Holomorphic one-forms without zeros on threefolds
Feng Hao, Stefan Schreieder

TL;DR
This paper characterizes smooth complex projective threefolds that admit a holomorphic one-form without zeros, linking this property to the topology of the underlying real 6-manifold and providing a complete classification, thus confirming a conjecture of Kotschick.
Contribution
It provides a full classification of threefolds with holomorphic one-forms without zeros, establishing a topological criterion and proving Kotschick's conjecture in dimension three.
Findings
Threefolds admit zero-free holomorphic one-forms iff their underlying 6-manifold fibers over the circle.
Complete classification of such threefolds.
Proof of Kotschick's conjecture in dimension three.
Abstract
We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.
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Holomorphic one-forms without zeros on threefolds
Feng Hao
Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany
KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
and
Stefan Schreieder
Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany
(Date: December 26th, 2019)
Abstract.
We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real -manifold is a -fibre bundle over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.
Key words and phrases:
Topology of algebraic varieties, one-forms, minimal model program, classification, generic vanishing, threefolds.
2010 Mathematics Subject Classification:
primary 14F45, 14J30, 32Q55; secondary 32Q57
1. Introduction
1.1. Holomorphic one-forms and fibre bundles over the circle
For a smooth complex projective variety we may consider the following conditions:
- (A)
admits a holomorphic one-form without zeros; 2. (B)
admits a real closed -form without zeros; or, by Tischler’s theorem [Ti70] equivalently, the underlying differentiable manifold is a -fibre bundle over the circle.
Note that while (A) is an algebraic condition on , condition (B) is a differential geometric one which by [Ti70] characterizes the smooth manifold which underlies as , where is a closed manifold of real dimension and where is identified with via some diffeomorphism of .
While (A) (B) is clear, Kotschick conjectured [Kot13] that both condition might be equivalent to each other. In [Sch19], the second author developed an approach to this conjecture, showing that (B) implies
- (C)
there is a holomorphic one-form such that for any finite étale cover , the sequence
[TABLE]
given by cup product with is exact for all .
Moreover, all three conditions above coincide in dimension two [Sch19, Theorem 1.3]. In this paper, we address the much more difficult case of threefolds.
Theorem 1.1**.**
Let be a smooth complex projective threefold. Then all three conditions above are equivalent to each other: (A) (B) (C).
In the appendix to this paper, we explain how the argument in [Sch19] can be generalized to show that (C) is also implied by the following weak version of (B):
- (B’)
is homotopy equivalent to a CW complex that admits a continuous map to the circle which is a finite -homology fibration, i.e. are local systems of finite-dimensional -vector spaces for all .
Since (B) (B’) is clear (c.f. [Ti70]), this implies by Theorem 1.1 the following:
Corollary 1.2**.**
Let be a smooth complex projective threefold. Then (A) (B’). In particular, the question whether carries a holomorphic one-form without zeros depends only on the homotopy type of .
1.2. Classifying threefolds whose underlying -manifolds fibre over the circle
Theorem 1.1 follows from the following strong classification result.
Theorem 1.3**.**
If is a smooth complex projective threefold, any of the conditions (A), (B), (B’) and (C) is equivalent to the following:
- (D)
the minimal model program for yields a birational morphism to a smooth projective threefold , such that:
- (1)
* is a sequence of blow-ups along smooth elliptic curves that are not contracted via the natural map to the Albanese variety .* 2. (2)
There is a smooth morphism to a positive-dimensional abelian variety . 3. (3)
If , then a finite étale cover splits into a product , where is smooth projective and is an abelian variety such that for all , the natural composition
[TABLE]
is a finite étale cover. 4. (4)
If , then one of the following holds:
- (i)
* admits a smooth del Pezzo fibration over an elliptic curve;* 2. (ii)
* has the structure of a conic bundle over a smooth projective surface which satisfies (A) (B) (C). Moreover, is either smooth, or is an elliptic curve and the degeneration locus of is a disjoint union of smooth elliptic curves on which are étale over (via the map induced by ).*
Note that (D) (A), because a general one-form on has no zeros by (D1) and (D2). Since (A) (B) (B’) is clear and (B’) (C) is known (see [Sch19] and Theorem A.1 in the Appendix), in order to prove Theorems 1.1 and 1.3, it thus suffices to show (C) (D). In the course of our proof, we will obtain the following refined version of (A) (C).
Theorem 1.4**.**
Let be a smooth complex projective threefold and let be a holomorphic one-form on . Then the following are equivalent:
- (1)
* has no zero on ;* 2. (2)
for any étale cover , the sequence given by cup product with
[TABLE]
is exact for all .
The implication (1) (2) is a result of Green and Lazarsfeld which holds in arbitrary dimensions, see [GL87, Proposition 3.4]. The converse implication (2) (1) has previously been proven in dimension two by the second author in [Sch19, Theorem 1.3].
Theorem 1.4 has the following interesting consequence.
Corollary 1.5**.**
For a holomorphic one-form on a smooth complex projective threefold , the condition that has no zeros on is a topological one which depends only on the cohomology class of and the homotopy type of .
By the above corollary, if is a holomorphic one-form without zeros on a smooth projective threefold , then for any smooth projective threefold which is deformation equivalent to , and for any one-form that is obtained via parallel transport of with respect to some path, has no zeros on . This is interesting already in the case where .
1.3. Around a theorem of Popa and Schnell
Recall that items (D1) and (D2) imply condition (A). Hence, Theorem 1.3 shows in particular that (D1) and (D2) imply conditions (D3) and (D4). For this reason, Theorem 1.3 has the following consequence; we give the details of the argument in Section 9 below.
Corollary 1.6**.**
Let be a smooth morphism from a smooth projective threefold to an abelian variety . If , then there is a smooth projective threefold with the structure of an analytic fibre bundle over , such that and are birational over .
Passage to a birational model of is necessary; the example of suitable minimal del Pezzo fibrations over elliptic curves shows that the assumption is necessary as well.
By [Ue75, Theorem 15.1], the Kodaira dimension is additive for analytic fibre bundles and so, in the situation of Corollary 1.6, we have , where denotes a fibre of . Hence, , which is a special case of a celebrated result of Popa and Schnell [PoSch14] (with earlier results in [Za97, LZ05, HK05]), who showed that a smooth projective variety has Kodaira dimension , if it admits a -dimensional linear subspace of holomorphic one-forms such that any nonzero has no zeros on .
The results in this paper lead us to the following two conjectures, which by the aforementioned additivity of Kodaira dimensions in analytic fibre bundles [Ue75, Theorem 15.1] would generalize Popa–Schnell’s result.
Conjecture 1.7**.**
Let be a smooth projective variety which admits a holomorphic one-form without zeros. Then is birational to a smooth projective variety which admits a smooth morphism to a positive-dimensional abelian variety .
Conjecture 1.8**.**
Let be a smooth morphism from a smooth projective variety to an abelian variety . If , then, up to birational equivalence, is an analytic fibre bundle.
Conjectures 1.7 and 1.8 hold for surfaces by [Sch19, Corollary 3.2] and for threefolds by item (D2) in Theorem 1.3 and Corollary 1.6. If the fibres of are of general type, a weak form of Conjecture 1.8 had been proven by Popa–Schnell in arbitrary dimensions, see [PoSch14, Corollary 3.2]. Moreover, [PoSch14, Corollary 3.2] easily implies Conjecture 1.8 if the fibres of are curves.
1.4. Why one-forms?
Theorem 1.3 yields a complete classification of all smooth complex projective threefolds with a one-form without zeros, and Corollary 1.2 shows that this is in fact a topological property. It is natural to wonder if such a classification is possible also for forms of higher degree. For top differential forms, this essentially amounts to classifying Calabi-Yau threefolds, and it is a famous open problem to show that such varieties come in finitely many topological types. The remaining case is that of two-forms without zeros. Two-forms on threefolds have previously been studied by Campana and Peternell [CaPe00] who found infinitely many examples of smooth projective threefolds of general type (see e.g. [CaPe00, Example 1.3.3]) which carry two-forms without zeros. This suggests that a classification is probably impossible in this case.
1.5. A remark on the Kähler case
It is conceivable that the methods of this paper allow to prove analogues of Theorems 1.1, 1.3 and 1.4 also in the case of Kähler threefolds. The main technical difficulty that one has to overcome is the fact that short exact sequences of abelian varieties always split after étale cover, while this is in general not true for short exact sequences of arbitrary complex tori. As a consequence, item (D3) in Theorem 1.3 does not remain true in the Kähler setting, but we expect that under the Kähler assumption, one can still prove that in item (D3) there is a smooth morphism as in item (D2) which is in fact an analytic fibre bundle.
Conventions and notation
We work over the field of complex numbers. A variety is an integral separated scheme of finite type over . A minimal model is a projective variety with terminal -factorial singularities such that is nef.
2. Preliminaries
2.1. Analytic fibre bundles
A proper morphism of complex manifolds (or smooth complex projective varieties), is an analytic fibre bundle, if it is analytically locally isomorphic to a product of the base with a typical fibre . The isomorphism type of is determined by a cocycle in . Moreover, by a well-known result of Fischer and Grauert [FG65], a proper morphism of complex manifolds is an analytic fibre bundle if and only if it is isotrivial, i.e. all fibres are isomorphic to each other.
2.2. Basic properties of condition (C)
In [Sch19, Theorems 1.2 and 1.5], the second author proved the following two theorems, which are the starting point of our investigation.
Theorem 2.1** ([Sch19]).**
For any compact Kähler manifold , we have (B) (C).
Theorem 2.2** ([Sch19]).**
Let be a compact Kähler manifold with a holomorphic one-form such that the complex given by cup product with is exact. Then the analytic space given by the zeros of has the following properties.
- (1)
For any connected component with ,
[TABLE]
In particular, does not have any isolated zero. 2. (2)
If is a holomorphic map to a complex torus such that , then is fibred by tori.
We will also use the following lemma, which is also crucial in [Sch19, Theorem A.1].
Lemma 2.3**.**
Let be a compact Kähler manifold and let be a morphism to a complex torus which is generated by the image . Assume that there is a one-form , such that is exact. If there is a prime divisor with , then is an elliptic curve and is linearly equivalent to some rational multiple of a general fibre of .
Proof.
Let be the Albanese morphism. Then there is a morphism with . Let the subtorus generated by . Since is a point, is contracted by to a point and so factors through . Hence, up to replacing by , we may assume that
[TABLE]
Then there is a natural short exact sequence
[TABLE]
Since , and so by exactness of we get
[TABLE]
Since is a real class, This implies
[TABLE]
That is, the pullback of the holomorphic one-form is identically zero on . Since is generated by , it follows that restricts to zero on and so it lies in the image of . That is, lies in the image of and so
[TABLE]
This implies that there is a line bundle on with for some integer . Let be the normalization and let be the natural map. Since is normal, induces a morphism . Applying the Stein factorization to , we conclude that there is a positive integer (equal to the degree of the finite map in the Stein factorization of ) and a natural isomorphism
[TABLE]
Since is effective, we find that admits a section whose pullback to vanishes along with multiplicity . Since is a point, is a point as well. Since is finite, is a point and so is a curve. This implies that must be a curve as well. Since is a curve, its normalization is an elliptic curve by exactness of . Since , it must be a smooth elliptic curve and since it generates , the latter is an elliptic curve as well. Since is a line bundle on the elliptic curve , implies that is linearly equivalent to a rational multiple of a general fibre of . This concludes the lemma. ∎
Corollary 2.4**.**
In the situation of Lemma 2.3, the Stein factorization of yields a morphism to an elliptic curve with irreducible fibres.
Proof.
Since is an elliptic curve, the Stein factorization of yields a morphism to a smooth projective curve . By assumptions, there is a one-form such that is exact. This implies . Applying Lemma 2.3 to the irreducible components of the fibres of then shows that has irreducible fibres, as we want. ∎
Lemma 2.5**.**
Let be a compact Kähler threefold which satisfies (C). Then
[TABLE]
Proof.
Condition (C) implies immediately for all and so the claim follows from Riemann–Roch. ∎
3. Reduction to minimal threefolds or Mori fibre spaces
Proposition 3.1**.**
Let be a smooth complex projective threefold with a one-form such that is exact. If is not nef and does not carry the structure of a Mori fibre space, then there is a smooth projective threefold such that is the blow-up of along a smooth elliptic curve . Moreover, if denotes the one-form induced by , then is exact and is nonzero.
Proof.
If is not nef and does not carry the structure of a Mori fibre space, then by [Mor82, Theorem 3.3], there is a divisorial contraction whose exceptional divisor has one of the following two properties:
- •
is a -bundle over a smooth curve , is smooth and ;
- •
is a point.
Moreover, in both cases, contains a curve which has negative self-intersection with . It thus follows from Lemma 2.3, applied to the Albanese map of , that cannot be contracted to a point and so must be the blow-up along a smooth curve . The formula for the cohomology of blow-ups shows that exactness of implies that is exact, is an elliptic curve and is nonzero. This concludes the proposition. ∎
Corollary 3.2**.**
Let be a smooth complex projective threefold which satisfies condition (C). Then there is a smooth projective threefold with a birational morphism , which is given as a sequence of blow-ups along smooth elliptic curves that are not contracted via the natural map to . Moreover, satisfies (C) and it is either minimal or a Mori fibre space.
Proof.
This is a direct consequence of Proposition 3.1, where we note that an elliptic curve on a smooth projective variety is contracted via the Albanese map of if and only if any holomorphic one-form on restricts trivially on . ∎
By Proposition 3.1 and Corollary 3.2, the proof of Theorems 1.3 and 1.4 reduce to the case where is either minimal, or it admits the structure of a Mori fibre space.
The following corollary of the above discussion generalizes the main result of Luo and Zhang in [LZ05].
Corollary 3.3**.**
A smooth projective threefold which satisfies condition (A), (B) or (C) is not of general type.
Proof.
Since (A) (B) is clear and (B) (C) by [Sch19] (see Theorem 2.1), we may assume that satisfies (C). For a contradiction, we assume that is of general type. By Corollary 3.2, we may additionally assume that is minimal. By the Miyaoka–Yau inequality,
[TABLE]
This contradicts Lemma 2.5, which concludes the corollary. ∎
4. 1-forms on threefolds of non-negative Kodaira dimension
In the case of non-negative Kodaira dimension, our main results will follow from:
Theorem 4.1**.**
Let be a smooth complex projective threefold of non-negative Kodaira dimension and with nef. Assume that satisfies condition (C).
Then there is a finite étale covering which splits into a product , where is an abelian variety of positive dimension.
The proof of the above theorem occupies the following three sections and the final arguments will be summarized in Section 7. Before we turn to the proofs, let us note the following consequence.
Corollary 4.2**.**
In the notation of Theorem 4.1, let be the one-form from condition (C). Then the following holds:
- (1)
up to passing to a finite étale covering of , we may assume that is simple and restricts to a nonzero form on for all ; 2. (2)
* is smooth projective with ;* 3. (3)
* has no zeros on and so has no zeros on ;* 4. (4)
There is a smooth morphism to an abelian variety such that for any , the composition
[TABLE]
is finite étale.
Proof.
We have
[TABLE]
To see that restricts to a nonzero form on for all , it suffices to show that it does not map to zero under the projection to . If it does map to zero, then satisfies the equivalent conditions (A) (B) (C). This implies by [Sch19, Corollary 3.2] that some étale cover of splits off a positive-dimensional simple abelian variety and the restriction of the pullback of to that factor is non-trivial, as we want. Hence, up to passing to another étale cover and replacing the decomposition by another one, we may assume that restricts to a nonzero form on for all . Up to passing to another finite étale cover of , we may by the complete reducibility theorem also assume that is simple and so item (1) of the corollary holds.
Since is étale and is smooth, so is . Since , is smooth of Kodaira dimension , as claimed in item (2). Moreover, item (3) is an immediate consequence of item (1).
It remains to prove (4). Consider the Albanese map . Since is simple and restricts non-trivially to for all , we find that for all , the image of the natural composition
[TABLE]
is the translate of an abelian subvariety of that is isogeneous to . Since is projective, there is a quotient map such that for all , the natural composition
[TABLE]
is finite étale. We then define as composition of the Albanese map of with the projection \operatorname{Alb}(X)\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.99997pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces A. With this definition, the composition (6) is finite étale for all . This implies that is smooth. Since is étale, it follows that is smooth as well, as we want. This proves (4), which concludes the corollary. ∎
5. Proof of Theorem 4.1 for
In this section we aim to prove Theorem 4.1 in the case where .
5.1. Preliminaries on elliptic threefolds
Definition 5.1**.**
An elliptic threefold is a normal projective threefold with a morphism to a normal projective surface whose general fibre is an elliptic curve. We say that has trivial (or no) monodromy, if restricts to a trivial local system over some non-empty (Zariski) open subset .
Lemma 5.2**.**
Let be an elliptic threefold with trivial monodromy and such that has rational singularities (e.g. terminal singularities). Then the general fibre of is not contracted via the Albanese morphism .
Proof.
Since has rational singularities, the Albanese morphism of any resolution of factors through , and so is defined. The lemma then follows from Deligne’s global invariant cycle theorem (see e.g. [Voi03, Theorem 4.24]) applied to , which implies that carries a holomorphic one-form which restricts nontrivially on the general fibre of the natural map . ∎
We will need the following result, c.f. [Gra94, Theorem 2.7].
Proposition 5.3**.**
Let be terminal threefold with nef of Kodaira dimension two and with Iitaka fibration . If , then has only finitely many singular fibers which are not multiples of a smooth elliptic curve. Moreover, the -invariants of the smooth fibres of are constant.
Proof.
The Iitaka fibration is a morphism by the abundance conjecture [Ka92]. In particular, there is a very ample divisor on such that for some positive rational number . Let be a general element of and let . Since is normal and is terminal, it follows from Bertini’s theorem that and are smooth. Moreover, is a minimal elliptic surface of Kodaira dimension one, because is minimal of Kodaira dimension two and Iitaka fibration . Since is smooth and contained in the smooth locus of , we have a short exact sequence of vector bundles on :
[TABLE]
Applying the Whitney sum formula, we deduce that the second Chern number of is given by
[TABLE]
By adjunction, , and so , as it is a multiple of . Since , we conclude from the above formula. Hence, is a minimal surface of Kodaira dimension one with . Since coincides with the sum of the Euler numbers of the singular fibres of , we find by Kodaira’s classification of singular fibres (see [BHPV]) that any singular fibre of is a multiple of a smooth elliptic curve. This implies that the j-invariant is not dominant and so it must be constant. This proves the proposition. ∎
We have the following important structure theorem of Nakayama. To state it, recall that an elliptic threefold is a standard elliptic fibration if is -factorial and terminal, is equi-dimensional and for an effective -divisor such that is log terminal.
Theorem 5.4** ([Nak02, Theorem A.1]).**
Let be an elliptic threefold. Then there is a proper birational morphism and a standard elliptic fibration that is birational to over , such that is semi-ample over .
Lemma 5.5**.**
Let be an elliptic threefold, such that is terminal, -factorial and is -nef. Let and be as in Theorem 5.4. Then there is a smooth open subset , whose complement in is zero-dimensional, and such that the base change is a smooth threefold. Moreover, for any such open subset , the natural birational map induces an isomorphism
[TABLE]
where denotes the smooth locus of .
Proof.
Since is a standard elliptic fibration, has only terminal -factorial singularities, is equi-dimensional and is normal. In particular, and have isolated singularities. Hence, there is an open subset whose complement in is zero-dimensional, and such that the base change is smooth.
Let now be any such subset. Since is semi-ample over , it is in particular nef over . Hence, and are birational minimal models over and so they are isomorphic in codimension one, see e.g. [KM08, Theorem 3.52(2)]. Since is zero-dimensional and is equi-dimensional, is at most one-dimensional. Since and are isomorphic in codimension one, we conclude the same for and , and hence also for and , because is terminal and so it has isolated singularities. Since and are smooth, this implies , as we want. ∎
5.2. Condition (C) implies trivial monodromy after étale cover
Lemma 5.6**.**
Let be a smooth projective threefold with nef of Kodaira dimension two and with Iitaka fibration . Assume that (C) holds for . If has non-trivial monodromy, then is equi-dimensional.
Proof.
Assume that has non-trivial monodromy. Since any variation of Hodge structure of weight one and rank two with a nonzero section is trivial, this implies that has no generically non-zero section. Hence, the general fibre of is contracted via the Albanese map . This implies that factors rationally through . Since is the base of the Iitaka fibration, it has at most klt singularities, hence rational singularities, and so any rational map from to is a morphism. That is, factors through . Hence, any prime divisor which maps to a point on is contracted to a point by the Albanese map of . Lemma 2.3 then shows that is an elliptic curve and is numerically equivalent to a rational multiple of a fibre of . Since is contracted by , also contracts a general fibre of . Hence factors through , which is impossible because is a surface. This proves the lemma. ∎
Remark 5.7**.**
The condition on the monodromy is necessary in Lemma 5.6. To see this, let be a canonical surface with ample canonical bundle and a single node as singularity, and with minimal resolution . Then for any elliptic curve , the product is a minimal threefold of Kodaira dimension two which satisfies (C), but the Iitaka fibration of is given by the natural map , which is not equi-dimensional.
We are now able to prove the following, which is the main result of Section 5.2.
Proposition 5.8**.**
Let be an elliptic threefold, with smooth and nef. Assume that satisfies (C). Then there is a finite étale cover , such that the elliptic fibration that is induced by via Stein factorization has trivial monodromy.
Proof.
By Lemma 5.6, we may assume that is equi-dimensional. By Proposition 5.3, is generically isotrivial and only finitely many singular fibers of are not multiples of a smooth elliptic curve. This implies that there is an open subset whose complement is zero-dimensional, such that is a local system (which above the multiple fibres can be checked via topological base change). Moreover, this local system has finite monodromy, because the identity component of acts trivially on .
It follows that there is a finite étale cover such that the base change is an elliptic threefold over with trivial monodromy. Note that is a finite étale cover of . Since is equi-dimensional, , and so this finite étale cover extends to a finite étale cover , such that the map , induced via Stein factorization of , is an elliptic threefold with trivial monodromy. This finishes the proof of the proposition. ∎
Remark 5.9**.**
The case where is a product of a curve with a bi-elliptic surfaces shows that the étale covering performed in Proposition 5.8 is really necessary.
5.3. Classification of minimal elliptic threefolds with trivial monodromy
Proposition 5.8 reduces the proof of Theorem 4.1 for to the case of minimal elliptic threefolds with trivial monodromy. Even though is smooth in that situation, it is not much harder to classify more generally such threefolds with terminal singularities. This is the content of the following theorem. To state the result, recall that a finite morphism between normal varieties is called quasi-étale if it is étale in codimension one, see e.g. [GKP16]; if is smooth, then this implies that is étale. In particular, is ramified at most at the singular points of .
Theorem 5.10**.**
Let be an elliptic threefold with trivial monodromy, where is terminal, -factorial and is nef. Then there is a finite quasi-étale covering with , where is an elliptic curve and is a smooth projective surface with a generically finite map to .
In the proof of the above theorem, we will use the following local result.
Lemma 5.11**.**
Let be a proper morphism of complex manifolds over the disc , which is a submersion over the punctured disc . Assume that the special fibre of is irreducible and of multiplicity . Assume that there is a morphism to a compact complex manifold which restricts to a finite morphism on a general fibre of . Let be a general fibre of . Then the normalization of the base change is smooth, the family has reduced fibres and the natural map is étale. If furthermore the reduced fibre is smooth, then is smooth.
Proof.
Since is a submersion over the punctured disc, it suffices to prove the lemma after shrinking , if necessary. By Sard’s theorem, is smooth. Moreover, the natural map is submersive away from the origin of . Let be a connected component of . Since is connected, up to shrinking , meets the central fibre in a single point and so is a cyclic cover and we denote its degree by . Since has multiplicity , divides . Conversely, the morphism induced by is generically smooth and so the preimage of in is given by disjoint reduced points. Since the intersection of with is a single point, we find that this intersection has multiplicity at most . Hence, and is a cyclic cover of degree by . It then follows from a well-known local computation, see e.g. [BHPV, Proposition III.9.1], that the normalization of the base change is smooth, the family has reduced fibres and the natural map is étale. Moreover, the central fibre of is an étale cover of . Hence, is smooth if is smooth. This concludes the lemma. ∎
Proof of Theorem 5.10.
By Theorem 5.4, there is a birational morphism and a standard elliptic fibration that is birational to over and such that is nef over . By Lemma 5.5, there is a smooth open subset whose complement is zero-dimensional and such that is smooth and the birational map induces an isomorphism
[TABLE]
Since has only terminal singularities, there is a well-defined Albanese map , obtained by observing that the Albanese map of any desingularization of factors through . By Lemma 5.2, the general fibre of is not contracted by . Hence, the general fibre of is via mapped to a translate of a fixed elliptic curve . Since is projective, we can dualize this inclusion to get a surjection \operatorname{Alb}(X^{\prime})\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.99997pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces E. Composing this with , we get a surjection
[TABLE]
which restricts to finite étale covers on general fibres of . Taking the Stein factorization, we may assume that has connected fibres. (Note that the target of the Stein factorization will receive a surjection from the general fibres of , which are elliptic curves, and so it cannot be a curve of genus .)
Since is terminal, it has isolated singularities. A general fibre of is thus smooth by Bertini’s theorem and we consider the normalization of the base change . We then get a commutative diagram
[TABLE]
Let be the preimage of . We consider the base change . Since has trivial monodromy by assumptions, the same holds for . Since additionally is nef over , the base change to a general hyperplane section is a smooth minimal elliptic surface with trivial monodromy. This implies that the singular fibres are multiples of smooth elliptic curves: they are (multiples of) one of the fibres in Kodaira’s table [BHPV, p. 201] and additionally have second Betti number at least , because of the triviality of the monodromy and topological proper base change. Hence, away from finitely many points in , all singular fibres of are multiples of smooth elliptic curves.
Let be a general hyperplane section and let be its preimage in . Applying Lemma 5.11 to the base change of and to and , respectively, we find the following: up to removing finitely many points from and , we may assume that is a smooth elliptic fibre bundle and is étale. Since this bundle has a section by construction, the existence of a fine moduli space for elliptic curves with level structure shows that for an elliptic curve .
By [GKP16, Theorem 3.8], any finite étale cover of extends to a finite quasi-étale cover of . Since , the finite étale cover is thus birational to a finite quasi-étale covering
[TABLE]
of . Since , we conclude that is birational to , where is a minimal surface and is an elliptic curve. Since is nef, so is . Moreover, is terminal by [KM08, Proposition 5.20], because it is a finite quasi-étale cover of a terminal threefold.
By [KM08, Theorem 6.25], there is a -factorialization , i.e. a proper birational morphism which is an isomorphism in codimension one such that is -factorial, terminal and is nef. Hence, and are birational minimal models and so they are connected by a sequence of flops (see [Kol89]). Since does not admit any non-trivial flop, On the other hand, the product does not admit a small contraction to a terminal threefold (because any rational curve on it maps to a point on the second factor and so it sweeps out a divisor on ). Hence, and so , as we want. This concludes the proof. ∎
5.4. Proof of Theorem 4.1 for
Since is a smooth projective threefold with nef, the Iitaka fibration is a morphism by the abundance conjecture for threefolds, see [Ka92]. This endows with the structure of an elliptic threefold. By Proposition 5.8, there exists a finite étale covering , such that induces an elliptic fibration without monodromy. Hence, by Theorem 5.10, there is a finite étale cover , such that for some smooth projective surface and an elliptic curve . This concludes Theorem 4.1 if .
6. Proof of Theorem 4.1 for
In this section we aim to prove Theorem 4.1 in the case where is a minimal smooth projective threefold with . By the abundance conjecture (which is known in dimension three, see [Ka92]), the Iitaka fibration of yields a morphism to a smooth projective curve . We may thus consider the diagram
[TABLE]
where denotes the Albanese morphism.
After collecting some preliminary results in Section 6.1, we treat in Sections 6.2, 6.3 and 6.4 below the cases where the Albanese image of a general fibre of has dimension two, one and zero, respectively.
6.1. Preliminaries
We begin with the following simple observation, cf. [Gra94, Theorem 1.5].
Lemma 6.1**.**
Let be a smooth projective threefold with Iitaka fibration to a smooth projective curve . Then, if and only if the smooth fibres of are bi-elliptic or abelian surfaces.
Proof.
Some multiple is linearly equivalent to the pullback of an ample divisor on . Hence, is numerically equivalent to a positive multiple of a general fibre of . Hence, if and only if restricts to zero on , which is to say that because has trivial normal bundle. Since is a minimal surface of Kodaira dimension zero, implies that is either bi-elliptic or an abelian surface, see e.g. [BHPV, Chapter VI.1]. This concludes the lemma. ∎
Next, we recall the following definition, see [CaPe00, Definition 1.9].
Definition 6.2**.**
Let be a smooth projective threefold with a surjective morphism to a smooth curve . A holomorphic two-form on is vertical (with respect to ) if the annihilator of on the tangent space of at a general point is tangential to the fibre of at . Equivalently, is vertical if and only if it has trivial image via the natural map .
With this terminology, Campana and Peternell proved the following:
Theorem 6.3** ([CaPe00, Theorem 4.2]).**
Let be a minimal smooth projective threefold with . Let be the Iitaka fibration. Assume that there is a holomorphic two-form on which is not vertical with respect to . Then there is a finite morphism such that the normalization of the base change splits into a product . In particular, is quasi-smooth, i.e. all singular fibres of are multiple fibres.
The multiplicities of the singular fibres define an effective divisor on and this divisor defines an orbifold structure on , see e.g. [FM94, Section 2.1.3]. We say that such an orbifold is good, if there is an orbifold étale covering with trivial orbifold structure.
Proposition 6.4**.**
Let be a smooth projective threefold with a morphism to a smooth projective curve . Assume that is quasi-smooth with typical fibre an abelian surface , i.e. the smooth fibres of are isomorphic to a fixed abelian surface and the singular fibres are multiples of an abelian surface isogeneous to . If the orbifold structure on that is induced by the multiple fibres of is good, then there is a finite étale covering such that , where is a finite cover of .
Proof.
If the given orbifold structure is good, then there is a branched cover , branched exactly at the orbifold points of with the prescribed multiplicities. The base change is singular along the preimage of the singular fibres of . Let be the normalization of . Then a local analysis shows that is smooth, the natural map is étale and is smooth. Since is projective, the same argument as in the proof of [CaPe00, Theorem 4.2] shows that up to a further étale base change , as we want.111Note that in loc. cit. it is claimed that this splitting holds even in the Kähler setting, but this seems to be incorrect, because even after étale cover, the extension (8) does in the non-polarized setting in general not split. We repeat the argument for convenience of the reader. Consider the natural map and note that is a local system. Since acts on via a finite quotient, has finite monodromy. Hence, after a suitable étale base change, we may assume that is trivial and so . We thus get a short exact sequence
[TABLE]
Since is projective, this sequence splits after a suitable étale cover of and so , as we want. This concludes the proposition. ∎
We finally recall the classification of all good orbifolds, see e.g. [CHK00, Corollary 2.29].
Theorem 6.5**.**
Let be a smooth projective curve with an effective divisor . The orbifold is good unless and one of the following holds:
- (1)
* consists of one point with some multiplicity;* 2. (2)
* consists of two points with different multiplicities.*
6.2. The general fibre of is not contracted via the Albanese map
By Lemmas 2.5 and 6.1, a general fiber of is an abelian or a bi-elliptic surface. Since in the present case, must be an abelian surface. In particular, is the translate of a fixed abelian subvariety of . This implies that the pullback of a general holomorphic two-form from to is not vertical with respect to . Hence, Theorem 6.3 applies and we see that is quasi-smooth. The multiplicities of the singular fibres define an effective divisor on and this divisor defines an orbifold structure on , see e.g. [FM94, Section 2.1.3]. We claim that this orbifold structure is good, i.e. there is an orbifold étale covering which has trivial orbifold structure. To this end, we may by Theorem 6.5 assume . Then the Leray spectral sequence yields . This implies , because a holomorphic one-form (hence also an anti-holomorphic one-form) on which vanishes on one smooth fibre of vanishes at any smooth fibre, since the image of these fibres in are translates of the same abelian subvariety. Hence, induces an isogeny on the general fibre of . For a general point , the preimage is a smooth projective curve and the natural map is branched exactly at the given points of the orbifold structure on . Hence, the orbifold structure is good, as we want. The proof of Theorem 4.1 in the case treated in Section 6.2 thus follows from Proposition 6.4.
6.3. The general fibre of is via the Albanese map contracted to a curve
Recall that the smooth fibres of are either bi-elliptic or abelian surfaces by Lemmas 2.5 and 6.1. In the present case, this implies that the fibres of are mapped via the Albanese morphism to translates of a fixed elliptic curve in . Since is projective, we can dualize the inclusion and get a surjection . This gives rise to a morphism
[TABLE]
which when restricted to the fibres of coincides with up to isogeny. Taking the Stein factorization, we may assume that has connected fibres. (Note that the target of the Stein factorization receives a surjection from the fibres of , which are bi-elliptic or abelian surfaces, and so it cannot be a curve of genus .)
We proceed in several steps.
Step 1. Up to replacing by a finite étale covering that is induced by a finite étale cover of , we may assume that there is a finite étale cover such that for general ,
[TABLE]
for an elliptic curve which might depend on and such that corresponds to the natural composition .
Proof.
Consider the restriction of to a smooth fibre of . Even though has connected fibres, this might a priori not have connected fibres, but taking the Stein factorization, we see that there is a finite étale cover , such that factors through a morphism with connected fibres. Since is bi-elliptic or abelian, there is a finite étale cover , such that the induced étale cover splits into a product
[TABLE]
where is an elliptic curve which might depend on .
Note that is an étale cover and so it is determined by a finite index subgroup of . In particular, there is a finite étale cover which is isomorphic to for all general points . Similarly, there is a finite étale covering which is isomorphic to for general .
Lemma 6.6**.**
Up to replacing by a further finite étale cover, there is a finite étale cover , such that is isomorphic to . That is, there is a Cartesian diagram
[TABLE]
Proof.
Consider the sequence of fundamental groups . Identifying all three groups with , this sequence is of the form
[TABLE]
where are invertible matrices with integer entries. Since is determined by a finite index subgroup of , the lemma is then equivalent to finding an invertible matrix with integer coefficients, such that
[TABLE]
the étale cover is then induced by the subgroup of given by the image of . The above condition is equivalent to asking that has integer entries. But this can easily be ensured by multiplying by a sufficiently divisible integer, which corresponds to replacing by a further finite étale cover. This proves the lemma. ∎
Let us now consider the finite étale cover of together with the natural maps and . The fibre of above a general point is given by the fibre product
[TABLE]
which coincides with from above and so it splits as in (10) into a product of elliptic curves. Hence, up to replacing by , we may assume that for general , for a finite étale cover and for an elliptic curve which might depend on . That is, (9) holds, which concludes step 1. ∎
Note that is an isomorphism if and only if has connected fibres. We show next that, up to a suitable base change, we may assume that this holds true.
Step 2. In the above notation, up to replacing by a finite étale cover, given by the normalization of for some finite map , we may assume that has connected fibres, i.e. in step 1 is an isomorphism.
Proof.
To prove the claim, let X\stackrel{{\scriptstyle h}}{{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces}}S\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces C\times E be the Stein factorization of . We claim that is smooth and the natural map is a minimal elliptic surface all of whose singular fibres are multiples of an elliptic curve.
To see this, note that is normal. By (9), the general fibre of is isomorphic to
[TABLE]
That is, is a normal elliptic surface whose general fibres are isomorphic to . Since has connected fibres, the fibres of the natural map are connected and their reductions are finite covers of . Since the arithmetic genus is constant in flat families, all fibres of must be irreducible. Let be a minimal resolution. Since has connected fibres, the induced map has connected fibres as well. Since is the minimal resolution and no fibre of has a rational component, it follows that is relatively minimal elliptic surface. Since the reductions of the fibres of are finite (étale) covers of , we find that all singular fibres of are multiple fibres (by Kodaira’s table of singular fibres of relatively minimal elliptic fibrations, see e.g. [BHPV]). But this implies that is an isomorphism, as we want.
Since is a minimal elliptic surface whose general fibres are isomorphic to and whose singular fibres are multiples of an elliptic curve, there is a finite étale cover , given by the normalization of for some finite morphism , such that
[TABLE]
The cover induces via a finite étale cover . Then we get the following commutative diagram
[TABLE]
where the composition of the lower horizontal arrow is and is the product of and the étale cover . Since has connected fibres by construction, has also connected fibres and so the Stein factorization of the natural map is given by the composition of with the second projection . Hence, up to replacing by , must be an isomorphism, as we want. This concludes step 2. ∎
By steps 1 and 2, we may from now on assume that for general
[TABLE]
where is an elliptic curve that might depend on and the restriction of to corresponds via the isomorphism in (12) to the first projection .
Step 3. is birational to for a minimal surface .
Proof.
By (12), we have for general a splitting for an elliptic curve which might depend on . To prove the claim in step 3, we need to show that such a splitting holds not only for a general point but also for the generic point of . That is, we need to show that the generic fibre of splits into a product of with an elliptic curve over .
It is well-known that the splitting in (12) implies that splits over the algebraic closure into a product of with an elliptic curve over . The latter must be defined over for some finite cover . That is, there is a Zariski open non-empty subset and a finite Galois cover such that splits into , where is an elliptic surface over . The base change fits into a diagram
[TABLE]
Here the outer square is Cartesian. The square on the right is Cartesian as well and so it follows that the square on the left must be Cartesian as well. By assumptions, is Galois and we denote its Galois group by . Then, acts faithfully on with . By base change, also acts on and with quotients and , respectively. Moreover, the upper horizontal arrows in the above diagram are -equivariant. In particular, the map is -equivariant and acts trivially on the first factor of . This implies that acts trivially on the first factor of and so
[TABLE]
where denotes the quotient of by the induced action of . This concludes step 3. ∎
By step 3, there is a minimal surface such that is birational to . Since has positive Kodaira dimension, the same holds for and so is nef. Hence, and are birational minimal models, and so they are connected by a sequence of flops, see [Kol89]. Since does not admit any non-trivial flop, , as we want. This concludes Theorem 4.1 in the case treated in Section 6.3.
6.4. The general fibre of is via the Albanese map contracted to a point
Since in the present case, the general fibre of is contracted via to a point, every fibre must be contracted to a point and so the Albanese map factors as . Since carries a holomorphic one-form such that is exact, and so . Moreover, is a pullback of a one-form from and so exactness of shows that . Hence, .
By Lemmas 2.5 and 6.1, the general fibre of is either an abelian surface or a bi-elliptic surface.
Case 1. .
In this case, carries a nontrivial holomorphic two-form . Since , exactness of shows that . Since is the pullback of a one-form on , the condition then implies that is not vertical. Hence, by Theorem 6.3, is quasi-smooth with typical fibre , a bi-elliptic or an abelian surface. Since is not vertical with respect to , it restricts to a nonzero form on the general fibre of and so must be an abelian surface. The multiple fibres of give rise to an orbifold structure on which is good because , see Theorem 6.5. Hence, Proposition 6.4 shows that splits into a product after a finite étale cover, as we want. This proves Theorem 4.1 in the case treated in Section 6.4 if .
Case 2. and the general fibre of is an abelian surface.
In this case, consider the sheaf . By Kollár’s theorem [Kol86, Theorem I.2.1], this is locally free of rank one, i.e. a line bundle on . Since vanishes, the Leray spectral sequence shows that
[TABLE]
Moreover, , as otherwise we get , which is impossible because and is exact on cohomology. It follows that the line bundle has no cohomology and so Riemann–Roch implies that it has degree zero. That is, .
Since , the cohomology support locus
[TABLE]
is a union of proper subtori of , translated by torsion points [Si93]. Since is an elliptic curve, the above set is in fact a union of torsion line bundles. This set contains , because
[TABLE]
Hence, is a torsion line bundle. But then up to passing from to a suitable étale covering (induced by a étale covering of ), we may (by flat base change) assume that is trivial. This implies and so by exactness of . Hence, we may conclude via Case 1.
Case 3. and the general fibre of is a bi-elliptic surface.
In this case, is locally free of rank one. Since , by the Leray spectral sequence (which degenerates by Kollár’s theorem or because is a curve). Similarly, implies . By relative Serre duality,
[TABLE]
where we used in the last step. Hence, by Serre duality on ,
[TABLE]
That is, is a line bundle on without cohomology and so it must be in .
Since , the cohomology support locus
[TABLE]
is a union of torsion line bundles (see [Si93]), where we use that is an elliptic curve. As before, the Leray spectral sequence degenerates at . Hence,
[TABLE]
is also a union of torsion line bundles. Since is a line bundle of degree zero on , we deduce that it must be torsion. Hence, after a suitable étale base change , we may (by flat base change) assume that . This implies , and so we are done by Case 1, treated above.
Cases 1, 2 and 3 above finish the proof of Theorem 4.1 in the case treated in Section 6.4. Together with the results in Sections 6.2 and 6.3, this finishes the proof of Theorem 4.1 in the case of Kodaira dimension one.
7. Proof of Theorem 4.1
Let be a smooth projective minimal threefold which satisfies (C) as in Theorem 4.1. By Corollary 3.3 cannot be of general type, and so we are left with the cases .
If , then Theorem 4.1 is proven in Section 5.4.
If , then the result is proven in Sections 6.2, 6.3 and 6.4 above.
If , is trivial for some by the abundance conjecture for complex projective threefolds, see [Ka92]. It then follows from the Beauville–Bogomolov decomposition theorem (see [Bea83]) and the fact that that some finite étale cover splits into a product with a positive-dimensional abelian factor , as required.
This concludes the proof of Theorem 4.1.
8. 1-forms on Mori fiber spaces
In this Section we prove Theorems 1.3 and 1.4 in the case of negative Kodaira dimension. The main results are Theorems 8.3 and 8.4 below.
8.1. Preliminaries
Recall the following well-known lemma.
Lemma 8.1**.**
Let be a surjective morphism from a compact complex manifold to an elliptic curve . Then the topological Euler characteristic of is given by
[TABLE]
where denotes a fixed smooth fibre of .
Lemma 8.2**.**
Let be a surjective morphism from a smooth complex projective threefold to a smooth projective surface with . Suppose that for , is exact. Then is exact.
Proof.
Assume first that there is a divisorial contraction with smooth. Then the composition contracts a divisor to a point, contradicting Lemma 2.3, applied to the Albanese morphism , because factors through by assumptions. Hence, a contraction as above does not exist and so is minimal. Next, we consider the following diagram
[TABLE]
By diagram chasing, we have that is exact at for . Also, is exact at . Assume is not exact at . Then and so is a ruled surfaces over a curve of genus , see [BHPV, Theorem VI.1.1]. The latter contradicts the fact that the Albanese image of (and hence that of ) is fibered by tori by item (2) in Theorem 2.2. ∎
8.2. Conic bundles over surfaces
Theorem 8.3**.**
Let be a smooth projective threefold which admits the structure of a Mori fibre space over a projective surface . Let be a holomorphic one-form on such that for any étale cover , is exact. Then the following holds:
- (1)
* is a smooth projective surface which satisfies (A) (B) (C);* 2. (2)
there is a smooth map to a positive-dimensional abelian variety ; 3. (3)
* is either smooth, or is an elliptic curve and the degeneration locus of is a disjoint union of smooth elliptic curves which are étale over (via the map induced by );* 4. (4)
* has no zero on .*
Proof.
By [Mor82, Theorem 3.5], is a conic bundle of relative Picard rank one, is smooth and the discriminant locus of is a curve with at worst ordinary double points.
Since induces an isomorphism on , and so we get a commutative diagram
[TABLE]
where and are the respective Albanese morphisms. Hence there is a one-form with .
Let be a finite étale cover, and let be the induced étale cover of . Then is a conic bundle and so . Hence, Lemma 8.2 implies that is exact. This proves item (1). In particular, admits a smooth morphism to an elliptic curve or abelian surface by [Sch19, Corollary 3.2], which proves item (2) in the case where is smooth. For the remainder of the proof of items (2) and (3), we may thus assume that .
By (1) and [Sch19, Corollary 3.1], is one of the following:
- (a)
a minimal ruled surface over an elliptic curve; 2. (b)
an abelian surface; 3. (c)
a minimal elliptic surface such that one of the following holds:
- (i)
is smooth, is an elliptic curve and ; 2. (ii)
is quasi-smooth, i.e. all singular fibres are multiple fibres, and the restriction of to a general fibre of is non-zero.
By [PaSh99, lemma 7.1.10],
[TABLE]
where denotes the arithmetic genus of . Since and by exactness of and , we deduce
[TABLE]
Since is a nodal curve, this implies that each connected component of is either a smooth elliptic curve with some attached trees of rational curves, or a rational curve with a single node with some attached trees of rational curves. Note that cannot contain a smooth which is attached to the remaining components at only one point , as the monodromy of the lines above would need to be trivial and so could not have relative Picard rank one, contradicting our assumptions. Moreover, since none of the surfaces in (a)–(cii) contains a rational curve with at least one node, we conclude that must be a disjoint union of smooth elliptic curves. By assumptions, and so we pick an irreducible component , which is automatically a smooth connected component of . Hence, each fibre of above a point is given by two distinct lines which meet in a point, see e.g. [PaSh99, Proposition 7.1.8(i)].
By [Sch19, Corollary 3.2], some étale cover is either a simple abelian surface, or it splits into the product of an elliptic curve with another curve. This implies that there is a finite étale cover whose restriction to induces an étale cover that trivializes the monodromy of the two lines above points of in the conic bundle . Let be the finite étale cover induced by . Then, has relative Picard rank at least two, because the monodromy of the two lines above is trivial. In particular, admits a divisorial contraction where exactly one of the two families of lines above is contracted, and we exhibit as the blow-up of a conic bundle which is smooth above . By Proposition 3.1, the pullback restricts nontrivially on . This implies that restricts nontrivially on . Hence, there is a surjective morphism p:A\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 1.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-1.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-1.99997pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces E to an elliptic curve , so that the composition restricts on to an isogeny .
Since splits into a product after some étale cover by [Sch19, Corollar 3.2(e)], one checks that must be a smooth morphism. Since the remaining components of are all smooth elliptic curves that are disjoint from , we conclude that in fact any component of is étale over via the morphism . This concludes item (3). Moreover, it shows that the composition is smooth, which proves item (2).
It remains to prove item (4). We have seen that the one-form comes from a form that has no zeros on by [Sch19, Corollary 3.1]. Hence, has no zeros on in the case where is smooth. If is not smooth, we have seen above that restricts nontrivially on each component of the ramification locus , and this implies that has no zeros along , and hence no zeros on . This proves item (4), which finishes the proof of Theorem 8.3. ∎
8.3. Del Pezzo fibrations
Theorem 8.4**.**
Let be a smooth complex projective threefold which admits a holomorphic one-form such that for any étale cover , is exact. Suppose that admits the structure of a Mori fibre space over a curve . Then is an elliptic curve and is smooth. In particular, has no zeros on .
Proof.
By [Mor82, Theorem 3.5.2], is a smooth projective curve and any fibre of is an irreducible reduced del Pezzo surface which is Gorenstein, as its canonical bundle is given by the restriction of the canonical bundle of . Moreover, since has its zeros exactly at the singular points of the fibres of , and since it cannot have isolated zeros by item (1) in Theorem 2.2, the singular fibres of are reduced and non-normal.
Note that induces an isomorphism . Since is exact, we deduce that is an elliptic curve, see e.g. item (2) in Theorem 2.2. It remains to prove that is smooth.
Since and , . Since and is exact, we then conclude
[TABLE]
We consider the Leray spectral sequence
[TABLE]
Since is a smooth projective curve, this spectral sequence degenerates at , see e.g. [PeSt08, Theorem 4.24]. Since the general fibre of is a smooth del Pezzo surface, is a skyscraper sheaf, which implies . We also have . Since and , we conclude
[TABLE]
In particular, is generated by the section of induced by an ample class on .
By topological proper base change, we have for any and any :
[TABLE]
We claim that this implies that is bounded from above by the second Betti number of a smooth fibre of . Indeed, if not, then there is a neighbourhood of and a section which is nonzero at the stalk at but zero at all other stalks. But then after taking the extension by zero, can be extended to a nontrivial global section of which is zero away from the point , contradicting (13).
Let be a general point, so that the fibre is smooth. For any , we have seen above that . Moreover, since is irreducible, and so the topological Euler characteristics satisfy
[TABLE]
for all . Since is an elliptic curve and has trivial topological Euler characteristic (cf. Lemma 2.5), we deduce from Lemma 8.1 that
[TABLE]
for all . In particular,
[TABLE]
for all and so the stalks of have the same dimension at all points of . This implies that is a local system, because we have seen above that any local section of vanishes if it vanishes generically.
We claim that in order to prove that is smooth, we may replace by any finite étale cover, induced by a finite étale cover . To prove this, let by a finite étale cover, induced by a finite étale cover , and assume that is smooth. Then the natural map is smooth as well and since is étale, is smooth, as claimed.
Since is generically a polarized variation of Hodge structure of type , its monodromy representation is finite. This implies that up to replacing by a finite étale covering, we may assume that is trivial. Applying Proposition 3.1 to this base change repeatedly, we may assume that we again arrive at the situation of a Mori fibre space over and so . Since is trivial, implies that it has rank one and so the general fibre of must be (because it is a smooth del Pezzo surface with ). It is not hard to prove directly that this implies that cannot have any non-normal fibres; alternatively, this claim also follows from [Fuj90, Theorem 3.1]. Since we have seen above that all singular fibres of are non-normal, this proves that is smooth. This finishes the proof of Theorem 8.4. ∎
9. Summary of the argument
Proof of Theorems 1.1 and 1.3.
Since 1.3 1.1, it suffices to prove Theorem 1.3. Since (D) (A) (B) (B’) is clear and (B’) (C) is proven in the appendix (see Theorem A.1), it suffices to prove (C) (D) for smooth projective threefolds. For this, let be a smooth projective threefold and assume that (C) holds for the holomorphic one-form on .
By Corollary 3.2, the minimal model program for yields a smooth projective threefold and a proper birational morphism which is a sequence of blow-ups along elliptic curves which are not contracted by the natural map to . This proves item (D1).
To prove items (D2), (D3) and (D4), we may replace by and assume that is either minimal or a Mori fibre space. The assertions then follow from Theorem 4.1, Corollary 4.2, and Theorems 8.3 and 8.4.
This concludes the proof of Theorems 1.1 and 1.3. ∎
Proof of Theorem 1.4.
By Proposition 3.1, it suffices to prove Theorem 1.4 for a smooth projective threefold which is either minimal or a Mori fibre space. In the latter case, Theorem 1.4 follows from Theorems 8.3 and 8.4; in the former case, Theorem 1.4 follows from item (3) in Corollary 4.2. This concludes the proof of Theorem 1.4. ∎
Proof of Corollary 1.6.
Let be a smooth morphism from a smooth projective threefold with to an abelian variety . If is zero-dimensional, the claim is trivial and so we may assume that . Hence, carries a holomorphic one-form without zeros and so the equivalent conditions (A) (B) (B’) (C) (D) hold by Theorem 1.3.
If is not minimal, then by Proposition 3.1, is the blow-up of a smooth projective threefold along an elliptic curve . Since factors through , this implies that must be an elliptic curve and must be étale over ( is non-flat if , and it has singular fibres if or if and is not étale over ). Hence, up to replacing by , we may inductively assume that is minimal. We claim that under this assumption, is an analytic fibre bundle. To prove this, we may replace by any finite étale cover. Hence, by (D3), we may assume that is a product of an abelian variety of positive dimension and a smooth projective variety . Then for any , maps to a translate of a fixed abelian subvariety . Let . If , then we are done. Otherwise, for any , consider the natural composition
[TABLE]
which is smooth, because is smooth. Since , [Sch19, Corollary 3.2] implies that up to a further étale covering, splits of a positive-dimensional abelian variety as a direct factor. Arguing as before therefore concludes the proof of the corollary by induction on the dimension. ∎
Appendix A
The purpose of this appendix is to show that the arguments in [Sch19] easily generalizes to prove the following, which is inspired by questions of Kieran Kedlaya and Burt Totaro.
Theorem A.1**.**
Let be a compact connected Kähler manifold. Then (B’) (C).
Proof.
Let be a compact connected Kähler manifold. In order to show that (C) holds, it thus suffices by the argument in [Sch19, Section 2.3] to show that there is a local system with stalk on whose first Chern class is trivial and such that for all . Since the space of such local systems is given by , it depends only on the homotopy type of . Moreover, the corresponding cohomology groups do not change when we replace by a homotopy equivalent CW complex, see e.g. [Wh78, VI.2.6*]. Hence, to prove (B’) (C), it suffices to show that any CW complex which admits a finite -homology fibration carries a local system with stalk , trivial first Chern class and no cohomology. Following a suggestion of Botong Wang (see [Sch19, Remark 2.3]), we claim that the local system on has this property, where is a generic local system on with monodromy given by a general element . To show this claim, consider the Leray spectral sequence with -term
[TABLE]
By our assumptions, the sheaf is a local system on with stalk a finite dimensional -vector space . Since is general, is different from all eigenvalues of the natural monodromy operator on . Hence, is a local system on which corresponds to a finite-dimensional -vector space together with a monodromy operator whose eigenvalues are all different from one. In particular, . Since the Euler characteristic of any local system of finite rank on is zero, we also get . Hence, for all and so for all . This concludes the proof. ∎
Acknowledgement
The second author thanks Dieter Kotschick for sending him the preprint [Kot13] in spring 2013, where he poses the problem about the equivalence of (A) and (B). We are grateful for useful comments and conversations to Antonella Grassi, Thomas Peternell, Mihnea Popa, Christian Schnell and Burt Totaro. Both authors are supported by the DFG grant “Topologische Eigenschaften von Algebraischen Varietäten” (project no. 416054549).
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