Chern numbers of uniruled threefolds
Stefan Schreieder, Luca Tasin

TL;DR
This paper proves that the Chern numbers of certain three-dimensional algebraic varieties are bounded by their underlying topology, extending previous results to cases with negative Kodaira dimension.
Contribution
It establishes bounds on Chern numbers for smooth Mori fibre spaces and generalizes existing theorems to broader classes of threefolds.
Findings
Chern numbers of smooth Mori fibre spaces are topologically bounded.
Extended boundedness results to threefolds with negative Kodaira dimension.
Provides new insights into the relationship between topology and algebraic geometry in threefolds.
Abstract
In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology
Chern numbers of uniruled threefolds
Stefan Schreieder
Mathematisches Institut, Ludwig–Maximilians–Universität München, Theresienstr. 39, 80333 München, Germany
and
Luca Tasin
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
Abstract.
In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension.
1. Introduction
One of the most basic numerical invariants of a compact complex manifold are its Chern numbers. While these numbers depend only on the topological type of the complex structure of the tangent bundle, they are in general not invariants of the underlying topological manifold, but really depend on the complex structure. In fact, answering a question of Hirzebruch from 1954, all linear combinations of Chern and Hodge numbers which are topological invariants of smooth complex projective varieties have recently been determined in [9, 10, 11].
Generalising Hirzebruch’s question, Kotschick asked [8] (see also [12]) whether the topology of the underlying smooth manifold determines the Chern numbers of smooth complex projective varieties at least up to finite ambiguity. In [19], we have shown that in dimension at least four, this question has in general a negative answer. That is, there are smooth real manifolds that carry infinitely many complex algebraic structures such that the corresponding Chern numbers are unbounded, except for , and which are known to be bounded (see [14] for the non-trivial one ). This result left however open the case of threefolds, where it remains unknown whether is determined up to finite ambiguity by the underlying smooth manifold.
In [2], Cascini and the second named author started to investigate the boundedness question for Chern numbers via methods from the minimal model program, see also [16, 20] for further developments. In dimension three, the approach in [2] is motivated by the Miyaoka–Yau inequality, which implies that for a minimal smooth complex projective threefold of non-negative Kodaira dimension, can be bounded in terms of the Betti numbers of , see e.g. [20, Proposition 9]. This observation makes it natural to approach the boundedness of by trying to bound the effect on of the steps in the minimal model program for . This leads to a positive answer for the boundedness question for many smooth projective threefolds of non-negative Kodaira dimension whose minimal model program is a composition of blow-downs to points and smooth curves in smooth loci, see [2, Corollary 1.5].
In this paper we focus on the case of threefolds of negative Kodaira dimension. The main difficulty that we face in this case is that the aforementioned Miyaoka–Yau inequality, which was essential for the case of non-negative Kodaira dimension, does not hold any longer. It is also known by examples of LeBrun [13], that the boundedness does not hold in the non-Kähler case. Nonetheless, for any smooth Kähler threefold we can run a minimal model program thanks to [6, 7]. If is uniruled then we arrive at a Mori fibre space , i.e. a Kähler threefold with at most terminal singularities together with a morphism of relative Picard rank one with connected fibres to a complex Kähler variety of smaller dimension whose general fibre is Fano.
The first result of this paper is the following.
Theorem 1**.**
Let be a sequence of Mori fibre spaces, where are smooth Kähler threefolds. If each is homeomorphic to , then the sequence of Chern numbers is bounded.
The above result should be compared to the fact that all known examples of sequences of homeomorphic varieties with unbounded Chern numbers are Mori fibre spaces, and in fact projective bundles (see [19]). We therefore believe that together with the aforementioned results from [2], the above theorem puts forward strong evidence for the conjecture that the Chern numbers of smooth projective threefolds are determined up to finite ambiguity by the underlying smooth manifold.
If is a Mori fibre space and is a smooth Kähler threefold, then there are three main cases to consider, depending on the dimension of . If is a point, then is a Fano variety and we conclude because Fano varieties of fixed dimension form a bounded family. If is a curve, then it is smooth projective and is also projective. Since the Pontryagin classes are up to torsion homeomorphism invariants by Novikov’s theorem [17], [20, Proposition 26] proves the above theorem in case all but finitely many of the are Mori fibre spaces over points or curves. Using Novikov’s theorem [17] once again, Theorem 1 thus follows from the following more precise result about Mori fibre spaces over surfaces, where we denote by the quotient of by the subgroup of all torsion classes.
Theorem 2**.**
Let be a sequence of smooth Kähler threefolds admitting a conic bundle structure f_{i}:X_{i}\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_{i} of relative Picard number 1 over a smooth Kähler surface . If there is an isomorphism of graded rings which respects the first Pontryagin classes, then the sequence of Chern numbers is bounded.
In view of Theorem 1, it is therefore natural to wonder if we can also bound the Chern numbers of certain threefolds of negative Kodaira dimension which are not necessarily Mori fibre spaces themselves. Our next result achieves this by generalising [2, Corollary 1.5] to the case of negative Kodaira dimension. To state it, recall that for any smooth complex projective threefold , there is a cubic form on , given by cup product. For technical reasons, we will assume that the discriminant of the cubic form is non-zero.
Theorem 3**.**
Let be a smooth complex projective threefold which is uniruled and let be its associated cubic. Assume that and that there exists a birational morphism onto a Mori fibre space , which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci.
Then there exists a constant depending only on the topology of the -manifold underlying such that
[TABLE]
A major step in proving Theorem 3 is Proposition 8 (cf. [2, Theorem 1.3(2)]), where we show that in the assumptions of Theorem 3, most of the topological invariants of are determined (up to finite ambiguity) a priori by the smooth manifold underlying . It would be interesting to understand to what extend this is true in general (see [4] for the case of Betti numbers):
Question 4**.**
Let be a smooth complex projective threefold with cubic form and first Pontryagin class . Let be the set of pairs , taken up to isomorphism, such that there exists an MMP . Is the set determined by the pair of invariants of up to finite ambiguity?
1.1. Conventions
All manifolds are closed and connected. A Kähler manifold is a complex manifold which admits a Kähler metric. For any (Kähler) manifold , we denote by the quotient , where denotes the torsion subgroup of .
2. Mori fibre spaces over surfaces
The starting point of our investigation is the following lemma.
Lemma 5**.**
([21, Sec. 7.1]) Let be a Mori fibre space such that is a smooth projective threefold and is a surface. Then
- (i)
* is a standard (i.e. relative Picard number 1) conic bundle and is smooth;* 2. (ii)
the discriminant of is either empty or a reduced curve with at worst ordinary double points; 3. (iii)
, and ; 4. (iv)
* and .*
We will also use the following lemma.
Lemma 6**.**
Let be a Mori fibre space such that is a smooth projective threefold and is a surface. Then,
[TABLE]
where is the first Pontryagin class of and denotes the discriminant curve of .
Proof.
Since is smooth projective by Lemma 5, the Néron–Severi group of is generated by very ample curves. Hence, it suffices to compute the intersection product with a general smooth projective curve . The preimage is then the blow-up of a minimal ruled surface over in many points. The normal bundle of in is given by . Since , we get
[TABLE]
Hence, using , we get
[TABLE]
By Lemma 5, . Using , we get
[TABLE]
which proves the lemma. ∎
Proof of Theorem 2.
By [15, Theorem 1.1], any standard conic bundle , where is a smooth Kähler threefold, has an algebraic deformation. To bound it thus suffices to assume that and are projective for any .
By assumptions, there is an isomorphism which respects the trilinear forms given by cup products. We use this isomorphism to identify degree two cohomology classes of with those of (up to torsion). Using Poincaré duality, we further identify classes of with linear forms on .
The codimension one linear subspace of is contained in the cubic hypersurface . Passing to a suitable subsequence we can therefore assume that
[TABLE]
does not depend on . Let be the class of a fibre of . The linear form determined by this class on has kernel , and so is independent of . Since is an integral class with , we may after possibly passing to another subsequence assume that does not depend on .
Since the natural cup product pairing on can be recovered from the pairing
[TABLE]
we get that the pairing on is determined by the cubic form on and so it does not depend on .
Since does not depend on , the same holds for the homomorphism
[TABLE]
By the projection formula, we have . Lemma 6 thus yields
[TABLE]
where is the discriminant curve of . This shows that the linear form determined by on does not depend on . Since the natural pairing is perfect by Poincaré duality, we get that the class does not depend on . Using again the fact that we know the pairing on , we finally get that the self-intersection does not depend on .
For any class , which does not lie in , we have
[TABLE]
In particular, for some and . Replacing by a suitable multiple of , we may thus assume that
[TABLE]
For any , we then get
[TABLE]
for some . Since
[TABLE]
it suffices to prove the boundedness of .
Since , the pushforward of via yields . Lemma 5 therefore implies that
[TABLE]
Hence,
[TABLE]
Since does not depend on and is bounded in terms of the Betti numbers of , the statement follows from the fact that is bounded by Lemma 5. ∎
3. Uniruled threefolds
Before we turn to the proof of Theorem 3, we state few preliminary facts about terminal -factorial threefolds.
3.1. Invariant triples
Let be a terminal -factorial threefold.
There exists a well-defined class obtained in the following way (see page 411 in [22]). For any set
[TABLE]
where is a resolution of .
We then define the Pontryagin class in terms of and in the same way as in the smooth case, where is the class of in :
[TABLE]
We also associate to its cubic form , which is induced by the cup product on . In this way we can associate to the triple . When is smooth, this triple encodes many geometrical properties of the 6-manifold underlying (see for instance [18] and [1]).
Definition 7**.**
We call the invariant triple of . Two triples and , where (resp. ) is a free abelian group, (resp. ) is a cubic form and is a linear form on (resp. is a linear form on ) are isomorphic if there exists a linear isomorphism which identifies with and its -extension identifies with .
3.2. Terminal singularities
We now recall few known facts about terminal singularities in dimension three.
Let be the germ of a three-dimensional terminal singularity. The index of is the smallest positive integer such that is Cartier. It follows from the classification of terminal singularities, that there exists a deformation of into a space with terminal singularities which are isolated cyclic quotient singularities of index (for details see [22, Remark 6.4]). The set is called the basket of singularities of at . As in [5, Section 2.1], we define
[TABLE]
Thus, if is a projective variety of dimension with terminal singularities and denotes the finite set of singular points of , we may define
[TABLE]
3.3. Proof of Theorem 3
The following result is interesting by itself and leads naturally to the problem of understanding what kind of topological invariants are determined up to finite ambiguity during a running of an MMP, see Question 4.
Proposition 8**.**
[Cf. [2, Theorem 1.3(2)]] Let be a finitely generated free abelian group of rank , be a cubic form such that and a linear form on . Consider the set of invariant triples up to isomorphism, such that there exist
- (1)
a terminal -factorial threefold with associated triple ; 2. (2)
a terminal -factorial threefold with associated triple ; 3. (3)
a birational morphism which is a divisorial contraction to a point or to a smooth curve contained in the smooth locus of .
Then the set is finite.
Proof.
Note that the proof of this case works also for . Consider the set of primitive elements such that is proportional to the exceptional divisor of some divisorial contraction to a point as in the statement. The elements of are points of rank 1 for the Hessian of the cubic form and so they are finite by [2, Proposition 3.3]. It follows from [2, Proposition 4.7] that for any sub-module there is such that and such that the index of in is at most , where . This implies that for all possible contractions to points as in the statement, the inclusion is determined up to finite ambiguity. This determines also up to finite ambiguity just restricting to .
To prove the finiteness of consider a divisorial contraction to a point and write
[TABLE]
where is the discrepancy of the exceptional divisor . Since we have that
[TABLE]
and so is given by the restriction of to . This means that also is determined up to finite ambiguity and we are done.
We now look at divisorial contractions to curves. Consider the set of pairs where is a primitive element in and is a submodule such that
[TABLE]
and the cubic assumes the form
[TABLE]
with respect to any basis with .
By [2, Thm. 3.1] there are only finitely many possible non-equivalent reduced forms for . In particular, up to finite ambiguity, we can assume that the coefficients of in the expression (1) are fixed. Since the isotropy group of a cubic with non-zero discriminant is finite ([18, Thm. 4]), we deduce that is finite.
If is a divisorial contraction which contracts a divisor to a smooth curve in the smooth locus of , then (see [18, Proposition 14] and [2, Proposition 4.8])
[TABLE]
and
[TABLE]
Recalling that for any we deduce that is determined by , and by the inclusion and we conclude using the finiteness of . ∎
Proposition 9**.**
Let be a sequence of terminal -factorial threefolds admitting a conic bundle structure f_{i}:X_{i}\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_{i} of relative Picard number 1 over a surface . Assume that
- (1)
the Euler characteristics are bounded and ; 2. (2)
the invariant triples of and are isomorphic for any ; 3. (3)
the sequence is bounded.
Then the sequence of Chern numbers is bounded.
Proof.
Let be an ample generator of and let be a primitive class proportional to . Then , and letting we can write
[TABLE]
From now on we will use the isomorphism to think about and as basis elements of . Note that since already and . Moreover, the space of elements in with zero cube is a union of three lines (through [math]) and so we may assume without loss of generality that for each , the pullback of the generator of to is a multiple of . In particular, is a multiple of the class of the general fibre of X_{i}\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_{i} for all .
We have
[TABLE]
for some . Since is bounded, there is a positive integer such that is Cartier for any . In particular, . Since where is a general fibre, we deduce that the sequence of is bounded.
We are going to bound the sequence of . By the singular version of Riemann–Roch [22, Corollary 10.3] we get
[TABLE]
where
[TABLE]
and the sum runs over all the points of all the baskets of . Note that is a bounded sequence since is bounded. This implies that
[TABLE]
and so the are also bounded, since and are bounded. ∎
Proof of Theorem 3.
Let be the birational contraction as in Theorem 3. By the proof of [2, Corollary 1.5], we know that is bounded by a constant depending only on the Betti numbers of and on the cubic form . To conclude we need to bound in terms of the topology of .
Since is a Mori fibre space and , we deduce that either is a Fano variety or has a conic bundle structure over a surface with second Betti number 1 (otherwise there would be an element in with square zero, which would imply that has a singular point and so ). Since terminal Fano threefolds are bounded, we are left with the conic bundle case.
Proposition 8 assures us that the invariant triple of is determined up to finite ambiguity by the invariant triple of . Moreover, the Euler characteristic is bounded in terms of the Betti numbers of and by [3, Prop. 3.3] we also have a bound for depending only on . The result follows then from Proposition 9. ∎
Acknowledgments
We thank P. Cascini for many useful conversations on the topic of this manuscript. The first author is supported by DFG grant “Topologische Eigenschaften von Algebraischen Varietäten” (project nr. 416054549). The second author was supported by the DFG grant “Birational Methods in Topology and Hyperkähler Geometry" and is a member of the GNSAGA group of INdAM.
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