Non-measure hyperbolicity of complex K3 surfaces

Gunhee Cho Department of Mathematics
University of California, Santa Barbara
South Hall, Room 6607
Santa Barbara, CA 93106. gunhee.cho@math.ucsb.edu and David R. Morrison Department of Mathematics
University of California, Santa Barbara
South Hall, Room 6607
Santa Barbara, CA 93106. drm@math.ucsb.edu

Abstract.

We show that the non-measure hyperbolicity of K3 surfaces—which M. Green and P. Griffiths verified for certain cases in 1980—holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of points on K3 surfaces. We also obtain a new proof of the non-measure hyperbolicity of any Enriques surface.

Key words and phrases:

Non-measure hyperbolicity, K3 surfaces, Enriques surfaces

1. Introduction and results

The Kobayashi pseudometric on a complex manifold is a generalization of the Kähler-Einstein metric with constant negative holomorphic sectional curvature. It corresponds to the Poincaré metric in complex hyperbolic space. According to Brody’s theorem, the non-existence of entire curves (i.e., non-constant holomorphic maps from $\mathbb{C}^{1}$ to $M$ ) on a compact complex manifold $M$ is equivalent to the non-vanishing of the Kobayashi pseudometric [MR0470252]. S. Kobayashi conjectured that all compact Calabi-Yau manifolds have a vanishing Kobayashi pseudometric [MR414940]. Numerous studies have focused on demonstrating the non-existence of entire curves based on Brody’s work (for example, [MR0726433, MR3209357, MR3644248, MR4124989, MR3649666]). For a comprehensive introduction to the subject, refer to the cited sources [MR1492539, MR3524135]. Notable problems in this area include the Green-Griffiths and Lang conjectures, which involve the Demailly-Semple vector bundle approach for hypersurfaces of high degree [MR0828820, MR609557]. Another method involves applying deformation to hyper-Kähler manifolds, including all complex K3 surfaces, as presented by Kamenova, Lu, and Verbitsky [MR3263959].

A more challenging problem than the vanishing Kobayashi pseudometric is the vanishing Kobayashi-Eisenman pseudovolume. This is because the vanishing Kobayashi-Eisenman pseudovolume implies the vanishing Kobayashi pseudometric. However, the non-existence of entire curves cannot be directly applied, necessitating a new approach. In the case of the Demaily-Semple vector bundle approach, there have been intermittent attempts to extend the domain of entire curves to $\mathbb{C}^{k}$ instead of $\mathbb{C}^{1}$ , in comparison to the study of entire curves (see, for example, [MR2818709]).

It is also worth noting that the following version of the Kobayashi conjecture [MR414940] is well-known in the classical literature. Recent updates on this conjecture can be found in [MR2132645] and [MR4068832]. This conjecture is based on the observation that measure hyperbolicity holds for any variety of general type over $\mathbb{C}$ . This implication follows from a result of F. Sakai [MR0590433], and result of Sakai was generalized to all intermediate Kobayashi–Eisenman measures in the paper by S. Kaliman [MR1722804] (also see the definition of measure hyperbolicity in Definition 5 below).

Conjecture 1.

An $n$ -dimensional algebraic variety $X$ over $\mathbb{C}$ is measure hyperbolic if and only if $X$ is of general type.

Complex K3 surfaces are a significant class of compact complex surfaces that have been extensively studied in both differential geometry and algebraic geometry (see, for example, [MR0707352, MR0728142] and the references therein). M. Green and P. Griffiths [MR609557] confirmed Conjecture 1 for certain classes of complex algebraic K3 surfaces $X$ , including 2-sheeted coverings $X\rightarrow\mathbb{P}^{2}$ that are branched over a smooth curve of degree six, and non-degenerate smooth surfaces $X\subset\mathbb{P}^{3}$ (respectively $X\subset\mathbb{P}^{4}$ ) with degrees 4 (respectively 6) and geometric genus $p_{g}\neq 0$ . In those three cases, they were able to show that the complex algebraic K3 surface is rationally swept out by elliptic curves. It is now known that all complex algebraic K3 surfaces have this property [MR3586372, Corollary 13.2.2]. It then follows (from [MR609557] or [MR414940]) that Conjecture 1 holds for all complex algebraic K3 surfaces.

In this paper, we verify conjecture 1 for all complex K3 surfaces:

Theorem 2 (Corollary 20).

Every complex K3 surface $X$ is non-measure hyperbolic.

The key idea is to deform the complex structure of a complex K3 surface to achieve one with vanishing Kobayashi-Eisenman pseudovolume. To accomplish this, we consider the moduli space of (“marked”) complex K3 surfaces and focus on the collection of elliptic K3 surfaces with a section, which forms a dense subset in this moduli space and satisfies the property of vanishing Kobayashi-Eisenman pseudovolume. Exploiting the upper semicontinuity of the Kobayashi-Eisenman pseudovolume (and pseudometric) under deformations of complex structures, we can establish the vanishing of volume for any complex K3 surface. Recent research by the first author has explored how the deformations of complex structure can be utilized to prove Kobayashi hyperbolic embedding through toric geometry [GCJY21].

An interesting consequence of Theorem 2 is the vanishing of the Kobayashi pseudovolume on any $n$ th punctual Hilbert schemes of a complex K3 surface, which appears to be previously unknown. Our result provides a stronger condition than the Kobayashi non-hyperbolicity for any complex K3 surface and for any $n$ th punctual Hilbert schemes of a complex K3 surface, which had been demonstrated by L. Kamenova, S. Lu, and M. Verbitsky [MR3263959].

Corollary 3 (Corollary 21).

Every Hilbert schemes of points on K3 surfaces is non-measure hyperbolic.

Also, Theorem 2 yields a new proof of the vanishing of the Kobayashi pseudovolume on Enriques surfaces, which had previously been proven in [MR609557].

This paper is structured as follows:

Section 2 provides the definitions of the Kobayashi pseudometric and Kobayashi-Eisenman pseudovolume. In Section 3, we discuss complex K3 surfaces and their moduli space, and describe the collection of elliptic K3 surfaces with a section, which forms a dense subset of this moduli space. Section 4 focuses on the proof of the vanishing Kobayashi-Eisenman pseudovolume specifically for elliptic K3 surfaces with a section. Section 5 establishes the vanishing of the Kobayashi-Eisenman pseudovolume on any complex K3 surface, leveraging the upper semicontinuity of the Kobayashi-Eisenman pseudovolume. In conclusion, we examines the vanishing properties of the Kobayashi-Eisenman pseudovolume for complex K3 surfaces and any $n$ th punctual Hilbert schemes of a complex K3 surface. Additionally, we address the non-measure hyperbolicity of Enriques surfaces.

Acknowledgments

We thank Ariyan Javanpeykar, Ljudmila Kamenova, and Steven Lu for helpful correspondence. We also thank Mikhail Zaidenberg for comments on upper-semicontinuity of Kobayashi-Eisenman pseudovolume and relevant references. GC is partially supported by Simons Travel funding. DRM is partially supported by National Science Foundation Grant #PHY-2014226.

2. Preliminaries

2.1. Kobayashi–(Royden) pseudometric, Kobayashi-Eisenman pseudovolume

We always denote $\triangle$ and $\triangle_{r}$ the unit disk and the disk of a radius $r$ in $\mathbb{C}$ . Let $M$ be a complex manifold of dimension $n$ . Let $p\in M$ . We denote by $T_{p}M$ (resp. $T_{M}$ ) the holomorphic tangent space to $M$ at $p$ (resp. the holomorphic tangent bundle).

Definition 4.

(1) The Kobayashi–Royden pseudo-metric $KR_{X}$ is defined by

KR_{X}(u)=\inf_{\phi}\left\{\frac{1}{|\mu|},\phi_{*}\left(\frac{\partial}{\partial t}\right)=\mu u\right\}

for $u\in T_{X,x}$ . Here $\phi$ runs over the set of holomorphic maps from the unit disk $\triangle$ to $X$ , such that $\phi(0)=x$ .

(2) The Kobayashi–Eisenman pseudo-volume $\Psi_{X}$ on the $n$ -dimensional complex manifold $X$ is the pseudo-volume form whose associated Hermitian pseudo-norm on $\bigwedge^{n}T_{X}$ is defined by

\Psi_{X}(\zeta)=\inf_{\phi}\left\{\frac{1}{|\mu|},\phi_{*}\left(\frac{\partial}{\partial t_{1}}\wedge\ldots\wedge\frac{\partial}{\partial t_{n}}\right)=\mu\zeta\right\},

for $\zeta\in\bigwedge^{n}T_{X,x},n=\operatorname{dim}X$ . Here $\phi$ runs over the set of holomorphic maps from $\triangle^{n}$ to $X$ such that $\phi(0)=x$ .

(3) More generally the Eisenman p-pseudo-volume form $\Psi_{X}^{p}$ is defined as in (2), replacing $n=\operatorname{dim}X$ by any $p$ comprised between 1 and $n$ . It is then defined only on p-vectors $\zeta\in\bigwedge^{p}T_{X,x}$ which are decomposable, that is $\zeta=u_{1}\wedge\ldots\wedge u_{p}$ .

We clearly have

\Psi_{X}^{1}=KR_{X},\Psi_{X}^{n}=\Psi_{X}.

Intuitively, $KR_{X}(p;v)$ measures the maximal radius of one-dimensional disk embedded holomorphically in $X$ in the direction of $v$ (see also [GunheeChoJunqingQian20] for the concrete formula for very simple Riemann surfaces). It is only a pseudo-metric since it is possible that $KR_{X}(p;v)=0$ for some non-zero tangent vector. The Kobayashi-Royden pseudometric is well defined as a pseudo-metric on arbitrary complex manifolds, and it coincides with the Poincaré metric in the case of a complex hyperbolic space. The Kobayashi pseudo-distance $d_{X}$ on $X$ is defined by

d_{X}(x,y)=\inf_{f_{i}\in\textup{Hol}(\triangle,X)}\left\{\sum_{i=1}^{n}\rho_{\triangle}(a_{i},b_{i})\right\},

where $x=p_{0},...,p_{n}=y,f_{i}(a_{i})=p_{i-1},f_{i}(b_{i})=p_{i}$ and $\rho_{\triangle}(a,b)$ denotes the Poincaré distance between two points $a,b\in\triangle$ . Royden proved that the infimum of the arc-length of all piecewise $C^{1}$ -curves with respect to the Kobayashi–Royden metric induces the Kobayashi pseudo-distance $d_{X}$ [MR0291494].

The upper semicontinuity of the Kobayashi-Eisenman measures on a variety was established as a lemma in [MR0367300] of Eisenman (publishing under the surname Pelles). One of the results there says that the $k$ th Kobayashi–Eisenman measure is insensible to removing an analytic subset of codimension k+1 or more [MR1321580]. For the Kobayshi-Royden pseudometric, this is a result of [MR0425186]. (Also, see [MR1323721, MR1363172]).

Definition 5.

A complex manifold $X$ is said to be measure hyperbolic if the Kobayashi–Eisenman pseudo-volume does not vainish on any open set, or equivalently $\Psi_{X}$ is nonzero almost everywhere.

The following decreasing property is clear from the definition of Kobayashi-Eisenman volume.

Lemma 6.

Let $\phi:X\rightarrow Y$ be a holomorphic map between two complex manifolds. Then

\phi^{*}\Psi_{Y}\leq\Psi_{X}.

Also, we have the following:

Lemma 7.

Let $X_{1},X_{2}$ be complex manifolds. Denote the $i$ th projection $pr_{i}:X_{1}\times X_{2}\rightarrow X_{i},i=1,2$ . Then

\Psi_{X_{1}\times X_{2}}\leq pr^{*}_{1}\Psi_{X_{1}}\otimes pr^{*}_{2}\Psi_{X_{2}}.

Proof.

[MR2132645, Lemma 1.12]. ∎

3. Complex K3 surfaces

Definition 8.

A complex K3 surface is a compact connected complex manifold $X$ of complex dimension $2$ such that the canonical line bundle is trivial and $H^{1}(X,\mathcal{O}_{X})=0$ . Note that the triviality of the canonical bundle is equivalent to the existence of a nowhere-vanishing holomorphic $2$ -form $\omega_{X}$ , and that such a $2$ -form is unique up to a (complex) scalar multiple.

In this paper, ‘K3 surface’ shall mean a complex K3 surface in this sense. Not all K3 surfaces are algebraic.

3.0.1. Moduli space of K3 surfaces

There is a well-known and well-studied moduli space for K3 surfaces which is formulated in terms of the periods of the holomorphic $2$ -form. There are several versions of this moduli space; the one of greatest interest to us will be the moduli space of marked K3 surfaces. To construct this, we declare a marking of a complex K3 surface $X$ to be a choice of isometry $\alpha:H^{2}(X,\mathbb{Z})\to L$ , where $L$ is a fixed free ${\mathbb{Z}}$ -module of rank $22$ carrying a $\mathbb{Z}$ -valued inner product $\langle\quad,\quad\rangle$ of signature $(3,19)$ such that $\langle v,v\rangle$ is an even integer for each $v\in L$ . (Such a module is known [MR0255476] to be isometric to the orthogonal direct sum of two copies of the sign-reversed $E_{8}$ lattice plus three copies of the “hyperbolic plane” $U$ (the even integral inner product space whose inner product has matrix $\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$ in an appropriate basis)). A marking induces a dual map (again denoted by $\alpha$ ): $\alpha:H_{2}(X,\mathbb{Z})\to L^{*}\cong L$ . Two marked K3 surfaces are said to be isomorphic if there is an isomorphism compatible with the markings; the set of isomorphism classes of marked K3 surfaces is known to be a complex manifold of complex dimension 20, although it is not Hausdorff (see [MR3586372, Section 7.2]).

The moduli space of marked K3 surfaces can also be regarded as the Teichmüller space of K3 surfaces. Since autorphisms of a K3 surface $M$ always act nontrivially on its second cohomology [MR3586372, Proposition 15.2.1], the mapping class group

\Gamma:=\operatorname{Diff}(M)/\operatorname{Diff}_{0}(M)

(where $\operatorname{Diff}_{0}(M)$ is the connected component of the diffeomorphism group) can be regarded as the group which permutes markings. Thus, if we let Comp denote the space of complex structures on $M$ , equipped with a structure of a Fréchet manifold, and we let the Teichmüller space be $\operatorname{Teich}:=\operatorname{Comp}/\operatorname{Diff}_{0}(M)$ then the quotient Comp / Diff $=$ Teich $/\Gamma$ is the moduli space of complex structures on $M$ , and $\operatorname{Teich}$ is the moduli space of marked K3 surfaces.

Given a marked K3 surface $X$ with marking $\alpha$ , one fixes a basis $\gamma_{1},\dots,\gamma_{22}$ of $L^{*}\cong L$ and for each nonzero holomorphic $2$ -form $\omega_{X}$ on $X$ defines the period point

\pi(X,\alpha):=(\int_{\alpha^{-1}(\gamma_{1})}\omega_{X},\dots,\int_{\alpha^{-1}(\gamma_{22})}\omega_{X})\in\mathbb{P}^{21}\

which is a well-defined point in projective space independent of $\omega_{X}$ since $\omega_{X}$ is unique up to a nonzero complex multiple. Each $\pi(X,\alpha)$ lies in the subset

\Omega:=\{\omega\in\mathbb{P}^{21}|\langle\omega,\omega\rangle=0\}

where we endow $\mathbb{P}^{21}=\mathbb{P}(L\otimes_{\mathbb{Z}}\mathbb{C})$ with the inner product $\langle\quad,\quad\rangle$ induced from that of $L$ . The period map $(X,\alpha)\mapsto\pi(X,\alpha)$ is known to be a local isomorphism (this is the “local Torelli theorem”), and this is what gives the moduli space the structure of a complex manifold of complex dimension 20.

3.1. Elliptic K3 surfaces with section

An elliptic K3 surface $X$ is a K3 surface which admits a holomorphic mapping $\pi:X\to\mathbb{P}^{1}$ whose general fiber $\pi^{-1}(t)$ is a complex curve of genus one. We say that $X$ is an elliptic K3 surface with a section if there is a curve $C_{0}\subset X$ (called the “section”) such that $\pi|_{C_{0}}$ establishes an isomorphism from $C_{0}$ to $\mathbb{P}^{1}$ .

It is known that a K3 surface $X$ is an elliptic K3 surface with a section if and only if there exists an embedding $U\rightarrow NS(X)$ of the hyperbolic plane into the Néron-Severi group $NS(X)$ of $X$ . (For a marked K3 surface, NS(X) can be identified with $\pi(X,\alpha)^{\perp}\cap\overline{\pi(X,\alpha)}^{\perp}\cap L$ .)

Elliptic K3 surfaces with section are quite common among K3 surfaces, as shown by the following proposition (proven in [MR3586372, Remark 14.3.9] or [MR749574, Chapter VIII]).

Proposition 9.

Elliptic K3 surfaces with a section are dense in the moduli space of all marked K3 surfaces and also in the moduli spaces $M_{d}$ of polarized K3 surfaces of fixed degree.

4. Vanishing Kobayashi-Einsenman volume on Elliptic k3 surfaces with a section

In this section, we prove

Proposition 10.

For any elliptic K3 surface $\pi:X\rightarrow\mathbb{P}^{1}$ with a section, the Kobayashi-Eisenman volume on $X$ is vanishing.

Proposition 10 follows from [MR1738063] which assumes dominability by $\mathbb{C}^{2}$ , but we proceed with the explicit construction.

For an elliptic K3 surface $\pi:X\rightarrow\mathbb{P}^{1}$ with a section, the section meets every fibre transversally, and it meets a singular fibre $X_{t}=\sum m_{i}C_{i}$ in precisely one of the irreducible components $C_{i_{0}}$ and in a smooth point for which $m_{i_{0}}=1$ . By Kodaira’s classfication of singular fibres the reduced curve $\sum_{i\neq i_{0}}C_{i}$ is an ADE curve which can be contracted to a simple surface singularity.

Due to the fact that $\pi:X\rightarrow\mathbb{P}^{1}$ always has a Weierstrass model $\overline{X}$ which is birational to $X$ , Proposition 10 immediately follows from Proposition 11 and Proposition 12.

Proposition 11.

Let $\tau:M\rightarrow M^{\prime}$ be a birational equivalence of complex manifolds. Suppose that the Kobayashi-Eisenman pseudo-volume on $M$ vanishes. Then it vanishes on $M^{\prime}$ .

Proof.

Apply [MR2132645, Lemma 1.14]. ∎

Proposition 12.

Consider an elliptic K3 surface $\pi:X\rightarrow\mathbb{P}^{1}$ with a section. Then the Kobayashi-Eisenman volume of the Weierstrass model $\overline{X}$ vanishes.

Proof of Proposition 12.

We first construct the Weierstrass model $\overline{X}$ of $X$ . The construction of Weirestrass model for an elliptic K3 surface with a section is well-known (for example, [MR3586372, Section 11.2]). We provide the proof for the sake of completeness.

Fix a section $C_{0}\subset X$ of $\pi:X\rightarrow\mathbb{P}^{1}$ . Then we construct the desired map from a Weierstrass model $\overline{X}$ of $X$ as follows: the exact sequence $0\rightarrow\mathcal{O}_{X}\rightarrow\mathcal{O}(C_{0})\rightarrow\mathcal{O}_{C_{0}}(-2)\rightarrow 0$ induces the long exact sequence

0\rightarrow\mathcal{O}_{\mathbb{P}^{1}}\rightarrow\pi_{*}\mathcal{O}(C_{0})\rightarrow\mathcal{O}_{\mathbb{P}^{1}}(-2)\rightarrow R^{1}\pi_{*}\mathcal{O}_{X}\rightarrow 0,

where the vanishing $R^{1}\pi_{*}\mathcal{O}(C_{0})=0$ follows from the corresponding vanishing on the fibres. Note that $\pi_{*}\mathcal{O}(C_{0})$ is a line bundle, as $h^{0}(X_{t},\mathcal{O}(p))=1$ for any point $p\in X_{t}$ in an arbitrary fibre $X_{t}$ . (It is enough to test smooth fibres, as $\pi_{*}\mathcal{O}(C_{0})$ is torsion free.) Thus, the cokernel of $\mathcal{O}_{\mathbb{P}^{1}}\rightarrow\pi_{*}\mathcal{O}(C_{0})$ is torsion, but also contained in the torsion free $\mathcal{O}_{\mathbb{P}^{1}}(-2)$ . Hence,

\mathcal{O}_{\mathbb{P}^{1}}\simeq\pi_{*}\mathcal{O}(C_{0})\text{ and }\mathcal{O}_{\mathbb{P}^{1}}(-2)\simeq R^{1}\pi_{*}\mathcal{O}_{X}.

Similarly, using the short exact sequences

0\rightarrow\mathcal{O}(C_{0})\rightarrow\mathcal{O}(2C_{0})\rightarrow\mathcal{O}_{C_{0}}(-4)\rightarrow 0,

and

0\rightarrow\mathcal{O}(2C_{0})\rightarrow\mathcal{O}(3C_{0})\rightarrow\mathcal{O}_{C_{0}}(-6)\rightarrow 0,

we can deduce that $\pi_{*}\mathcal{O}(2C_{0})\simeq\mathcal{O}_{\mathbb{P}^{1}}(-4)\oplus\mathcal{O}_{\mathbb{P}^{1}}$ and $\pi_{*}\mathcal{O}(3C_{0})\simeq\mathcal{O}_{\mathbb{P}^{1}}(-4)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-6)\oplus\mathcal{O}_{\mathbb{P}^{1}}$ . Let $F:=\mathcal{O}_{\mathbb{P}^{1}}(-4)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-6)\oplus\mathcal{O}_{\mathbb{P}^{1}}$ . Thus, the linear system $\mathcal{O}(3C_{0})_{|_{X_{t}}}$ on the fibres $X_{t}$ (or rather the natural surjection $\pi_{*}F=\pi^{*}\pi_{*}\mathcal{O}(3C_{0})\rightarrow\mathcal{O}(3C_{0})$ ) defines a morphism

\varphi:X\rightarrow\mathbb{P}(F^{*})

with $\varphi^{*}\mathcal{O}(1)\simeq\mathcal{O}_{p}(3C_{0})$ , which is a closed embedding of the smooth fibres and contracts all components of singular fibres $X_{t}$ that are not met by $C_{0}$ (since $\mathcal{O}(3C_{0})$ is indeed base point free on all fibres). The image $\overline{X}$ is the Weierstrass model of the elliptic surface $X$ . Using the Riemann-Roch theorem on the fibers, one learns that there is an equation satisfied by the image $\overline{X}\subset\mathbb{P}(F^{*})$ , which can be regarded as a section

f\in H^{0}(\mathbb{P}(F^{*}),\mathcal{O}_{p}(3)\otimes p^{*}\mathcal{O}_{\mathbb{P}^{1}}(6d)).

for some degree $d$ .

In order to determine which $d$ corresponds to a K3 surface, we use the adjunction formula $\omega_{\overline{X}}\simeq(\omega_{\mathbb{P}(F^{*})}\otimes\mathcal{O}(\overline{X}))|_{\overline{X}}$ and the relative Euler sequence expressing $\omega_{\mathbb{P}(F^{*})}$ to show that $\omega_{\overline{X}}=O_{\overline{X}}$ if and only if $\mathcal{O}(\overline{X})\simeq\mathcal{O}_{p}(3)\otimes p^{*}\mathcal{O}_{\mathbb{P}^{1}}(12)$ , i.e., $d=2$ .

Now use $H^{0}(\mathbb{P}(F^{*}),\mathcal{O}_{p}(3)\otimes p^{*}\mathcal{O}_{\mathbb{P}^{1}}(12))\simeq H^{0}(\mathbb{P}^{1},Sym^{3}(F)\otimes\mathcal{O}_{\mathbb{P}^{1}}(12))$ and view $x,y$ , and $z$ as the local coordinates of the direct summands $\mathcal{O}_{\mathbb{P}(-4)},\mathcal{O}_{\mathbb{P}(-6)}$ , and $\mathcal{O}_{\mathbb{P}}$ of $F$ . By a change of coordinates, the equation $f$ can be put into Weierstrass form (cf. [silverman]):

y^{2}z=4x^{3}-g_{2}xz^{2}-g_{3}z^{3}

(4.1)

with coefficients

g_{2}\in H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(8)),\text{ }g_{3}\in H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(12)),

The terms in the Weierstrass equation can be seen as a sections of $p_{*}\mathcal{O}_{p}(3)\otimes\mathcal{O}_{\mathbb{P}^{1}}(12)$ , which implies, for example, that $g_{2}$ can be interpreted as a section of $\mathcal{O}_{\mathbb{P}^{1}}(8)=[xz^{2}]\mathcal{O}(-4)_{\mathbb{P}^{1}}\otimes\mathcal{O}_{\mathbb{P}^{1}}(12)$ . The discriminant is the non-trivial section $\triangle:={g_{2}}^{3}-27{g_{3}}^{2}\in H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(24))$ . Applying the standard coordinate changes, one can always reduce to the situation that $f$ has this form and $(g_{2},g_{3})$ is unique up to passing to $(\lambda^{4}g_{2},\lambda^{6}g_{3})$ , where $\lambda$ is a function which is non-vanishing away from the singular fibers.

For a given elliptic K3 surface with section $X$ , we let $g_{2}(t)$ and $g_{3}(t)$ be Weierstrass coeffients, and $\triangle(t)=g_{2}(t)^{3}-27g_{3}(t)^{2}$ be the discriminant locus. Let $S_{X}$ be the set of zeroes of the discriminant $\triangle(t)$ for $t\in\mathbb{P}^{1}$ , and $R_{X}=\mathbb{P}^{1}-S_{X}$ . We define a holomorphic map $F:\mathbb{C}\times R_{X}\rightarrow\overline{X}$ by using the Weierstrass $\wp$ -function associated with (4.1), as follows (see [MR1027834]). First, to a pair $(g_{2},g_{3})$ of complex numbers satisfying $g_{2}^{3}-27g_{3}^{2}\neq 0$ , we associate a lattice $\Lambda\subset\mathbb{C}$ which is the set of possible values

\omega=\int_{\gamma}\frac{dx}{\sqrt{4x^{3}-g_{2}x-g_{3}}},

where $\gamma$ ranges over all possible contours of integration defined on the double cover of the complex $x$ -plane defined by $\sqrt{4x^{3}-g_{2}x-g_{3}}$ . (The condition $g_{2}^{3}-27g_{3}^{2}\neq 0$ ensures that the cubic $4x^{3}-g_{2}x-g_{3}$ does not have any repeated roots, so that the double cover is ramified at precisely $4$ points, including infinity.) Note that the elements $\omega\in\Lambda$ depend holomorphically on $g_{2}$ and $g_{3}$ .

The Weierstrass $\wp$ -function is defined by

\wp_{\Lambda}(z)=\frac{1}{z}+\sum_{0\neq\omega\in\Lambda}\left\{\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right\},

which converges uniformly on compact subsets of $\mathbb{C}-\Lambda$ to a doubly-periodic meromorphic function with poles at $z=\omega\in\Lambda$ and periods $\wp_{\Lambda}(z+\omega)=\wp_{\Lambda}(z)$ for $\omega\in\Lambda$ . Note that this function also depends holomorphically on $g_{2}$ and $g_{3}$ .

Using the Weiestrass $\wp$ -function, we define a holomorphic map $F:\mathbb{C}\times R_{X}\rightarrow\overline{X}$ starting from the meromorphic map $F=[\wp_{\Lambda}(z),\wp^{\prime}_{\Lambda}(z),1]$ for $z\in\mathbb{C}$ and extending holomorphically over the poles of $\wp_{\Lambda}$ . The image of $F$ satisfies (4.1) with Weierstrass coefficients

g_{2}(\Lambda)=60\sum_{0\neq\omega\in\Lambda}\frac{1}{\omega^{4}};\quad g_{3}(\Lambda)=140\sum_{0\neq\omega\in\Lambda}\frac{1}{\omega^{6}}

(see for example [MR1027834]).

Now apply Lemma 6, i.e., the decreasing property of the Kobayashi-Eisenman volume. Since the Kobayashi-Eisenman (in fact Kobayashi-Royden metric) pseudovolume on the first component $\mathbb{C}$ is zero and by Lemma 7, the proof is complete.

∎

5. Vanishing Kobayashi-Eisenman pseudo-volume on K3 surfaces

5.1. Upper semi-continuity of Kobayashi-Eisenman volume

We are interested in the upper semicontinuity of $\Psi_{X_{t}}$ in the variable $t$ for a proper smooth fibration $\pi:\mathcal{M}\rightarrow T$ , i.e., $\pi$ is holomorphic, surjective, having everywhere of maximal rank and connected fibers $X_{t}=\pi^{-1}(t)$ . We say a function $F$ on a topological space $X$ with values $\mathbb{R}\cup\left\{\infty\right\}$ is upper semi-continuous if and only if $\left\{x\in X:F(x)<\alpha\right\}$ is an open set for every $\alpha\in\mathbb{R}$ . It is upper semi-continuous at a point $x_{0}\in X$ if for all $\epsilon>0$ there is a neighbourhood of $x_{0}$ containing $\left\{x\in X:F(x)<F(x_{0})+\epsilon\right\}$ . If $X$ is a metric space, this is equivalent to

\displaystyle\limsup_{t_{i}\rightarrow t_{0}}F(t_{i})\leq F(t_{0}),

(5.1)

for all sequence $(t_{i})$ converging to $t_{0}$ .

We will be interested in the upper semicontinuity of $\Psi_{M_{t}}$ in the variable $t$ for a proper smooth fibration $\pi:\mathcal{M}\rightarrow\Delta$ , i.e., $\pi$ is holomorphic, surjective, having everywhere of maximal rank and connected fibers $M_{t}=\pi^{-1}(t)$ by the local Torelli Theorem. We apply the following result of M. Zaidenberg.

Proposition 13.

[MR791317, Theorem 4.4] $f:M\rightarrow\Delta$ is a surjective holomorphic mapping with smooth fibers $D_{c}=f^{-1}(c)(c\in\Delta)$ . We fix an arbitrary tubular neighborhood $U$ of the fiber $D_{0}$ and a smooth retraction $\pi:U\rightarrow D_{0}$ . Fix an Hermitian metric $h$ on $D_{0}$ and denote the associated volume form by $\omega$ . We set $\pi_{c}=\pi\mid U\cap D_{c}$ . If the Kobayashi-Eisenman pseudo volume form $\mathrm{\Psi}_{D_{0}}$ is continuous, then for each domain $V_{0}\Subset D_{0}$ and every $\varepsilon>0$ there is a $\delta_{0}>0$ such that for all $c\in\Delta_{\delta_{0}}^{*}$ the following inequality holds:

{\Psi_{D_{c}}}\leq\pi_{c}^{*}\left(\left[(1+\varepsilon){\Psi_{D_{0}}}+\varepsilon|\omega|\right]\right).

For the moduli space of marked K3 surfaces, since in our case the central fiber is an elliptic K3 surface with a section, the pseudovolume is the constant $0$ . Thus, the continuity at the central fiber implies having the upper semicontinuity with respect to $t$ .

Corollary 14.

For $M$ a compact complex manifold, let $\operatorname{Vol}(M):=\int_{M}\Psi_{M}\wedge\overline{\Psi_{M}}$ be the volume of $M$ with respect to $\Psi_{M}$ . Then $\operatorname{Vol}(M)$ is upper semicontinuous with respect to the variation of the complex structure on $M$ .

The proof is based on an adjustment of an argument of L. Kamenova, S. Lu, and M. Verbitsky [MR3263959] to our setting.

Proof.

We need to show that $\operatorname{Vol}\left(M_{t}\right)$ is upper semicontinuous with respect to $t$ for a family as given above, i.e. for all $t_{0}\in T$ and sequences $\left(t_{i}\right)$ converging to $t_{0}$ ,

\limsup_{t_{i}\rightarrow t_{0}}\operatorname{Vol}\left(M_{t_{i}}\right)\leqslant\operatorname{Vol}\left(M_{t_{0}}\right).

If the inequality is false, then after replacing the sequence $\left(t_{i}\right)$ by a subsequence there is an $\varepsilon>0$ such that $\operatorname{Vol}\left(M_{t_{i}}\right)>\operatorname{Vol}\left(M_{t_{0}}\right)+\varepsilon$ for all $i$ . Then by Proposition 13,

\operatorname{Vol}\left(M_{t_{0}}\right)+\varepsilon\leqslant\limsup_{i\rightarrow\infty}\operatorname{Vol}\left(M_{t_{i}}\right)=\limsup_{i\rightarrow\infty}\int_{M_{t_{i}}}\Psi_{M_{t_{i}}}\wedge\overline{\Psi_{M_{t_{i}}}}\leqslant\int_{M_{t_{0}}}\Psi_{M_{t_{0}}}\wedge\overline{\Psi_{M_{t_{0}}}}=\operatorname{Vol}\left(M_{t_{0}}\right),

which is a contradiction. ∎

Remark 15.

In general, the Kobayashi-Eisenman pseudovolume does not satisfy the lower-semicontinuity under deformation of complex structures. i.e., the “jumping phenomenon” under deformation. For such an example, [MR0776396, Proposition 9.7].

5.2. Teichmüller spaces and Ergodicity

We summarize the definition of the Teichmüller space of hyperkähler manifolds, following [MR3413979]:

Definition 16.

[MR3413979, Definition 1.4, 1.6] Let $M$ be a compact complex manifold and $\operatorname{Diff}_{0}(M)$ a connected component of its diffeomorphism group (the group of isotopies). Denote by Comp the space of complex structures on $M$ , equipped with a structure of Fréchet manifold. We let Teich $:=\operatorname{Comp}/\operatorname{Diff}_{0}(M)$ and call it the Teichmüller space of $M$ . Let $\operatorname{Diff}(M)$ be the group of orientable diffeomorphisms of a complex manifold $M$ . Consider the mapping class group

\Gamma:=\operatorname{Diff}(M)/\operatorname{Diff}_{0}(M)

acting on Teich. The quotient Comp / Diff $=$ Teich $/\Gamma$ is called the moduli space of complex structures on $M$ . The set Comp / Diff corresponds bijectively to the set of isomorphism classes of complex structures.

Definition 17.

[MR3413979, Definition 1.17] Let $M$ be a complex manifold, Teich its Teichmüller space, and $I\in$ Teich a point. Consider the set $Z_{I}\subset$ Teich of all $I^{\prime}\in$ Teich such that $(M,I)$ is biholomorphic to $\left(M,I^{\prime}\right)$ . Clearly, $Z_{I}=\Gamma\cdot I$ is the orbit of I. A complex structure is called ergodic if the corresponding orbit $Z_{I}$ is dense in Teich.

For the proof of Corollary 20, we use the following theorem:

Theorem 18.

[MR3413979, Theorem 1.16, 4.11] Let $M$ be a maximal holonomy hyperkähler manifold (which includes K3 surfaces) or a compact complex torus of dimension $\geq 2$ , and $I$ a complex structure on $M$ . Then $I$ is non-ergodic iff the Neron-Severi lattice of $(M,I)$ has maximal rank (i.e., the Picard number of $(M,I)$ is maximal).

Proposition 19.

Let $M$ be a complex manifold with vanishing Kobayashi-Eisenman pseudovolume. Then the volume of the Kobayashi-Eisenman pseudovolume $\Psi_{M}$ defined in Corollary 14 vanishes for all ergodic complex structures in the same deformation class.

Proof.

Let Vol : Teich $\longrightarrow\mathbb{R}^{\geqslant 0}$ map a complex structure $I$ to the volume of the Kobayashi-Eisenman pseudovolume on $(M,I)$ . By Corollary 14, this function is upper semi-continuous. Let $I$ be an ergodic complex structure. The set of points $I^{\prime}\in$ Teich such that $\left(M,I^{\prime}\right)$ is biholomorphic to $(M,I)$ is dense, because $I$ is ergodic. By upper semi-continuity, $0=\operatorname{Vol}(I)\geqslant$ $\sup_{I^{\prime}\in\text{Teich}}\operatorname{Vol}\left(I^{\prime}\right)$ . ∎

For the following Corollary, we adapt the similar argument of [MR3263959, Corollary 2.2].

Corollary 20.

Let $X$ be a complex K3 surface. Then the Kobayashi-Eisenman pseudovolume on $X$ vanishes almost everywhere (with respect to the associated volume form of any hermitian metric on $X$ ).

Proof.

For any elliptic K3 surface $X^{\prime}$ with a section, $\Psi_{X^{\prime}}\equiv 0$ follows from Proposition 12 and the decreasing property of the Kobayashi-Eisenman pseudovolume.

For general K3 surfaces $X$ , we use the following well-known fact: when the Picard number of a K3 surface is greater or equal to $5$ , then such a K3 surface admits an elliptic fibration, and when the Picard number is greater or equal to $12$ , then such a fibration exists having a section (see for instance [MR3586372, Chapter 11.1]). So, if the K3 surface $(X,I)$ does not have an elliptic fibration with section, then at least the Picard number cannot be $20$ (the maximal value). By Theorem 18, $I$ is an ergodic complex structure. Since any two complex structures of K3 surfaces are deformation equivalent to each other, we can find an ergodic complex structure which is an elliptic K3 surface with a section. Therefore for general K3 surfaces, Proposition 19 implies $\operatorname{Vol}=\int_{X}\Psi_{X}\wedge\overline{\Psi_{X}}=0$ , and thus $\Psi_{X}=0$ almost everywhere. ∎

One another vanishing consequence is on the Hilbert schemes of points on any complex K3 surface. Let $S$ be a smooth quasi-projective algebraic variety over a field $k$ , denotes the symmetric group $\Sigma_{n}$ , and symmetric powers $\operatorname{Sym}^{n}(S):=S^{n}/\Sigma_{n}$ , which is the moduli space of effective $0$ -cycles on $S$ that we only record the multiplicity when points come together. When $X=S$ is a surface, it is well-known that the Hilbert scheme $Hilb^{n}(S)$ , which is the moduli of 0-dimensional subschemes of length $n$ in $S$ , is a smooth, irreducible variety which is a resolution of singularities of $\operatorname{Sym}^{n}(S)$ .

Corollary 21.

Let $Hilb^{n}(X)$ be a Hilbert schemes of points on any complex K3 surface $X$ for any $n$ . Then the Kobayashi-Eisenman pseudovolume on $Hilb^{n}(X)$ vanishes almost everywhere.

Proof.

For any elliptic K3 surface $X^{\prime}$ with a section, from the proof of Proposition 12, we have a non-constant holomorphic function $F:\mathbb{C}\times R_{X^{\prime}}\rightarrow\overline{X^{\prime}}$ . Consequently, from the fact that $Hilb^{n}(X^{\prime})$ is a resolution of singularities of $\operatorname{Sym}^{n}(X^{\prime})$ , we have a non-constant holomorphic map $f:\mathbb{C}\times Z\rightarrow Hilb^{n}(X^{\prime})$ , where $Z$ is a complex manifold of dimension one less than the dimension of $Hilb^{n}(X^{\prime})$ . By Lemma 6, the Kobayashi-Eisenman pseudovolume vanishes almost everywhere.

For general K3 surfaces $X$ , since $Hilb^{n}(X)$ is deformation equivalent to $Hilb^{n}(X^{\prime})$ with some elliptic K3 surface $X^{\prime}$ with a section, as the proof of Corollary 20, the result follows. ∎

The following Corollary concerning Enriques surfaces is also proven in [MR609557].

Corollary 22.

Let $Y$ be a Enriques surface. Then the Kobayashi-Eisenman pseudovolume on $Y$ vanishes.

Proof.

Every Enriques surface $Y$ admits a complex K3 surface $X$ as a holomorphic double covering $\phi:X\rightarrow Y$ . The conclusion follows from Lemma 6 and Corollary 20. ∎

Non-measure hyperbolicity of complex K3 surfaces

Abstract.

Key words and phrases:

1. Introduction and results

Conjecture 1.

Theorem 2 (Corollary 20).

Corollary 3 (Corollary 21).

Acknowledgments

2. Preliminaries

2.1. Kobayashi–(Royden) pseudometric, Kobayashi-Eisenman pseudovolume

Definition 4.

Definition 5.

Lemma 6.

Lemma 7.

Proof.

3. Complex K3 surfaces

Definition 8.

3.0.1. Moduli space of K3 surfaces

3.1. Elliptic K3 surfaces with section

Proposition 9.

4. Vanishing Kobayashi-Einsenman volume on Elliptic k3 surfaces with a section

Proposition 10.

Proposition 11.

Proof.

Proposition 12.

Proof of Proposition 12.

5. Vanishing Kobayashi-Eisenman pseudo-volume on K3 surfaces

5.1. Upper semi-continuity of Kobayashi-Eisenman volume

Proposition 13.

Corollary 14.

Proof.

Remark 15.

5.2. Teichmüller spaces and Ergodicity

Definition 16.

Definition 17.

Theorem 18.

Proposition 19.

Proof.

Corollary 20.

Proof.

Corollary 21.

Proof.

Corollary 22.

Proof.

References