Rational connectedness and order of non-degenerate meromorphic maps from \({{\mathbb C}}^n\)

The purpose of this paper is the following result.

(Here a meromorphic map \(f:{\mathbb C}^n\dasharrow X\) is called non-degenerate if there is a point v such that f is defined in v and Open image in new window is surjective.)

This result belongs to a series of similar statements relating the existence and growth of maps from \({\mathbb C}^n\) to algebro-geometric properties of the target space X. These statements are better expressed by introducing the following invariant \(\rho (X)\), which suggests many questions, some of which are raised in the last section.

The invariant \(\rho \) is easily seen to be bimeromorphic, preserved by finite étale covers, and increasing (i.e., \(\rho (X)\geqslant \rho (Y)\) if there exists a dominant meromorphic map \(g:X\rightarrow Y\)). It is obvious that \(\rho (X)=0\) if X is unirational, because Open image in new window for every rational map f.

Kobayashi and Ochiai proved that the existence of a non-degenerate meromorphic map from \({\mathbb C}^n\) to a projective manifold X implies that X is not of general type [9]. Thus Open image in new window if X is of general type. It is proved in [4], more generally, that X is ‘special’ if there exists a non-degenerate meromorphic map Open image in new window . In particular, X must be special if Open image in new window .

If X is Kähler and \(K_X\) is pseudo-effective of numerical dimension \(\nu \in \{0,1,\dots , n\}\), then Open image in new window  [6]. This implies the previous result of Kobayashi–Ochiai (since X is of general type if and only if \(\nu =n\)). Using [2], it also implies that if X is projective, then X is uniruled if \(\rho (X)<2\).

If \(h^0\bigl (X,\mathrm{Sym}^k(\mathrm{\Omega }^p_X)\bigr )\ne 0\), for some \(k,p>0\), then \(\rho (X)\geqslant 2\) [11]. The Kähler condition is not required here. If, in addition, X is assumed to be a Kähler surface, then the condition \(h^0\bigl (X,\mathrm{Sym}^k(\mathrm{\Omega }^p_X)\bigr )=0\) for all p, k implies that X is rational. Thus, if X is a Kähler surface and if \(\rho (X)<2\), then X must be rational [11]. On the other hand, some Hopf (thus non-rational, non-Kähler) surfaces have \(\rho (X)= 1\) [11].

Mumford conjectured that for a projective manifold (of arbitrary dimension) the condition “\(h^0\bigl (X,\mathrm{Sym}^k(\mathrm{\Omega }^p_X)\bigr )\ne 0\) for some \(k,p>0\)” is equivalent to X being not rationally connected. However, it is still unknown whether this conjecture holds.

The present result generalizes these two results from [6, 11], avoiding the deep methods of [6]. The estimate Open image in new window is optimal, since for an Abelian variety A we have \(\rho _\tau =2\) for the universal covering map \(\tau :{\mathbb C}^g\rightarrow A\).

All these results provide lower bounds for \(\rho (X)\) deduced from the geometry of X. Producing upper bounds for \(\rho (X)\) (i.e., the existence of non-degenerate f) turns out to be a completely open topic, except in the trivial case of X unirational or a torus.

The case of rationally connected vs unirational manifolds when \(n\geqslant 3\) is of great interest. For example, what is \(\rho (X)\) if X is a ‘general’ smooth quartic in \({\mathbb {P}}^4\)? If \(\rho (X)>0\), then X is not unirational. More generally, are there projective manifolds X such that \(\rho (X)\in (0,2)\)? Such a manifold X would necessarily be rationally connected, but not unirational. See the last section for some more questions.

We start with some preparations. Let X be a compact complex manifold and let Open image in new window be a meromorphic map. The map f is said to be (differentiably) non-degenerate if there is a point \(p\in {\mathbb C}^n\) such that f is holomorphic at p and Open image in new window is surjective.

Let Open image in new window on \({\mathbb C}^n\), and let \(\omega \) be a positive (1, 1)-form on X. The characteristic function of f (with respect to \(\omega \)) is defined as

Open image in new window

Here \(B_t=\{z\in {\mathbb C}^n:\Vert z\Vert <t\}\). Observe that the integral over \(B_t\) is well defined even if f is only meromorphic, not necessarily holomorphic.

If \(\omega \) and \(\widetilde{\omega }\) are any two positive (1, 1)-forms on X, then (by the compactness of X) there are constants \(C_1,C_2>0\) such that

Open image in new window

(1)

and, consequently,

Open image in new window

The characteristic function Open image in new window is a very fine tool for measuring the growth of a map. A coarser invariant is the order Open image in new window which is defined as

Open image in new window

where \(\omega \) is a positive (1, 1)-form on X. Due to inequalities (1) this number Open image in new window is independent of the choice of the positive (1, 1)-form \(\omega \) used in the definition of Open image in new window .

The well-known Crofton’s formula (see e.g. [8]) implies that Open image in new window equals the average of Open image in new window taken over all complex lines \(L\subset {\mathbb C}^n\). This permits to extend many properties of characteristic functions from the case of entire curves to the higher-dimensional domains. For instance, let \(\tau :X'\rightarrow X\) be a bimeromorphic holomorphic map between compact complex manifolds. Then Open image in new window is again a non-degenerate meromorphic map, and Open image in new window .

Now let \(\alpha :X\rightarrow Y\) be a dominant holomorphic map. Then it follows directly from the definitions that Open image in new window (if \(\omega _X\geqslant \alpha ^*(\omega _Y)\)) and, consequently, Open image in new window .

The order is easily seen to behave nicely with respect to products. Let Open image in new window , \(\omega _i\) be positive (1, 1)-forms in \(X_i\) and let \(f_i:{\mathbb C}^{n_1}\rightarrow X_i\) be non-degenerate meromorphic maps. Define Open image in new window , \(n=n_1+n_2\) and \(f:{\mathbb C}^n\rightarrow X\) as \(f(v_1,v_2)=(f_1(v_1),f_2(v_2))\). Then Open image in new window . This implies Open image in new window , because for any \(t_1,t_2>0\) we have

Open image in new window

As usual, a compact complex manifold X (usually Kähler) is called rationally connected (RC) if every two points can be linked by a (possibly singular) irreducible rational curve. (Equivalently in the Kähler case: can be linked by a chain of rational curves). Unirational manifolds are RC. There are unirational threefolds which are not rational, e.g. smooth cubic hypersurfaces in \({\mathbb P}_4\). However, it is not known whether there exist RC manifolds which are not unirational, although it is expected that these should exist (the case of ‘general’ quartics in \({\mathbb P}_4\) being one of the first open cases) .

Rationally connected compact Kähler manifolds are projective (since \(h^{2,0}=0\), using Kodaira’s criterion).

Let X be a compact connected Kähler manifold. Then there exists an “almost holomorphic” rational dominant map Open image in new window (called the RC-reduction, or rational quotient [3], or the MRC-fibration [10] if X is projective), such that the fibers are RC, and maximal with this property.

When X is projective, it is known (by [7]) that the base Y is not uniruled. In fact, [7] shows that if Open image in new window is a surjective meromorphic map with fibres and base Y which are both RC, then X is RC if it is compact Kähler (remark first that \(h^{2,0}(X)=0\), and that X is thus projective). The base Y of the RC-reduction Open image in new window is not uniruled also when X is compact Kähler. Let indeed \(r:Y\dasharrow Z\) be the RC-reduction of Y. The fibres of Open image in new window are thus RC, and so \(Y=Z\), which means that Y is not uniruled.

Due to [2] a projective manifold is uniruled if and only if \(K_X\) is not pseudo-effective. Based on this, another recent criterion is the following (see [5]): Let X be a compact Kähler manifold. Then X is rationally connected if and only if there is no pseudo-effective invertible subsheaf \(F\subset \mathrm{\Omega }^p_X (\)for some \(p\in {\mathbb N})\).

The proof consists in observing that X is not RC precisely if its RC-reduction Open image in new window has \(\dim Y=p>0\). Define then \(F=\rho ^*(K_Y)\), which is pseudo-effective since Y is not uniruled.

It is conjectured that a compact Kähler (or equivalently, projective) manifold X is RC if (and only if) there is no \({\mathbb Q}\)-effective (instead of pseudo-effective) invertible subsheaf \(F\subset \mathrm{\Omega }^p_X\) (for some \(p\in {\mathbb N}\)). By means of the RC-reduction as above, this conjecture is equivalent to the ‘non-vanishing conjecture’, claiming that if \(K_X\) is pseudo-effective, it is \({\mathbb Q}\)-effective.

A singular hermitian metric h on a complex line bundle is given in the form Open image in new window , where s is the standard metric for some local holomorphic trivialization and \(\phi \) is a \(L^1_\mathrm{loc}\)-function. The \(L^1_\mathrm{loc}\)-condition ensures that the curvature Open image in new window makes sense (in the sense of currents) and represents the Chern class of the line bundle.

A line bundle L on a compact complex manifold is called pseudo-effective if there is a singular hermitian metric h with semipositive curvature \(\mathrm{\Theta }_h\geqslant 0\). This condition means that the metric is locally given via a weight function \(e^{-\phi }\) with \(\phi \) being plurisubharmonic. Now let \(F^*\) be the dual line bundle equipped with the dual metric \(h^*\), i.e., the metric with weight factor \(e^\phi \). If F is pseudo-effective and s is a holomorphic section in \(F^*\), plurisubharmonicity of \(\phi \) implies that \(\log \Vert s\Vert _{h^*}\) is plurisubharmonic.

Assume by contradiction that X is not rationally connected. Then there is a pseudo-effective invertible subsheaf \({\mathscr {F}}\subset \mathrm{\Omega }^p_X\) (for some \(p\in {\mathbb N}\)). We fix a hermitian metric h on X. The hermitian metric on \(T_X\) induces a hermitian metric \(h_p\) on \(\mathrm{\Omega }_X^p\).

The injection of sheaves \({\mathscr {F}}\hookrightarrow \mathrm{\Omega }^p_X\) corresponds to a non-zero vector bundle homomorphism \(\xi _0:F\rightarrow \mathrm{\Omega }^p_X\), where F is a pseudo-effective line bundle on X. Since F is pseudo-effective, there is a singular hermitian metric g on F such that \(\mathrm{\Theta }_g\geqslant 0\), i.e, with positive curvature current.

Indeed, locally Open image in new window may be written as Open image in new window for some plurisubharmonic function u. Since u is upper semicontinuous with values in Open image in new window , the map \(v\mapsto \Vert v\Vert _g\) is lower semicontinuous with values in Open image in new window for \(v\ne 0\). Hence

$$\begin{aligned} v\mapsto \frac{\Vert \xi _0(v)\Vert _{h_p}}{\Vert v\Vert _g} \end{aligned}$$

is upper semicontinuous with values in Open image in new window for \(v\ne 0\). Since upper semicontinuous functions with values in \({\mathbb R}\) assume a maximum on compact sets, compactness of X now implies the claim.

Next we consider the pull-back bundles Open image in new window and \(f^*\mathrm{\Omega }^p_X\). They are defined on Open image in new window , where I(f) denoting the indeterminacy set of the meromorphic map f. We consider these bundles equipped with the pull-backed metrics, which we denote by \(\widetilde{g}=f^*g\) and \(\widetilde{h}_p=f^*h_p\).

The meromorphic map Open image in new window and the vector bundle homomorphism \(\xi _0:F\rightarrow \mathrm{\Omega }^p_X\) induce vector bundle homomorphisms

Open image in new window

on Open image in new window . Let Open image in new window . Now \(\beta \) is a holomorphic section in

Open image in new window

and Open image in new window is the dual of a pseudo-effective line bundle. Therefore \(\log \Vert \beta \Vert _{\widetilde{g}}\) defines a plurisubharmonic function on Open image in new window . Plurisubharmonic functions extend through closed analytic subsets of codimension at least two. Hence \(\log \Vert \beta \Vert _{\widetilde{g}}\) extends to a plurisubharmonic function \(\zeta _0\) defined on the whole \({\mathbb C}^n\).

Let Open image in new window and Open image in new window . Let \(w=\xi (v)\). Then Open image in new window combined with inequality (3) implies

Open image in new window

Observe also (see inequality (2)) that

Open image in new window

Define Open image in new window . The plurisubharmonicity of \(\zeta _0\) implies the plurisubharmonicity of \(\zeta \). Thus we obtain the existence of a plurisubharmonic function \(\zeta \) on \({\mathbb C}^n\) such that \(\Vert Df\Vert _{h}\geqslant \zeta \).

Using Proposition 5.1, we may deduce that Open image in new window . It follows that

$$\begin{aligned} T_f(r;\omega )\geqslant \int ^r_1\! \frac{dt}{t^{2n-1}}\int _{B_t}\!\zeta \alpha ^n\!. \end{aligned}$$

Since moving the origin, i.e., replacing f by Open image in new window , where \(\tau \) denotes a translation, does not affect the order Open image in new window , we may assume that \(c=\zeta (0,\ldots ,0)>0\). Using the sub-mean value property of plurisubharmonic functions, it follows that

Open image in new window

where \(\nu \) denotes the volume of the unit ball. Therefore,

Open image in new window

Our result is not valid for non-Kähler manifolds. In fact, there are Hopf surfaces X admitting a non-degenerate holomorphic map \(f:{\mathbb C}^n\rightarrow X\) of order Open image in new window (see [11]). Of course, these Hopf surfaces are non-Kähler and not rationally connected; they do not contain any rational curve at all. More precisely, in [11] the following is proved

Theorem 7.1

Let \(\lambda \in {\mathbb C}\) with \(|\lambda |>1\) and let \(\sim \) denote the equivalence relation on Open image in new window given by

$$\begin{aligned} v\sim w \qquad \iff \qquad \text {there exists }k\in {\mathbb Z}v=\lambda ^k w. \end{aligned}$$

Let Open image in new window and let \(f:{\mathbb C}^2\rightarrow X\) be the map induced by \((z,w)\mapsto (z,1+zw)\). Then Open image in new window .

This result has been generalized by Amemiya [1] to the class of Hopf surfaces defined by an equivalence \((z,w)\sim (\lambda ^kz,\mu ^kw)\), \(k\in {\mathbb Z}\), where \(\lambda \) may be different from \(\mu \) (but \(|\lambda |,|\mu |>1\)).