Non-measure hyperbolicity of K3 surfaces and Hilbert schemes of points on K3 surfaces

The Kobayashi pseudometric is the intrinsic complex-analytic counterpart of the Poincaré metric on the unit disk and extends the notion of a Kähler–Einstein metric with constant negative holomorphic sectional curvature. By Brody’s theorem [3], the absence of non-constant entire curves \({\mathbb {C}}\rightarrow M\) on a compact complex manifold \(M\) is equivalent to the positivity of the Kobayashi pseudometric. S. Kobayashi conjectured that every compact Calabi–Yau manifold has vanishing Kobayashi pseudometric [21].

Classical problems in this area include the Green–Griffiths and Lang conjectures, which motivate the Demailly–Semple jet-bundle approach for varieties of general type [10, 22], and more recent deformation-theoretic approaches for hyperkähler manifolds (including K3 surfaces) developed by Kamenova et al. [17]. Numerous studies have refined the analysis of entire curves in these contexts [2, 18,19,20, 23]; see also [6, 8] for broad surveys.

A deeper invariant is the Kobayashi–Eisenman pseudovolume, whose vanishing automatically implies the vanishing of the Kobayashi pseudometric. However, because this pseudovolume cannot be controlled by entire curves alone, new techniques are required. Attempts to extend Demailly–Semple theory from entire maps \({\mathbb {C}}\rightarrow X\) to maps \({\mathbb {C}}^k\rightarrow X\) illustrate some of these difficulties [25].

These considerations lead to the classical Kobayashi measure-hyperbolicity conjecture [21], which predicts that an algebraic variety is measure hyperbolic precisely when it is of general type. Recent discussions appear in [7, 30]. The conjecture is consistent with the theorem of Sakai [27] that varieties of general type are measure hyperbolic, and with S. Kaliman’s extension to all intermediate Kobayashi–Eisenman measures [16] (see Definition 7).

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, central objects of study in both differential and algebraic geometry [24, 28], provide a natural testing ground for this conjecture. Green and Griffiths [10] verified Conjecture 1 for certain algebraic K3 surfaces, including double covers of \({\mathbb {P}}^2\) branched along a smooth sextic and smooth quartic or sextic surfaces in \({\mathbb {P}}^3\) and \({\mathbb {P}}^4\) with nonzero geometric genus. Subsequent work has shown that every algebraic K3 surface is rationally swept out by elliptic curves [12, Corollary 13.2.2], implying the conjecture for all algebraic K3 surfaces [10, 21].

The present paper proves that the conjecture holds for all complex K3 surfaces, algebraic or not.

Theorem 2

(Corollary 19) Every complex K3 surface \(X\) is non-measure-hyperbolic.

Our proof deforms the complex structure of a given K3 surface to an elliptic K3 surface (no section required) lying in a dense subset of the moduli of marked K3’s and known to be dominated by \({\mathbb {C}}^2\); hence, its Kobayashi–Eisenman pseudovolume vanishes. Upper semicontinuity of the Kobayashi–Eisenman pseudovolume and pseudometric under deformation then forces the vanishing for the original K3 surface. Related deformation arguments have appeared in the first author’s work on Kobayashi hyperbolicity and toric geometry [5].

An immediate corollary is a new vanishing result for all punctual Hilbert schemes of points on K3 surfaces, strengthening the Kobayashi non-hyperbolicity established in [17].

Corollary 3

(Corollary 20) For every complex K3 surface \(X\) and every \(n\ge 1\), the Hilbert scheme of \(n\) points \(\textrm{Hilb}^n(X)\) is non-measure-hyperbolic.

Furthermore, Theorem 2 yields an independent proof of the vanishing of the Kobayashi–Eisenman pseudovolume on Enriques surfaces, complementing the classical result of [10].

Organization of the paper. The paper has three main sections and one appendix. Section 1 (this section) states the results and outlines the strategy. Section 2 recalls the Kobayashi pseudometric and the Kobayashi–Eisenman pseudovolume, together with basic functorial properties and the upper semicontinuity input we need. Section 3 proves the vanishing on elliptic K3 surfaces and extends it to all K3 surfaces via deformation; the corollaries for Hilbert schemes and Enriques surfaces are derived here. "Appendix" presents the classical Weierstrass model construction for readers who prefer an explicit analytic picture.

Throughout this paper, we write

for the unit disk in \({\mathbb {C}}\). For a complex manifold X, we denote by \(T_xX\) the holomorphic tangent space at \(x\in X\), and by \(T_X\) the holomorphic tangent bundle.

2.1 Kobayashi–Royden pseudometric and Kobayashi–Eisenman pseudovolume

Definition 4

Let X be a complex manifold of dimension n.

1. (1)

   The Kobayashi–Royden pseudometric of X is the function

   $$\begin{aligned} \textrm{KR}_X(x;v)&:= \inf \bigl \{\lambda >0 \;\big |\; \exists ,円 f\in \text {Hol}({\mathbb {D}},X),,円 f(0)=x,\\&\quad f'(0)=v/\lambda \bigr \}, \end{aligned}$$

   for \(x\in X\) and \(v\in T_xX\).

2. (2)

   The Kobayashi–Eisenman pseudovolume of X is the (possibly degenerate) n-form \(\varPsi _X\) whose pointwise norm on decomposable n-vectors \(\zeta \in \bigwedge ^n T_xX\) is

   $$\begin{aligned} \varPsi _X(x;\zeta )&:= \inf \bigl \{\lambda >0 \;\big |\; \exists ,円 F\in \text {Hol}({\mathbb {D}}^n,X),\ F(0)\!=\!x,\\&\quad \det \textrm{d} F_0(\partial /\partial t_1\wedge \cdots \wedge \partial /\partial t_n) = \zeta /\lambda \bigr \}. \end{aligned}$$
3. (3)

   More generally, for \(1\le p\le n\), the Eisenman p-pseudovolume \(\varPsi _X^p\) is defined analogously, restricting to decomposable p-vectors \(\zeta = u_1\wedge \cdots \wedge u_p\in \bigwedge ^p T_xX\).

By construction one has

Remark 5

Different normalizations of constants appear in the literature; in this article, we consistently adopt the above normalization.

Remark 6

The Kobayashi–Royden pseudometric coincides with the Poincaré metric on the complex unit disk and is well defined on any complex manifold. Intuitively, \(\textrm{KR}_X(x;v)\) is the infinitesimal Poincaré norm of v measured via holomorphic disks.

The Kobayashi pseudodistance \(d_X\) is the largest pseudodistance on X not exceeding the pullback of the Poincaré distance under holomorphic disks. Equivalently,

where \(\rho _{{\mathbb {D}}}\) is the Poincaré distance on \({\mathbb {D}}\). Royden proved that \(d_X\) is the length metric induced by \(\textrm{KR}_X\) [11].

The upper semicontinuity of the Kobayashi–Eisenman measures under holomorphic variation of complex structures was established by Eisenman (publishing as Pelles) [26]. Moreover, the kth Kobayashi–Eisenman measure is insensitive to removing analytic subsets of codimension \(\ge k+1\) [13]; for \(k=1\), i.e., the Kobayashi–Royden pseudometric, see also [4, 14, 15].

Definition 7

A complex manifold X is said to be measure hyperbolic if the Kobayashi–Eisenman pseudovolume \(\varPsi _X\) does not vanish on any nonempty open subset; equivalently, \(\varPsi _X\ne 0\) almost everywhere with respect to any smooth volume form.

2.2 Basic functorial properties

The following facts are immediate from the definition of Kobayashi–Eisenman pseudovolume and will be used repeatedly.

Lemma 8

(Holomorphic monotonicity) Let \(\phi : X \rightarrow Y\) be a holomorphic map of complex manifolds of the same dimension n. Then,

Lemma 9

(Product inequality) Let \(X_1\) and \(X_2\) be complex manifolds of dimensions \(n_1\) and \(n_2,\), respectively,, and let \(n=n_1+n_2\). Write \(\operatorname {pr}_i: X_1\times X_2 \rightarrow X_i\) for the projections. Then,

Proof

See [30, Lemma 1.12]. \(\square \)

Lemma 10

(Dominance implies vanishing) Let \(f: M \rightarrow X\) be a dominant holomorphic map between complex manifolds of the same dimension n, with \(\textrm{d}f\) generically surjective. If \(\varPsi _M \equiv 0,\) then \(\varPsi _X \equiv 0\).

Proof

By Lemma 8, one has \(f^*\varPsi _X \le \varPsi _M\equiv 0\). At generic points where \(\textrm{d}f\) is surjective, this forces \(\varPsi _X=0\), and the vanishing extends to all of X by analytic continuation. \(\square \)

These lemmas will be key in later sections, for instance when passing from vanishing results on auxiliary spaces (such as \(\mathbb C^2\)) to K3 surfaces, Hilbert schemes of points, and their Enriques quotients.

3.1 Dominance and vanishing

A key ingredient is the functorial property of Kobayashi–Eisenman pseudovolume recorded in Lemma 10.

Proposition 11

(Elliptic K3’s are dominated by \({\mathbb {C}}^2\)) Let \(X\) be an elliptic K3 surface (a K3 surface admitting a holomorphic map \(\pi :X\rightarrow {\mathbb {P}}^1\) whose general fiber is a genus-one curve), with or without a section. Then, there exists a holomorphic dominant map \(F:{\mathbb {C}}^2\rightarrow X\) with Zariski-dense image.

Proof

This is classical; see [1, Chap. VIII, Theorem 24.1], where a holomorphic map from \({\mathbb {C}}^2\) onto the Weierstrass model is constructed, yielding dominance of \(X\) after resolving the (at worst) rational double points. \(\square \)

Since \(\varPsi _{{\mathbb {C}}^2}\equiv 0\), Lemma 10 immediately gives

for every elliptic K3 surface \(X\), without any assumption on sections. This observation will be the starting point for all subsequent vanishing results.

3.2 Upper semicontinuity of Kobayashi–Eisenman volume

We now recall the upper semicontinuity properties of \(\varPsi _{X_t}\) in smooth holomorphic families. Let \(\pi : {\mathcal {M}} \rightarrow T\) be a proper holomorphic submersion with connected fibers \(X_t=\pi ^{-1}(t)\) of dimension \(n\). A function \(F: T \rightarrow {\mathbb {R}}\cup \{\infty \}\) is upper semicontinuous if the sets \(\{ t \in T \mid F(t)<\alpha \}\) are open for every \(\alpha \in {\mathbb {R}}\). Equivalently, in a metric space,

The following result of Zaidenberg provides an upper bound for the Kobayashi–Eisenman form in smooth families.

Proposition 12

[31, Theorem 4.4] Let \(f: M \rightarrow \varDelta \) be a surjective holomorphic map with smooth fibers \(D_c=f^{-1}(c)\). Fix a tubular neighborhood \(U\) of \(D_0\) and a smooth retraction \(\pi :U\rightarrow D_0\), and endow \(D_0\) with a Hermitian metric \(h\) and associated volume form \(\omega \). For each relatively compact domain \(V_0\Subset D_0\) and every \(\varepsilon >0\), there exists \(\delta _0>0\) such that for all \(c\in \varDelta _{\delta _0}^*\), one has

where \(\pi _c=\pi |_{U\cap D_c}\).

In our application, the central fiber \(X_0\) is an elliptic K3 surface dominated by \({\mathbb {C}}^2\) (cf. Proposition 11), hence \(\varPsi _{X_0}\equiv 0\). Continuity at the central fiber therefore implies upper semicontinuity of \(t\mapsto \varPsi _{X_t}\).

Corollary 13

Let \(M\) be a compact complex manifold and set

Then, \(\operatorname {Vol}(M)\) is upper semicontinuous under deformation of the complex structure: for any family \(\pi :\mathcal M\rightarrow T\) as above and every \(t_0\in T\),

Proof

If the inequality failed for some sequence \(t_i\rightarrow t_0\), one could find \(\varepsilon >0\) with \(\operatorname {Vol}(M_{t_i})>\operatorname {Vol}(M_{t_0})+\varepsilon \) for all \(i\). Applying Proposition 12 gives

a contradiction. \(\square \)

Remark 14

In general, the Kobayashi–Eisenman pseudovolume need not vary lower-semicontinuously with the complex structure; examples of such "jumping" phenomena are given in [9, Proposition 9.7].

3.3 Teichmüller spaces and ergodicity

We next recall the Teichmüller-theoretic background, following [29].

Definition 15

[29, Definitions 1.4, 1.6] Let \(M\) be a compact complex manifold. Let \(\operatorname {Comp}\) be the Fréchet manifold of complex structures on \(M\), and \(\operatorname {Diff}_0(M)\) the connected component of the diffeomorphism group containing the identity. The Teichmüller space of \(M\) is

The mapping class group

acts on \(\textrm{Teich}\), and the quotient \(\textrm{Teich}/\varGamma \) is the moduli space of complex structures on \(M\).

Definition 16

[29, Definition 1.17] For \(I\in \textrm{Teich}\) let \(Z_I:= \{I'\in \textrm{Teich}\mid (M,I')\cong (M,I)\}\), the orbit of \(I\) under \(\varGamma \). A complex structure \(I\) is ergodic if \(Z_I\) is dense in \(\textrm{Teich}\).

The following theorem, combining work of Verbitsky and others, is fundamental.

Theorem 17

[29, Theorems 1.16 and 4.11] Let \(M\) be a maximal holonomy hyperkähler manifold (in particular, a K3 surface) or a compact complex torus of dimension \(\ge 2\), and \(I\) a complex structure on \(M\). Then, \(I\) is non-ergodic if and only if the Néron–Severi lattice of \((M,I)\) has maximal rank [equivalently, the Picard number of \((M,I)\) is maximal].

Proposition 18

Let \(M\) be a compact complex manifold with \(\varPsi _M\equiv 0\). Then, for every ergodic complex structure \(I\) deformation equivalent to \(M\), the Kobayashi–Eisenman volume satisfies \(\operatorname {Vol}(M,I)=0\).

Proof

The assignment

is upper semicontinuous by Corollary 13. If \(I\) is ergodic, the set of points in \(\textrm{Teich}\) corresponding to complex structures biholomorphic to \((M,I)\) is dense. Upper semicontinuity therefore implies \( 0 = \operatorname {Vol}(M,I) \ge \sup _{I'\in \textrm{Teich}} \operatorname {Vol}(M,I'), \) so \(\operatorname {Vol}(M,I)=0\). \(\square \)

3.4 Applications to K3 surfaces and their Hilbert schemes

Corollary 19

Let \(X\) be a complex K3 surface. Then, the Kobayashi–Eisenman pseudovolume on \(X\) vanishes almost everywhere (with respect to the volume form of any Hermitian metric on \(X\)).

Proof

By Proposition 11, every elliptic K3 surface (section not required) admits a dominant holomorphic map \({\mathbb {C}}^2\rightarrow X\), so \(\varPsi _X\equiv 0\). For a general K3 surface \(X\), if the Picard number of \((X,I)\) is not maximal, then \(I\) is ergodic by Theorem 17. All K3 surfaces are deformation equivalent, and the set of elliptic K3 surfaces is dense in the moduli space. Applying Proposition 18 to an elliptic K3 surface with \(\varPsi \equiv 0\) yields \(\operatorname {Vol}(X)=0\), hence \(\varPsi _X=0\) almost everywhere. \(\square \)

The same reasoning applies to Hilbert schemes of points.

Corollary 20

Let \(X\) be any complex K3 surface and \(n\in {\mathbb {N}}\). Then, the Kobayashi–Eisenman pseudovolume on \(\operatorname {Hilb}^n(X)\) vanishes almost everywhere.

Proof

Let \(X'\) be an elliptic K3 surface as in Proposition 11. Since \(\operatorname {Hilb}^n(X')\) is a resolution of \(\operatorname {Sym}^n(X')\), there exists a dominant holomorphic map

for some complex manifold \(Z\) of dimension \(\dim \operatorname {Hilb}^n(X')-1\). By Lemmas 8 and 9 and the fact that \(\varPsi _{{\mathbb {C}}}\equiv 0\), we obtain \(\varPsi _{\operatorname {Hilb}^n(X')}\equiv 0\). Since \(\operatorname {Hilb}^n(X)\) is deformation equivalent to \(\operatorname {Hilb}^n(X')\), upper semicontinuity (Corollary 13) implies \(\varPsi _{\operatorname {Hilb}^n(X)}=0\) almost everywhere. \(\square \)

Finally, we record the well-known case of Enriques surfaces (see also [10]).

Corollary 21

Let \(Y\) be an Enriques surface. Then, the Kobayashi–Eisenman pseudovolume on \(Y\) vanishes.

Proof

Every Enriques surface \(Y\) admits a holomorphic double covering \(\phi :X\rightarrow Y\) where \(X\) is a K3 surface. By Corollary 19 and Lemma 8, \(\varPsi _Y\equiv 0\). \(\square \)