Weakly Special Manifolds with no rational curves

Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi–Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an étale quotient of an Abelian variety. The same conclusion holds true if $X$ contains a Zariski dense entire curve, assuming Lang’s conjecture. This implies that any non-hyperbolic complex projective manifold contains the image of an Abelian variety, according to another conjecture of Lang. We illustrate this last conjecture by producing examples of canonically polarised submanifolds of abelian varieties containing no subvariety of general type, except for a finite number of disjoint copies of some simple abelian variety, which can be chosen arbitrarily. We also show, more generally, that any projective manifold containing a Zariski dense entire curve appears as the `exceptional set’ in Lang’s sense of some general type manifold.

💡 Research Summary

The paper investigates the interplay between “weakly special” complex projective manifolds, the presence or absence of rational curves, and Lang’s conjectures on hyperbolicity. A weakly special manifold is defined as one that does not admit, after any finite étale cover, a rational fibration onto a positive‑dimensional variety of general type. This notion is weaker than Campana’s “special” condition and coincides with it only in dimensions ≤2; in higher dimensions the two diverge.

The central result (Theorem 2.1) assumes two major conjectures: (1) the abundance conjecture (Conjecture 2.5), which guarantees a terminal model for any non‑uniruled projective manifold, and (2) a density conjecture (Conjecture 2.6) asserting that on a terminal Calabi–Yau or hyperkähler model with torsion canonical bundle, the union of rational curves is Zariski‑dense unless the variety is a quotient of an Abelian variety. Under these hypotheses, the authors prove that for a smooth projective variety (X) the following are equivalent:

I. (X) is weakly special and contains no rational curves.
II. (X) is an étale quotient of an Abelian variety.

If Lang’s conjecture (that varieties of general type contain no Zariski‑dense entire curves) is added, a third equivalent condition appears:

III. (X) contains a Zariski‑dense entire curve and no rational curves.

The proof proceeds in two stages. First, Theorem 2.2 establishes that for a Moishezon‑Iitaka fibration (f\colon X\to Z) whose generic fibers are étale quotients of Abelian varieties, the absence of rational curves forces the existence of a finite étale cover (u\colon X’\to X) such that (X’) admits an Abelian group scheme structure over a base (Z’). This step relies on deep results: Kawamata’s equidimensionality theorem, Kollár’s finiteness of local monodromies, and Deligne’s theory of variations of Hodge structures. The authors adapt Höering’s argument (originally for manifolds with nef cotangent bundle) to the present setting, handling the discriminant locus of the fibration and using cyclic covers to eliminate multiple fibers.

Second, using the abundance conjecture, one replaces (X) by a terminal model (X’). Conjecture 2.6 then forces a dense family of rational curves on any non‑Abelian terminal Calabi–Yau or hyperkähler model. Since (X) is assumed to have no rational curves, the only possibility is that the terminal model is itself an étale quotient of an Abelian variety, which descends to (X). In dimension three the abundance conjecture is known, yielding an unconditional Corollary 2.3: a smooth projective threefold without rational curves is either canonically polarized, Calabi–Yau (which would contradict the rational‑curve density conjecture), or after a finite étale cover becomes an Abelian group scheme over a canonically polarized base.

The paper also studies the “exceptional locus” in Lang’s sense: the Zariski closure of all entire curves in a variety of general type. Theorem 2.7 shows that, assuming the three conjectures, a projective manifold without rational curves or images of simple Abelian varieties must be hyperbolic and canonically polarized.

To illustrate these abstract results, the authors construct explicit examples. Theorem 2.8 produces, for any simple Abelian variety (V\subset\mathbb{P}^n) and any simple Abelian variety (A) of dimension (n+1), a canonically polarized submanifold (X\subset A\times V) with the following properties:

1. Every subvariety of (X) is of general type except for a smooth divisor (W) consisting of finitely many disjoint copies of (V).
2. Every entire curve in (X) is contained (and Zariski‑dense) in a component of (W).
3. The Kobayashi pseudo‑distance is a genuine distance on (X\setminus W) and vanishes on (W).
4. (X) has a nef and big cotangent bundle, admits a Kähler metric with non‑negative bisectional curvature, and supports a pseudo‑convex Finsler metric with quasi‑negative holomorphic sectional curvature.

When (\dim V\ge2), (X) contains no rational or elliptic curves, and all entire curves are transcendental. Theorem 2.9 generalizes this construction: given any projective manifold (V) that already carries a Zariski‑dense entire curve and a simple Abelian variety (A) of dimension (\dim V+1), one builds a submanifold (X\subset A\times V) of general type whose exceptional locus is precisely a divisor made of copies of (V). Consequently, any projective manifold that carries a Zariski‑dense entire curve appears as the exceptional set of a “generically hyperbolic” general‑type manifold.

These constructions answer questions raised by Diverio concerning the Wu–Yau theorem and its extensions to quasi‑negative holomorphic sectional curvature. While the examples satisfy many curvature properties, it remains open whether they admit Kähler (or even Hermitian) metrics with genuinely quasi‑negative holomorphic sectional curvature.

In summary, the paper establishes a striking rigidity: a weakly special projective manifold without rational curves must be an étale quotient of an Abelian variety, provided standard conjectures in the minimal model program and Lang’s hyperbolicity conjecture hold. Moreover, the authors exhibit concrete families of general‑type manifolds whose exceptional loci are precisely varieties that support dense entire curves, thereby giving a geometric realization of Lang’s exceptional set conjecture. This work deepens the understanding of the boundary between special and hyperbolic geometry in the algebraic setting and opens new avenues for exploring the structure of exceptional loci in higher dimensions.