The extended locus of Hodge classes
We introduce an “extended locus of Hodge classes” that also takes into account integral classes that become Hodge classes “in the limit”. More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is an analytic subspace that contains the usual locus of Hodge classes, but is finite and proper over the base manifold. The construction uses Saito’s theory of mixed Hodge modules and a small generalization of the main technical result of Cattani, Deligne, and Kaplan. We study the properties of the resulting analytic space in the case of the family of hyperplane sections of an odd-dimensional smooth projective variety.
Key words and phrases:Hodge class, locus of Hodge classes, variation of Hodge structure, mixed Hodge module
2000 Mathematics Subject Classification:14D07; 32G20; 14K30
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a.1. Hodge loci on Calabi-Yau threefolds
The purpose of this paper is to describe the construction of the extended locus of Hodge classes for polarized variations of -Hodge structure of weight zero. Before defining things more precisely, we shall consider a typical example that shows why this is an interesting problem, and what some of the issues are.
Let be a smooth projective Calabi-Yau threefold; this means that , and that . We fix an embedding of into projective space, with the corresponding very ample line bundle, and consider the family of hyperplane sections of . These are parametrized by the linear system
and we let denote the open subset that corresponds to smooth hyperplane sections. Given a cohomology class on a smooth hyperplane section , we can use parallel transport along paths in to move to other hyperplane sections; this operation is of course purely topological and does not preserve the Hodge decomposition. The Hodge locus of is the set
Most of these loci are non-empty: in fact, Voisin [Voisin] has proved that the union of the Hodge loci of all classes is a dense subset of . Since Hodge classes on surfaces are algebraic, the Hodge locus is an algebraic subvariety of ; in basic terms, what we are looking at are curves (or algebraic one-cycles) on that lie on hyperplane sections.
We observe that the expected dimension of the Hodge locus is zero. Indeed, a class is Hodge exactly when it pairs to zero against every holomorphic two-form on ; because is a Calabi-Yau threefold, we have
The number of conditions is the same as the dimension of the parameter space, and the Hodge locus of should therefore have a “virtual” number of points; those numbers are of interest in Donaldson-Thomas theory [KMPS]. But there are two issues that need to be dealt with:
If the Hodge locus actually has finitely many points, one can of course just count them. But there may be components of positive dimension, and before one can use excess intersection theory (or some other method) to assign them a number, one has to compactify such components.
An obvious idea is to take the closure of the Hodge locus inside the projective space ; but this is not the right thing to do because there are interesting limit phenomena that one cannot see in this way.
Here is a typical example. Consider a family of hyperplane sections , parametrized by , with smooth for , and having a single ordinary double point. In this case, contains a vanishing cycle , namely the class of an embedded two-sphere with self-intersection number . The vanishing cycle is not a Hodge class on , but becomes one “in the limit”. On the one hand, one has the limit mixed Hodge structure, which is pure of weight two in this case; is a Hodge class in this Hodge structure. On the other hand, one can blow up at the node; the exceptional divisor satisfies , and in a sense, is the limit of the as .
a.2. Statement of the problem
Abstracting from the example above, we now let be an arbitrary polarized variation of -Hodge structure of weight zero, defined on a Zariski-open subset of a smooth projective variety . The assumption about the weight is of course just for convenience: if has even weight , we can always replace it by the Tate twist , which has weight zero. Let denote the Hodge bundles, and let denote the underlying local system. Although it is not strictly necessary for what follows, we shall assume that the polarization form is defined over .
Let us first recall the definition of the usual locus of Hodge classes. The local system determines a (not necessarily connected) covering space
whose sheaf of holomorphic sections is isomorphic to . The points of are pairs , with a class in the fiber over the point .
The locus of Hodge classes of is the set
We consider the locus of Hodge classes (a subset of ) instead of the individual Hodge loci (subsets of ) because it is useful to keep track of the Hodge classes themselves: over any given point, there may be more than one such class, and all of them may be permuted by the monodromy action. On the face of it, is just an analytic subset of the complex manifold ; the following remarkable theorem by Cattani, Deligne, and Kaplan [CDK] shows that it is actually a countable union of algebraic varieties.
Theorem 1 (Cattani, Deligne, Kaplan).
Every connected component of is an algebraic variety, finite and proper over .
This theorem is one of the best results in Hodge theory. When comes from a family of smooth projective varieties, the Hodge conjecture predicts that should be a countable union of algebraic varieties; the point is that Cattani, Deligne, and Kaplan were able to prove this without assuming the conjecture.
Theorem 1 shows that every connected component of the locus of Hodge classes can be extended (more or less uniquely) to a projective algebraic variety that is finite and proper over . As in the example of Calabi-Yau threefolds, this suggests that we should look for a natural compactification of that also takes into account those integral classes that only become Hodge classes “in the limit”. The purpose of this paper is to solve that problem with the help of Saito’s theory of mixed Hodge modules. The idea is to construct a complex analytic space that extends , and to use it for defining the extended locus of Hodge classes. As far as I know, it was Clemens who first suggested working directly with limits of integral classes; in any case, I learned this idea from him.
a.3. The case of a Hodge structure
To motivate the construction, let us first look at the case of a single Hodge structure . We assume that is polarized and integral of weight zero; we denote the underlying -module by ; the polarization by ; and the Hodge filtration by . Let be the set of Hodge classes in . According to the bilinear relations, a class is Hodge exactly when it is perpendicular (under ) to the space ; this says that is precisely the kernel of the linear mapping
At first, it may seem that is not good for much else, because its image is not a nice subset of . In fact, the dimension of the vector space can be much smaller than the rank of , and so will typically have dense image. But it turns out that the restriction of to the subset
is well-behaved. The idea of bounding the self-intersection number of the integral classes already occurs in the paper by Cattani, Deligne, and Kaplan. To back up this claim, we have the following lemma; note that the estimate in the proof will play an important role in our analysis later on.
The mapping is finite and proper, and its image is a discrete subset of the vector space .
We have to show that the preimage of any bounded subset of is finite. It will be convenient to measure things in the Hodge norm: if
is the Hodge decomposition of a vector , then its Hodge norm is
Now suppose that satisfies and ; it will be enough to prove that is bounded by a quantity depending only on and . The assumption on means that for every . If we apply this inequality to the vector
we find that , and hence that
Because is invariant under conjugation, it follows that . This leads to the conclusion that , because
In particular, there are only finitely many possibilities for , which means that is a finite mapping, and that the image of is a discrete subset of . ∎
a.4. The general case
Now let us return to the general case. As in [CDK], it is not actually necessary to assume that is projective; we shall therefore consider a polarized variation of -Hodge structure of weight zero, defined on a Zariski-open subset of an arbitrary complex manifold . By performing the construction in §A.3 at every point of , we obtain a holomorphic mapping
here is the holomorphic vector bundle on whose sheaf of holomorphic sections is . The locus of Hodge classes is then exactly the preimage of the zero section in . For any rational number , we consider the submanifold
It is a union of connected components of the covering space , because the quantity is obviously constant on each connected component. More or less directly by Lemma 2, the holomorphic mapping
is finite and proper, with complex-analytic image; moreover, one can show that the mapping from to the normalization of the image is a finite covering space. For the details, please consult §C.1 below.
To construct an extension of to an analytic space over , we use the theory of Hodge modules [Saito-MHM]. Let be the polarized Hodge module of weight with strict support , canonically associated with . We denote the underlying filtered left -module by the symbol . The point is that
in particular, the coherent sheaf is an extension of the Hodge bundle to a coherent sheaf of -modules. Now consider the holomorphic mapping
where the analytic space on the right-hand side is defined as before as the spectrum of the symmetric algebra of the coherent sheaf . We have already seen that is an analytic subset of ; since we are interested in limits of integral classes, we shall extend it to the larger space by taking the closure. The main result of the paper is that the closure remains analytic.
The closure of is an analytic subset of .
The proof consists of two steps: (1) We reduce the problem to the special case where is a divisor with normal crossings and has unipotent local monodromy; this reduction is similar to [Schnell-N]. (2) In that case, we prove the theorem by a careful local analysis, using the theory of degenerating variations of Hodge structure. In fact, we deduce the theorem from a strengthening of the main technical result of Cattani, Deligne, and Kaplan, which we prove by adapting the method introduced in [CDK]. Rather than just indicating the necessary changes in their argument, I have chosen to write out a complete proof; I hope that this will make Chapter B useful also to those readers who are only interested in the locus of Hodge classes and the theorem of Cattani, Deligne, and Kaplan.
Once Theorem 3 is proved, it makes sense to consider the normalization of the closure of . The mapping from to its image in the normalization is a finite covering space; it can therefore be extended in a canonical way to a finite branched covering by appealing to the Fortsetzungssatz of Grauert and Remmert.
There is a normal analytic space containing the complex manifold as a dense open subset, and a finite holomorphic mapping
whose restriction to agrees with . Moreover, and are unique up to isomorphism.
Since each is a union of connected components of the covering space , we can take the union over all the ; this operation is well-defined because of the uniqueness statement in the theorem. In this way, we get a normal analytic space , and a holomorphic mapping
with discrete fibers that extends . Now the preimage of the zero section in gives us the desired compactification for the locus of Hodge classes.
The extended locus of Hodge classes is the closed analytic subscheme ; by construction, it contains the locus of Hodge classes.
a.5. The family of hyperplane sections
The construction above can be applied to the family of hyperplane sections of a smooth projective variety of odd dimension. In this case, one has a good description of the filtered -module in terms of residues [Schnell-R], and it is possible to say more about the space . The fact that is the quotient of an ample vector bundle leads to the following result; it was predicted by Clemens several years ago.
The analytic space is holomorphically convex. Every compact analytic subset of dimension lies inside the extended locus of Hodge classes.
In writing this paper, I have benefited a lot from a new survey article by Cattani and Kaplan [CK-survey] that explains the results of [CDK] in the case . I thank Eduardo Cattani for letting me read a draft version, and for answering some questions. Several years ago, Davesh Maulik asked me about the case of hyperplane sections of a Calabi-Yau threefold; I thank him for many useful conversations, and for his general interest in the problem. Most of all, I thank my former thesis adviser, Herb Clemens, for suggesting that one should study limits of integral classes with the help of residues; as in many other cases, his idea contained the seed for the solution of the general problem.
B. Local analysis in the normal crossing case
b.1. Main result
The purpose of this chapter is to prove the following special case of Theorem 3. We shall see later how the general case can be reduced to this one.
Let be a complex manifold, and let be the complement of a divisor with normal crossing singularities. Let be a polarized variation of -Hodge structure of weight zero on whose local monodromy at each point of is unipotent. Then the closure of the image of the holomorphic mapping
is a complex-analytic subspace of .
Here is a brief outline of the proof. The assertion is local on , and unaffected by enlarging the divisor . We may therefore assume that , with coordinates , and that is the divisor defined by . Denote by the generic fiber of the local system , by the pairing on giving the polarization, and by the logarithms of the unipotent monodromy transformations. Define
After pulling back to the universal covering space , with coordinates (related by to the coordinates on ), we get a holomorphic mapping
The main point is to show the following: Suppose that is a sequence such that are going to zero along a bounded sector, and remains bounded. Then after passing to a subsequence, is constant and partially monodromy invariant. Roughly speaking, this means that if we let denote the smallest analytic subvariety containing all the points , then is monodromy invariant on . As explained in Theorem 14 below, such a result quickly leads to a proof of Theorem 7.
For technical reasons, we prove a slightly more general result. Let denote the period mapping associated with the variation of Hodge structure on . Since if and only if , it is reasonable to expect that a bound on should control the distance between and . In fact, we show that if the sequence remains bounded in , then
Here is a bounded sequence with property that, for some , every is in . This uses the description of the -module in terms of Deligne’s canonical extension , and the fact that , compared to , contains additional sections with poles along the divisors . In Theorem 22 below, we prove that even under this weaker assumption, a subsequence of is constant and partially monodromy invariant.
In the special case when is in , this result is due to Cattani, Deligne, and Kaplan [CDK, Theorem 2.16]. Their proof is an application of the theory of degenerating variations of Hodge structure, especially the multi-variable -orbit theorem [CKS]. We prove Theorem 22 by adapting their method; there are several difficulties, caused by the fact that the sequence is not necessarily going to zero, but these difficulties can be overcome. As in the original, we argue by induction on the dimension of ; the description of period mappings in [CKS] lends itself very well to such an approach.
A subtle point is that the assumption is needed in many places: it ensures that certain terms that would only be going to zero when are actually equal to zero after passing to a subsequence. Rather than giving an abstract description of the proof, I have decided to include (in §B.6) a careful discussion of the special case . All the interesting features of the general case are present here, but without the added complications of having several nilpotent operators and several variables . Hopefully, this will help the reader understand the proof of the general case.
b.2. Local description of the problem
Since Theorem 7 is evidently a local statement, we shall begin by reviewing the local description of polarized variations of Hodge structure [Schmid, Kashiwara, CKS]. Fortunately, Cattani and Kaplan have written a beautiful survey article, where they describe all the major results [CK]. Rather than citing the original sources, I will only quote from this article.
Let , with coordinates , be the product of copies of the unit disk; then is the complement of the divisor defined by . Let be a polarized variation of -Hodge structure of weight zero on ; we assume that the underlying local system of free -modules has unipotent monodromy around each of the divisors . Let , with coordinates , be the product of copies of the upper half-plane; the holomorphic mapping
makes it into the universal covering space of . If we pull back the local system , it becomes trivial; let denote the free -module of its global sections, and the symmetric bilinear form coming from the polarization on . By assumption, the monodromy transformation around is of the form , where is a nilpotent endomorphism of that satisfies . It is clear that commute.
We now review the description of that results from the work of Cattani, Kaplan, and Schmid. Let denote the parameter space for filtrations that satisfy whenever ; let denote the subset of those that define a polarized Hodge structure on with polarization . Recall that is a closed subvariety of a flag variety, and that the so-called period domain is an open subset of .
The variation of Hodge structure can be lifted to a period mapping
which is holomorphic and horizontal. It is known that every element of the cone
defines the same monodromy weight filtration [CK, Theorem 2.3]; we denote the common filtration by . In the limit, determines another filtration for which the pair is a mixed Hodge structure on , polarized by and every element of . According to the nilpotent orbit theorem [CK, Theorem 2.1], the period mapping is approximated (with good bounds on the degree of approximation) by the associated nilpotent orbit
One can use the mixed Hodge structure to express in terms of the nilpotent orbit and additional holomorphic data on . Denote by
the Lie algebra of infinitesimal isometries of . The mixed Hodge structure determines a decomposition of with the following properties:
A formula for the subspaces can be found in [CK, (1.12)]. The decomposition leads to a corresponding decomposition of the Lie algebra
with consisting of those that satisfy . In this notation, we have ; moreover, the restriction of to the subspace is nondegenerate if and , and zero otherwise.
The more precise version of the nilpotent orbit theorem [CK, Theorem 2.8] is that the period mapping of can be put into the normal form
for a unique holomorphic mapping
with . When we write , it is of course understood that for every . The horizontality of the period mapping has the following very useful consequence [CK, Proposition 2.6].
Let be the normal form of a period mapping on . Then for every , the commutator
vanishes along the divisor .
The presentation of the period mapping in (8) is also convenient for describing the polarizable Hodge module that we obtain by taking the intermediate extension of to . Here is a brief explanation of how this works.
Let be the flat vector bundle on underlying the variation of Hodge structure. The monodromy being unipotent, this bundle admits a canonical extension to a vector bundle on , on which the connection has a logarithmic pole along each of the divisors with nilpotent residue [Deligne, Proposition 5.2]. Explicitly, for each , the holomorphic mapping
descends to a holomorphic section of on , and is the locally free subsheaf of generated by all such sections [CK, (2.2)]. The Hodge bundles extend uniquely to holomorphic subbundles of the canonical extension; concretely, is generated by those sections in (10) with . Now let denote the filtered -module underlying . Then is simply the -submodule of generated by . Moreover, the Hodge filtration on is given by
It satisfies , and each is a coherent sheaf on whose restriction to agrees with . This is a translation of Saito’s results in [Saito-MHM, §3.10]; note that Saito uses right -modules. For the purposes of our construction, the important point is that has more sections than ; the following lemma exhibits the ones that we will use.
For any vector , and any index , the formula
defines a holomorphic section of the coherent sheaf on .
It is clear from the description above that is a holomorphic section of . By [CK, (2.7)], the horizontality of the period mapping is equivalent to
Using this identity and the fact that , we compute that
This section belongs to by the definition of the filtration; we now obtain the result by noting that is a holomorphic mapping from into . ∎
We close this section by describing the mapping in coordinates. With the conventions in [CK, (1.8)], the étalé space of the local system can be obtained as the quotient of by the following -action:
As in the general problem, we define, for any integer , a set
Then is the quotient of by the action in (12). Now is a subsheaf of , and so we have a commutative diagram
The concrete description of shows that . We therefore obtain a holomorphic mapping
which, in coordinates, is given by the formula
As usual, the relation is implicit in the notation.
b.3. Reformulation of the problem
We continue to use the notation introduced in the previous section. Our goal is to deduce Theorem 7 from the following more precise local statement.
Suppose we are given a sequence of points
with bounded and going to infinity for every . If remains bounded inside , then there is a subsequence with the following properties:
The sequence is constant and equal to .
One has for certain positive integers ; in particular, .
There is a vector such that
Each is a rational Hodge class in the mixed Hodge structure
Let us prove that Theorem 14 implies Theorem 7. It suffices to show that the closure of the image of is analytic in a neighborhood of any given point in . After choosing local coordinates, we may therefore assume without loss of generality that and , and consider the behavior of the closure over the origin. For every , we have a holomorphic mapping
We only need to prove that the closure of the image of is analytic; this is because, by assertion 1 in Theorem 14, any bounded subset of can intersect only finitely many of the sets . Furthermore, we may assume that there is a vector such that each is a rational Hodge class in the mixed Hodge structure , and that for certain positive integers ; otherwise, the closure of does not actually contain any points over the origin, according to Theorem 14.
Under these assumptions on , we can prove the stronger result that the image of has an analytic closure; here denotes the holomorphic mapping induced by ; see (13). This suffices to conclude the proof, because the image of is then contained in the closed analytic subset
As is an isomorphism over , it follows that the closure of is also analytic – in fact, it is a connected component of the above set.
Let be an element with for all . Then the image of the holomorphic mapping
has an analytic closure (where as usual).
Let be Deligne’s decomposition of the mixed Hodge structure . Since
we may replace by and by the expression in parentheses, and assume without essential loss of generality that . We then have for every . Under the isomorphism
induced by , the linear functional corresponds to
Here we have used the fact that and . Since is holomorphic on , it is therefore enough to prove that the image of
has an analytic closure. This is what we are going to do next. We denote by the direct sum of the with ; given a vector , we write for its component in the summand .
Let ; note that embeds into , and is therefore a free -module, say of rank . We can thus find a matrix whose last columns give a basis for the submodule . If we now introduce new coordinates by defining
we have . The vectors are linearly independent, while . The mapping in (16) therefore has the same image as