $\sigma_{x}(x,y,z)=(-x,y,z)$, and analogously for $\sigma_{y},\sigma_{z}$. Therefore, let $K_{\sigma}\subset\Gamma_{\sigma}$ be the group from § 2.2.6 obtained as the kernel of this action; it is a free group on five generators $\gamma_{i}$ and acts nontrivially on any smooth and strict $(2,2,2)$-surface.

Even for smooth surfaces $X_{u}\subset(\mathbb{P}^{1})^{3}$, we will be interested only in their intersection with the affine chart $\mathbb{C}^{3}$, and specifically a neighborhood of $X_{0}(\mathbb{R})$. We will thus restrict to a neighborhood in $\mathcal{U}$ of $u=0$ for which no additional real components arise.

Choose a finite cover of $X_{0}(\mathbb{R})\subset\mathbb{R}^{3}$ by open sets $V_{i}\subset\mathbb{C}^{3}$ such that we have biholomorphisms $\phi_{i}\colon V_{i}\to B_{0}(1)\subset\mathbb{C}^{3}$ to a ball of radius $1$ around $0$, and the preimages of the smaller balls $V_{i}^{\prime}:=\phi_{i}^{-1}\left(B_{0}(\tfrac{1}{4})\right)$ still cover $X_{0}(\mathbb{R})$.

Choose now a sufficiently small open neighborhood of the origin $\mathcal{U}_{0}\subset\mathbb{R}^{27}$ such that the following property is satisfied: For each of the five generators $\gamma_{j}$ of $K_{\sigma}$ and their inverses, we have for every chart $V_{i}$ that $\gamma_{ij}^{\prime}:=\phi_{i}\circ\gamma_{j}\circ\phi^{-1}_{i}$ satisfies:

Require also that for any $u\in\mathcal{U}_{0}$ that $X_{u}(\mathbb{R})$ is nonempty and still covered by the sets $\{V_{i}^{\prime}\}$.