Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.


Introduction
1.1.The purpose of this paper is to prove a rather unexpected new finiteness result for polarized integral variations of Hodge structure, containing the theorem of Cattani, Deligne, and Kaplan for the locus of Hodge classes [CDK95] as a special case.Instead of integral Hodge classes, we consider integral classes that are "self-dual", meaning that they are preserved by the action of the Weil operator; the motivation for this comes from considerations in theoretical physics.Analyzing such classes using the methods in [CDK95] becomes rather complicated, so our main tool is going to be the definability of period mappings in the o-minimal structure R an,exp , recently proved by Bakker, Klingler, and Tsimerman [BKT20].This transforms the problem into a pleasant set of exercises about certain algebraic groups.

1.2.
We begin by describing a toy case of the problem, to set up the notation.Suppose that H is a polarized integral Hodge structure of even weight 2k.We denote by the Hodge decomposition, and by Q : H Z ⊗ Z H Z → Z the symmetric bilinear form giving the polarization.If we define the Weil operator by the formula Cv = i p−q v for v ∈ H p,q , then C ∈ End(H R ) and C 2 = id, and the expression puts a positive definite inner product on H R = H Z ⊗ Z R. We usually write the resulting Hodge norm simply as ∥v∥ 2 = ⟨v, v⟩.
We shall be interested in integral vectors v ∈ H Z with the property that Cv = v.Since C 2 = id, any vector v ∈ H R can of course be decomposed uniquely as with Cv + = v + and Cv − = −v − ; concretely, v + is the sum of all the "even" components in the Hodge decomposition of v, and v − the sum of all the "odd" ones.By analogy with the action of the Hodge * -operator on the cohomology of four-manifolds, we may call v + and v − the self-dual respectively anti-self-dual part of v.We say that an integral class is self-dual if v = v + ; note that Hodge classes (in H Z ∩ H k,k ) are obviously self-dual.With this notation, we have with equality exactly when Cv = v.Among integral vectors with a fixed value of Q(v, v), those with Cv = v therefore have the smallest possible Hodge norm ∥v∥ 2 .In this setting, we have the following completely trivial finiteness result: the set -dual integral vectors with a given self-intersection number q ≥ 1 is finite.Our main theorem is a generalization of this fact to arbitrary polarized integral variations of Hodge structure of even weight.

1.3.
We now turn to the main result.Let X be a nonsingular complex algebraic variety, not necessarily complete, and let H be a polarized integral variation of Hodge structure on X, of even weight 2k.Let p : E → X be the underlying complex vector bundle, whose sheaf of holomorphic sections is isomorphic to O X ⊗ Z H Z ; here H Z denotes the underlying local system of free Z-modules.At each point x ∈ X, the complex vector space E x = p −1 (x) is equipped with a polarized integral Hodge structure of weight 2k; the set of integral vectors coincides with the stalk H Z,x .We denote by C x ∈ End(E x ) the Weil operator, and by Q x the polarization; it is the stalk of the pairing Q : H Z ⊗ Z H Z → Z X that defines the polarization on H.We shall think of the points of E as pairs (x, v), where x ∈ X and v ∈ E x .
Recall that E is actually an algebraic vector bundle [Del70]; the algebraic structure is uniquely determined by H.We shall give both E and X the R an,exp -definable structure extending their algebraic structure; then the projection p : E → X becomes a morphism of definable spaces.Our main result is that the set of all self-dual integral classes with fixed self-intersection number is a definable subspace of E.
Theorem 1.1.Let H be a polarized integral variation of Hodge structure of even weight on a nonsingular complex algebraic variety X.For each q ≥ 1, the set is a definable, closed, real-analytic subspace of E, and the restriction of p : E → X to this set is proper with finite fibers.

1.4.
By analogy with the theorem of Cattani, Deligne, and Kaplan, it is a natural to ask whether the locus of self-dual classes is actually real (semi-)algebraic, meaning actually definable in the much smaller structure R alg .Simple examples in dimension one show that semi-algebraic is the best one can hope for in general.We do not know the answer to this question.

Several useful variants of the main result can be obtained by tensoring with certain auxiliary Hodge
structures.The first one is the analogue of Theorem 1.1 for integral classes that are anti-self-dual.
Corollary 1.2.Let H be a polarized integral variation of Hodge structure of even weight on a nonsingular complex algebraic variety X.For each q ≥ 1, the set is a definable, closed, real-analytic subspace of E, and the restriction of p : E → X to this set is proper with finite fibers.
The second one is a generalization to polarized integral variations of Hodge structure of arbitrary weight, where we now consider pairs of integral classes that are related by the Weil operator.
Corollary 1.3.Let H be a polarized integral variation of Hodge structure on a nonsingular complex algebraic variety X.For each q ≥ 1, the set is a definable, closed, real-analytic subspace of E × X E, and the restriction of p : E × X E → X to this set is again proper with finite fibers.
Note that when H is an integral Hodge structure of odd weight, the Weil operator C ∈ End(H R ) satisfies C 2 = − id, and so its eigenvalues are the two complex numbers ±i.The condition v = Cw is equivalent to is an eigenvector of the Weil operator that is integral with respect to the Gaussian integers Z[i].
1.6.Unlike in the case of Hodge classes, the locus of self-dual (or anti-self-dual) classes is in general not a complex analytic subset of the vector bundle E, hence in particular not algebraic.The reason is that the Weil operator C x ∈ End(E x ) depends real analytically -but not complex analytically -on the point x ∈ X, which means that C x v = ±v is not a holomorphic condition.We intend to discuss both the local structure of the locus of self-dual classes, and its more precise behavior near a divisor with normal crossing singularities, in a future paper.Here we only give two examples to show what these loci can look like in practice.

1.7.
Our first example concerns anti-self-dual classes on K3 surfaces; these show up naturally in Verbitsky's study of ergodic complex structures on hyperkähler manifolds [Ver15,Ver17].
Example 1.1.Let S be a (not necessarily algebraic) K3 surface and let Λ Z = H 2 (S, Z) together with the cup product pairing.The period domain D parametrizing the Hodge structures of K3 surfaces is a complex manifold of dimension 20; concretely, if H 2,0 (S) = Cσ , one has The set of points where a given integral class v ∈ Λ Z is anti-self-dual, of Hodge type (2, 0) + (0, 2), is easily seen to be This is a totally real submanifold of real dimension 20.
Using Ratner theory, Verbitsky shows that for any finite index subgroup Γ ⊂ O(Λ Z ), orbits Γ p of elements p ∈ Λ Z come in three flavors: closed orbits, dense orbits, and orbits whose closures are the Γ -orbit of an anti-self-dual locus.The three behaviors correspond to the three possibilities rk((H 2,0 ⊕H 0,2 )∩Λ Z ) = 2, 0, 1, respectively, for the Hodge structure associated to p.The same analysis holds more generally for the period domain associated to the degree two cohomology of any (possibly singular) hyperkähler variety, and is important for instance in the proof of the global Torelli theorem in the singular case [BL18].

1.8.
Further assume that F ♯ = e iN F is the Hodge filtration of an integral Hodge structure of even weight on H Z , polarized by the pairing Q.Then it is known [CK89, Proposition 3.9] that the nilpotent orbit

Background on definability
2.1.The theory of o-minimal structures provides a precise notion of tameness for subsets of euclidean space and functions on them.It is flexible enough to allow for complicated constructions but restrictive enough to imply strong finiteness properties.A general reference for this section is [vdD98].

2.2.
To formalize this notion, we first introduce a way to describe collections of subsets of R n which are closed under a variety of natural operations.
Definition 2.1.A structure S is a collection (S n ) n∈N where each S n is a set of subsets of R n satisfying the following conditions: (1) Each S n is closed under finite intersections, unions, and complements; (2) The collection (S n ) n∈N is closed under finite Cartesian products and coordinate projections; (3) For every polynomial P ∈ R[x 1 , . . ., x n ], the zero set We refer to an element U ∈ S n as an S-definable subset U ⊂ R n .For U ∈ S n , and V ∈ S m , we say a map f : U → V of S-definable sets is S-definable if the graph is as a subset of R m+n .When the structure S is clear from context, we will often just refer to "definable" sets and functions.

2.3.
For each n taking S n to be the Boolean algebra generated by real algebraic subsets (P = 0) of R n , the resulting S is not a structure.For example, the algebraic set (x 2 − y = 0) ⊂ R 2 projects to the semialgebraic set (y ≥ 0) ⊂ R. On the other hand, if we take S n to be the Boolean algebra generated by real semialgebraic subsets (P ≥ 0) of R n , then by the Tarski-Seidenberg theorem the resulting S = R alg is a structure.
Note.Tarski-Seidenberg is usually phrased as quantifier elimination for the real ordered field.Indeed, the above axioms for a structure say that definable sets are closed under first order formulas, as intersections, unions, and complements correspond to the logical operators "and", "or", and "not", while the projection axiom corresponds to universal and existential quantifiers.For this reason, structures have been studied extensively in model theory.

2.4.
Surprisingly, a good notion of tame structure can be achieved by simply restricting the definable subsets of the real line.
Definition 2.2.A structure S is said to be o-minimal if S 1 = (R alg ) 1 -that is, if the S-definable subsets of the real line are exactly finite unions of intervals.
Sets and functions which are definable in an o-minimal structure have very nice properties, including the following.Here we fix an o-minimal structure S and by "definable" we mean "S-definable." • For any definable function f : U → V with finite fibers, the fiber size |f −1 (v)| is a definable function.
In particular, it is uniformly bounded, and for any n the set admits a definable triangulation: it is definably homeomorphic to a finite simplicial complex.• Any definable subset U ⊂ R n has a well-defined dimension, namely, the dimension as a simplicial complex for any definable triangulation.Moreover, for any definable map f : U → V and any n the set is definable.• For any k and any definable function f : U → R, there is a definable triangulation of U such that f is C k on each simplex.As a consequence, any definable U ⊂ R n can be partitioned into finitely many C k -submanifolds of R n .

2.5.
We give some examples of o-miminal structures.As discussed above, the structure R alg is o-minimal; in fact, it is the smallest o-minimal structure.Given a collection Σ = (Σ n ) n∈N of subsets of R n for each n, we say the structure generated by Σ is the smallest structure in which each set in each Σ n is definable.It is the structure whose definable sets are given by first order formulas involving real polynomials, inequalities, and the sets in Σ.The structure R exp generated by the graph of the real exponential exp : R → R is o-minimal by a result of Wilkie [Wil96].However, the function sin : R → R is not definable in any o-minimal structure, as the definable set πZ = sin −1 (0) is both discrete and infinite.

2.6.
To get a much larger o-minimal structure, let R an be the structure generated by the graphs of all restrictions f | B(R) , where f : B(R ′ ) → R is a real analytic functions on a finite radius R ′ < ∞ open euclidean ball (centered at the origin) and B(R) ⊂ B(R ′ ) is a ball of strictly smaller radius R < R ′ .Via the embedding R n ⊂ RP n , this is equivalent to the structure of subsets of R n that are subanalytic in RP n .As observed by van-den-Dries [vdD98], Gabrielov's theorem of the complement implies that R an is o-minimal.Note that while the sine function is not R an -definable, its restriction to any finite interval is.
Finally, let R an,exp be the structure generated by R an and R exp .Then R an,exp is o-minimal by a result of van-den-Dries-Miller [vdDM96].Most of the applications to algebraic geometry currently use the structure R an,exp , and this will be the default structure we work with.

2.7.
For applications, we typically wish to discuss definability for manifolds that don't arise as subsets of R n , and for this we need an appropriate notion of definable atlas.Definition 2.3.Let M be a topological space and S a structure.
• A (S-)definable atlas {(U i , φ i )} consists of a finite open covering U i of M, and homeomorphisms The V i and the pairwise intersections V ij := φ i (U i ∩ U j ) are definable sets; (2) The transition functions topological space is a topological space M equipped with an equivalence class of definable atlases.A morphism of (S-)definable topological spaces is a continuous map f : M → M ′ which is definable for any choice of atlases in the equivalence classes on the source and target.• We likewise define (S-)definable manifolds (resp.(S-)definable complex manifolds) by in addition requiring that the charts map to open subsets of R n (resp.C n ) and that the transition functions are smooth (resp.holomorphic).Here we make sense of definability in C n via the identification C n R 2n by taking real and imaginary parts.
Example 2.1.Any complex algebraic variety X naturally has the structure of an S-definable topological space for any structure S. It admits a finite covering U i by affine algebraic varieties, each of which is a complex (hence real) algebraic subset of some C n i , and the transition functions are given by algebraic functions.Likewise, any nonsingular complex algebraic variety has a natural structure as an S-definable complex manifold.We denote this definable complex manifold by X def .

2.8.
One common method of producing interesting definable topological spaces is by taking quotients of other definable topological spaces by definable group actions.
Definition 2.4.Let X be a locally compact Hausdorff definable topological space and Γ a group acting on X by definable homeomorphisms.A definable fundamental set for the action of We shall require the following proposition: If F is a definable fundamental set for the action of Γ on X, then there exists a unique definable structure on Γ \X such that the canonical map F → Γ \X is definable.

Background on Siegel sets
3.1.In this section, we give some background on Siegel sets for symmetric spaces of the type that appear in the study of period mappings.We begin by recalling the definition of a Siegel set; a good general discussion is [BJ06, § 2].Let G be a reductive Q-algebraic group.The set of real points G(R) is a real Lie group, and we fix a maximal compact subgroup K ⊆ G(R).By [BS73, Proposition 1.6], this determines a Cartan involution θ of the R-algebraic group G R , whose fixed point set is K. Let P ⊆ G be a minimal parabolic Q-subgroup, and let U ⊆ P be its unipotent radical.The dimension of any maximal split Q-torus in P is called the Q-rank of G; we shall denote it by rk G.The Levi quotient P /U is isomorphic to the product of a split Q-torus of dimension rk G and a maximal anisotropic Q-subgroup.According to [BS73, Corollary 1.9], there is a unique Levi subgroup L = S × M ⊆ P R that maps isomorphically to the Levi quotient P R /U R and is stable under the Cartan involution θ.In particular, S is a split R-torus of dimension rk G that is conjugate over G R to a maximal Q-split torus of G, such that θ(g) = g −1 for every g ∈ S(R); compare [Orr18, Lemma 2.1].Moreover, M is contained in the centralizer of S in G R .The adjoint action of S on the Lie algebra of P R determines a root system, and we write ∆ for the set of simple roots.
We use the definition of Siegel sets in [Orr18, § 2.2]; for a discussion of how it relates to the original definition in [Bor19, § 12], see [Orr18,§ 2.3].For our purposes, a Siegel set in G(R), with respect to the maximal compact subgroup K and the minimal parabolic Q-subgroup P , is any set of the form where Ω ⊆ U (R)M(R) + is a compact set, t > 0 is a positive real number, and Note.More generally, it is known that any set of the form S(Ω, t) with Ω ⊆ P (R) compact is contained in a Siegel set in the above sense [Orr18, § 2.3].

3.2.
We are only going to be interested in Siegel sets with respect to a fixed maximal compact subgroup.To emphasize this, we usually talk about Siegel sets for K, meaning that the maximal compact subgroup K in the definition is fixed, whereas the minimal parabolic Q-subgroup P is allowed to be arbitrary.For later use, let us briefly recall how Siegel sets for different minimal parabolic Q-subgroups are related to each other.Let K ⊆ G(R) be a fixed maximal compact subgroup, and P ⊆ G be a minimal parabolic Q-subgroup.
Since all minimal parabolic Q-subgroups are conjugate to each other, any other choice P ′ ⊆ G has the form P ′ = gP g −1 for a suitable element g ∈ G(Q).Write g = kp with k ∈ K and p ∈ P (R) + , so that Now suppose that S ⊆ G(R) is a Siegel set for K and P .By [Bor19, § 12.4], the translate pS is contained in a larger Siegel set ΩA t K ⊆ G(R) with respect to K and P ; here and the right-hand side is now a Siegel set with respect to K and P ′ = gP g −1 .This shows that any Siegel set for K is contained in a G(Q)-translate of a Siegel set with respect to K and a fixed minimal parabolic Q-subgroup P .
3.3.Now we specialize to the case that is of interest in the study of period mappings.The general setting is as follows.Let H Q be a finite-dimensional Q-vector space, of dimension n = dim H Q , equipped with a nondegenerate symmetric bilinear form Further suppose that there is an endomorphism is a positive definite inner product on H R .By analogy with the case of Hodge structures, we shall say that C is a Weil operator for the pair (H Q , Q).

The orthogonal group
whose set of real points is the real Lie group Evidently, C ∈ G(R).It is easy to see that an element g ∈ G(R) preserves the inner product ⟨−, −⟩ C if and only if gC = Cg; therefore the subgroup [Sch73,(8.4)]that K is actually a maximal compact subgroup of G(R), and that the associated Cartan involution is given by the simple formula The following result is well-known.
Lemma 3.1.The symmetric space G(R)/K parametrizes Weil operators for (H Q , Q), with the coset gK corresponding to the Weil operator gCg −1 ∈ End(H R ).
Proof.All elements in the coset gK give us the same operator gCg −1 ∈ End(H R ), which is a Weil operator for the pair Pick a basis e 1 , . . ., e n ∈ H R that is orthonormal for the inner product ⟨−, −⟩ C , in such a way that e 1 , . . ., e p ∈ E 1 (C) and e p+1 , . . ., e n ∈ E −1 (C).Pick a second basis e ′ 1 , . . ., e ′ n ∈ H R that is similarly adapted to ⟨−, −⟩ C ′ and C ′ , and let g ∈ Aut(H R ) be the unique automorphism such that ge i = e ′ i for i = 1, . . ., n.Then and therefore g ∈ G(R) by the nondegeneracy of Q.By construction, C ′ g = gC, which makes C ′ = gCg −1 equal to the image of the coset gK.□ 3.5.Now let us turn our attention to Siegel sets in G(R).The Q-rank of G and the collection of minimal parabolic Q-subgroups P ⊆ G can be described concretely as follows.Let r ≥ 0 be the Witt rank of Q, meaning the dimension of a maximal Q-isotropic subspace of be a maximal flag of isotropic subspaces, with dim V i = i.As explained in [Bor19, § 11.16], the stabilizer P of this flag is a minimal parabolic Q-subgroup of G, and every minimal parabolic Q-subgroup arises in this way; moreover, the

3.6.
The unique Levi subgroup S × M ⊆ P R that is stable under the Cartan involution can be described concretely as follows.Using the Gram-Schmidt process, construct an orthonormal basis e 1 , . . ., e r ∈ V r relative to the inner product ⟨−, −⟩ C , in such a way that is isotropic, the vectors e 1 , . . ., e r , Ce r , . . ., Ce 1 are still orthonormal, and we get an embedding by letting s(λ 1 , . . ., λ r ) act as multiplication by λ i on the vector e i , as multiplication by λ −1 i on the vector Ce i , and as the identity on the orthogonal complement of e 1 , . . ., e r , Ce r , . . ., Ce 1 .The image of this embedding is the desired R-torus S. The other factor of the Levi subgroup S × M has as its set of real points which is clearly stable under the Cartan involution θ(g) = CgC.Note in particular that M(R) preserves the orthogonal complement of e 1 , . . ., e r , Ce r . . ., Ce 1 .

3.7.
We also need to know the set of simple roots ∆ for the action of S on the Lie algebra of P R .These are computed in [Bor19,§ 11.16].There are two cases, depending on the value of the integer n − 2r ≥ 0: (1) If n = 2r, the simple roots are λ 1 /λ 2 , . . ., λ r−1 /λ r and λ r−1 λ r ; this is the case where (H Q , Q) is split, hence isomorphic to a sum of hyperbolic planes.(2) If n > 2r, the simple roots are λ 1 /λ 2 , . . ., λ r−1 /λ r and λ r ; this is the case where (H Q , Q) has a nontrivial anisotropic summand.

3.8.
As in the case of the general linear group, Siegel sets in G(R) are closely related to the reduction theory of positive definite quadratic forms.We are going to make this idea precise by comparing Siegel sets for the two In G, we have the maximal compact subgroup The associated Cartan involution is g → C(g t ) −1 C, where g t means the adjoint of g with respect to the nondegenerate pairing Q.
. ., r, and then filling in the n − 2r steps in between V r and V ⊥ r .Let P ⊆ G be the stabilizer of this maximal flag, and let Ũ ⊆ P be its unipotent radical; then Using the Gram-Schmidt process, construct an orthonormal basis e 1 , . . ., e n ∈ H R with the property that In this case, the Levi subgroup S × M reduces to the split R-torus S ⊆ PR consisting of all diagonal matrices diag(λ 1 , . . ., λ n ) with respect to the basis e 1 , . . ., e n ; with a little bit of work, one can show that S ∩ G(R) = S.

3.9.
We can now compare Siegel sets in Proof.We need to consider the two cases n > 2r and n = 2r separately.Let us first deal with the easier case n > 2r (where H Q has a nontrivial anisotropic summand).Without loss of generality, we can assume that the Siegel set in G(R) has the form and Ω M ⊆ M(R) + are compact subsets and t > 0 is a real number.From the description of the simple roots above, we know that It will be convenient to write elements of G(R) as matrices with respect to our fixed orthonormal basis e 1 , . . ., e n ∈ H R .With this convention, the set A t consists of all diagonal matrices of the form diag(λ 1 , . . ., λ r , 1, . . ., 1, λ −1 r , . . ., λ −1 1 ) with λ 1 /λ 2 ≥ t, . . ., λ r−1 /λ r ≥ t and λ r ≥ t.The crucial point is that every such matrix belongs to Ãt ⊆ S(R), because the number of 1's in the middle is n − 2r ≥ 1.Consider an arbitrary element As a matrix, u is upper triangular with all diagonal elements equal to 1, and m is block-diagonal, with the first r and last r diagonal entries equal to 1; in particular, we have ma = am.Since m ∈ Ω M varies in a compact set, the components of the polar decomposition m = pm • km ∈ P (R) • K also belong to compact subsets of P (R) and K; moreover, pm is again block-diagonal, and therefore a pm = pm a.This gives The first factor lies in a compact subset of P (R), and we have already noted that a ∈ Ãt ; consequently, our Siegel set is contained in a Siegel set in G(R).
The split case n = 2r is less straightforward.Here the subgroup M is trivial, and therefore our Siegel set takes the form where Ω ⊆ U (R) is compact.The simple roots for the action of S on the Lie algebra of P R are now λ 1 /λ 2 , . . ., λ r−1 /λ r , and λ r−1 λ r ; with respect to our orthonormal basis e 1 , . . ., e 2r ∈ H R , the set A t consists of all diagonal matrices of the form with λ 1 /λ 2 ≥ t, . . ., λ r−1 /λ r ≥ t, and λ r−1 λ r ≥ t.As long as λ r ≥ 1, this matrix belongs to Ãmin(1,t) ; but if λ r ≤ 1, this only holds after we swap λ r and λ −1 r , which amounts to conjugating by the permutation matrix In a nutshell, this is the reason why we need two translates of a Siegel set.Getting down to the details, consider again an arbitrary element If a ∈ A t is such that λ r ≥ 1, then a ∈ Ãmin(1,t) , and we can argue as before to show that this part of S(Ω, t) is contained in a Siegel set in G(R).Let us therefore suppose that λ r ≤ 1.We can rewrite our element in the form where a σ = σ aσ etc.Now the crucial point is that u σ ∈ Ũ (R), which puts this part of S(Ω, t) into the translate by σ of a Siegel set in G(R).
Here is the reason why u σ = σ uσ ∈ Ũ (R).The matrix for u is upper triangular with all diagonal entries equal to 1; in particular, there is some x ∈ R such that ue r+1 ≡ e r+1 + xe r mod ⟨e 1 , . . ., e r−1 ⟩.
It remains to argue that the translate is actually by an element of G(Q); note that σ ∈ K is not rational in general.To that end, we define an involution g ∈ G(Q) by requiring that gv i = v i and gv ′ i = v ′ i for i = 1, . . ., r − 1, and gv r = v ′ r and gv ′ r = v r .Then it is easy to check that the matrix for gσ in the basis e 1 , . . ., e 2r is upper triangular, which means that gσ ∈ P (R).Using [Bor19, § 12.4], it follows that the translate by σ of a Siegel set in G(R) with respect to K and P is contained in the translate by g of a bigger Siegel set in G(R), which is enough for our purposes.□

3.10.
Let us now relate Siegel sets in G(R) to reduction theory for quadratic forms.Following [Kli90, Section I.2], we shall say that a positive definite inner product ⟨−, −⟩ on the vector space H R is t-reduced relative to an ordered basis v 1 , . . ., v n ∈ H Q (where t > 0 is a real number) if the following three conditions hold: (a) For every 1 (c) The matrix of the quadratic form satisfies the inequality , where c 1 (n) the optimal constant in Minkowski's inequality.
For a given basis v 1 , . . ., v n ∈ H Q and a given number t > 0, consider the set of elements g ∈ GL(H R ) such that the inner product is t-reduced relative to the basis v 1 , . . ., v n .It is known that every Siegel set in GL(H R ) for the maximal compact subgroup K is contained in a set of this type; conversely, every set of this type is contained in a Siegel set (for K).

3.11.
To simplify the discussion, let us denote by the set of elements g ∈ G(R) such that the inner product Proof.This is an easy consequence of reduction theory, and all that is required is collecting some results from [Bor19].Let Γ G ⊆ G(Q) be an arithmetic subgroup; the intersection According to [Bor19, Theorem 15.5], there exists a Siegel set S H ⊆ H(R) for the maximal compact subgroup K H , and a finite set C H ⊆ H(Q), such that Since the intersection S G ∩ H(R) is of course contained in Γ H CS H , it is therefore enough to prove finiteness of the set After enlarging S G , if necessary, our assumption about Siegel sets in H(R) implies that there is a finite set which is finite because S G has the Siegel property [Bor19, Theorem 15.4].It remains to justify our claim that S H ⊆ AS G .By assumption, S H is contained in many G(Q)-translates of a Siegel set in G(R), but probably with respect to a different minimal parabolic Q-subgroup.After translation by an element of G(Q), we can assume that the minimal parabolic Q-subgroup is the same as for S G ; and then we can enlarge S G and assume that the Siegel set in question is actually S G .This completes the proof.□

3.13.
Let us return to the setting considered in § 3.3, but add one additional piece of data.We still assume that G = O(H Q , Q), and that we have a fixed Weil operator C ∈ End(H R ) for which (v, w) → ⟨v, w⟩ C = Q(v, Cw) is a positive definite inner product on the R-vector space H R .Let us now assume in addition that we have a nonzero element a ∈ H Q with the property that Ca = a.In particular, The stabilizer of this vector is a reductive We are interested in Weil operators C ′ ∈ End(H R ) with the property that C ′ a = a.The following result says that all such Weil operators are conjugate to C by elements of the real group G a (R).
Proof.The proof is similar to that of Lemma 3.1.We have The unit vector e 1 = a/∥a∥ C can be completed to a basis e 1 , . . ., e n ∈ H R that is orthonormal for the inner product ⟨−, −⟩ C , in such a way that e 1 , . . ., e p ∈ E 1 (C) and e p+1 , . . ., e n ∈ E −1 (C).Choose a second basis e ′ 1 , . . ., e ′ n ∈ H R with e ′ 1 = a/∥a∥ C ′ that is similarly adapted to ⟨−, −⟩ C ′ and C ′ , and let g ∈ Aut(H R ) be the unique automorphism such that ge i = e ′ i for i = 1, . . ., n.Then obviously ga = a.As in the proof of Lemma 3.1, one shows that C ′ = gCg −1 and g ∈ G(R), which then implies that g ∈ G a (R) because ga = a.□

3.14.
The orthogonal complement of a relative to Q is the subspace the resulting maximal compact subgroup; note that K a = K ∩ G a (R).
Proposition 3.6.Any Siegel set in G a (R) for the maximal compact subgroup K a is contained in finitely many G(Q)-translates of a Siegel set in G(R) (for K).
Proof.The criterion in Theorem 3.3 reduces the problem to the following concrete statement: suppose that g ∈ G a (R) is an element with the property that is t-reduced relative to an ordered basis v 1 , . . ., v n−1 ∈ H ′ Q ; then it is possible to add the vector a to the basis (in one of the n possible places) and still keep the inner product t-reduced.This is completely elementary.To simplify the notation, let us agree to write ⟨v, w⟩ and ∥v∥ instead of ⟨v, w⟩ gCg −1 and ∥v∥ gCg −1 .Without loss of generality, we may assume that t ≥ 1. Recall that ∥a∥ 2 = Q(a, a).Since the second and third condition in the definition are trivially satisfied.For the first condition, note that for every i = 1, . . ., n − 1, at least one of the inequalities ∥a∥ 2 ≤ t∥v i ∥ 2 or ∥v i ∥ 2 ≤ t∥a∥ 2 will be true (because t ≥ 1).Consequently, there is some value of i ∈ {1, . . ., n − 1} with the property that But this is saying exactly that our inner product is t-reduced relative to the ordered basis

Definable structures on flat bundles
4.1.Let X be a nonsingular complex algebraic variety, and let E be a locally free O X -module of finite rank with a flat holomorphic connection ∇ : E → Ω 1 X ⊗ O X E .We denote by p : E → X the associated holomorphic vector bundle.Recall that E is actually an algebraic vector bundle; the algebraic structure on E is uniquely determined by the flat connection.Let us briefly review the construction.Choose an embedding X → Y into a complete nonsingular variety, such that D = Y \ X is a simple normal crossing divisor.Let ( Ẽ , ∇) be Deligne's canonical extension of the pair (E , ∇); up to isomorphism, it is determined by the following two conditions: (1) Ẽ is a locally free O Y -module with a flat logarithmic connection (2) For each irreducible component D j of the divisor D, the pointwise eigenvalues of the residue operator are complex numbers whose real part is contained in the interval [0, 1).
The canonical extension has the following simple description in local coordinates.Let U ∆ n be an open neighborhood of a point y ∈ Y , with local holomorphic coordinates t 1 , . . ., t n centered at y, such that the divisor D ∩ U is defined by the equation t 1 • • • t k = 0. Let V be the fiber of the vector bundle Ẽ at the point y.Then there is a unique holomorphic trivialization that restricts to the identity on V at the point y, such that the logarithmic connection takes the form for commuting operators R 1 , . . ., R k ∈ End(V ), all of whose eigenvalues have real part in [0, 1); here R j is the residue operator along t j = 0.It is easy to see that defines a multivalued flat section of (E , ∇) on U ∩ X.The monodromy transformation around the divisor t j = 0 is therefore described by the operator e −2πiR j ∈ GL(V ).
In particular, the following two conditions are equivalent: Since Y is complete, the holomorphic vector bundle p : Ẽ → Y associated to the locally free sheaf Ẽ has a unique algebraic structure; the algebraic structure on the bundle E is obtained by restriction.As before, we give Y and Ẽ the structure of R an,exp -definable complex manifolds extending their algebraic structures; this induces definable complex manifold structures on X and E. The former is the canonical algebraic definable structure X def on X and we call the latter the algebraic definable structure on E. It is uniquely described as that for which any holomorphic section s of p : Ẽ → Y over an open subset U ′ ⊂ Y restricts to a definable map on any definable U ⊂ U ′ ∩ X which has compact closure in U ′ .

4.2.
The holomorphic vector bundle E naturally comes equipped with another R an,exp -definable structure coming from the flat coordinates which we construct as follows.The subsheaf E ∇ ⊂ E of flat sections is a complex local system.By definable triangulation, the definable topological space X def admits a definable atlas {(U i , φ i )} with each U i simply connected.Choosing a basis of flat sections s 1 , . . ., s r of E ∇ | U i , we therefore obtain a holomorphic trivialization Moreover, on intersections U ij := U i ∩ U j the transition functions are constant: there are g ij ∈ GL r (C) such that ψ i = ψ j • (id ×g ij ).The flat definable structure on E is then given by the definable complex manifold atlas It is uniquely characterized by the property that any flat section s of p : E → X over a definable open subset U ⊂ X is definable.

4.3.
From now on, we assume that the local monodromy transformations around the components of D have eigenvalues of absolute value 1; in this case we say that E has norm one eigenvalues at infinity.Under this assumption, the two definable structures from the previous two paragraphs are equivalent.Proposition 4.1 (cf.[BM22, Theorem 1.2]).Let X be a nonsingular complex algebraic variety and E a holomorphic flat vector bundle over X with norm one eigenvalues at infinity in the above sense.Then the flat and algebraic definable complex manifold structures on E are equivalent.
The idea of the proof can be seen from the construction of the Deligne canonical extension as above.An algebraic frame for E extends to an algebraic frame for the canonical extension Ẽ on the compactification Y .Thus, locally on the boundary it is related by a matrix of restricted analytic functions to a basis of sections of the form e −2πi k j=1 log t j R j (1 ⊗ v).This basis is in turn related to the flat basis by the matrix e −2πi k j=1 log t j R j , which is R an,exp -definable on bounded angular sectors.Indeed, the functions log t j are R an,exp -definable on bounded angular sectors in polydisk neighborhoods of the boundary in X def .If R j = R ss j + R u j is the Jordan decomposition, then e −2πi k j=1 log t j R u j is polynomial in the log t j , while e −2πi k j=1 log t j R ss j is R an,exp -definable since the R ss j are real.In particular, we have the following concrete corollary, which we will use.
Corollary 4.2.Let Z be a complex manifold, and let f : Z → X be a holomorphic mapping that is moreover R an,exp -definable.Let σ ∈ Γ (Z, f * E ) be a holomorphic section with ∇σ = 0. Then the resulting function σ : Z → E is R an,exp -definable with respect to the algebraic definable structure.

Proof of the main theorem
5.1.We now come to the proof of Theorem 1.1.Let X be a nonsingular complex algebraic variety, H a polarized integral variation of Hodge structure on X, of even weight 2k.Fix a base point x 0 ∈ X and let H Z = H Z,x 0 ; this is a free Z-module of finite rank, which comes with a symmetric bilinear pairing

As usual, we set
for simplicity, we shall use the notation C = C x 0 ∈ End(H R ) for the Weil operator of the Hodge structure at the point x 0 .In particular, is a positive definite inner product on the vector space H R .

5.2.
Let D be the period domain parametrizing integral Hodge structures of weight 2k on H Z that are polarized by Q.Since the statement of Theorem 1.1 only involves the Weil operator (instead of the full Hodge structure), it makes sense to consider not the period domain D, but rather the associated symmetric space (which is a quotient of D).Consider the Q-algebraic group whose set of real points is the real Lie group Recall that G(R) acts transitively on the period domain D, and that D = G(R)/V , where V ⊆ G(R) is the stabilizer of the Hodge structure at x 0 .Clearly, the Weil operator satisfies C ∈ G(R) and C 2 = id.It is easy to see that is a maximal compact subgroup of G(R) containing the compact subgroup V ; by Lemma 3.1, the points of the quotient G(R)/K can be identified with Weil operators for the pair (H Q , Q), with the coset gK corresponding to the Weil operator gCg −1 .

Consider now the arithmetic subgroup
(5.1) Note that Γ is quite a bit larger than the monodromy group of the variation of Hodge structure; this will be important in what follows.Instead of the usual period mapping to Γ \D, we consider the (Weil operator) period mapping to the arithmetic quotient of the symmetric space G(R)/K.It associates to every point x ∈ X the Weil operator C x ∈ Aut(E x ), viewed as an automorphism of the fixed vector space H C by parallel transport; this is well-defined in the quotient Γ \G(R)/K since Γ contains the monodromy group of the variation of Hodge structure.According to [BKT20, Theorem 1.3], the mapping Φ in (5.2) is R an,exp -definable.The main result is stated for the usual period mapping into Γ \D, but what is actually proved in [BKT20, Theorem 4.1] is the R an,exp -definability of (5.2).

5.4.
Let π : X → X be the universal covering space of X.Since the period mapping is locally liftable, there is a real-analytic mapping Φ : X → G(R)/K, unique up to a choice of base point, making the following diagram commute: We now extend the definability result to the vector bundle p : E → X.On G(R)/K, consider the trivial complex vector bundle G(R)/K × H C , where The arithmetic group Γ acts on this bundle via the formula γ and the quotient gives us a "universal" complex vector bundle with fiber H C .The pullback π * E has a canonical trivialization by ∇-flat sections, hence π * E X × H C .The trivial morphism of vector bundles therefore descends to a morphism of complex vector bundles Proposition 5.1.The morphism of complex vector bundles Proof.Let Y be a complete nonsingular variety containing X, such that X = Y \ D for a simple normal crossing divisor D. By the same argument as in [BKT20, § 4.1], the problem is local on Y , and so we may assume that Y = ∆ n , with holomorphic coordinates t 1 , . . ., t n , and that the divisor D is defined by the equation there is a subalgebraic Siegel set S ⊆ G(R) for the maximal compact subgroup K, and a finite set A ⊆ G(Q), such that the image of We get the following commutative diagram: -definable, and so Corollary 4.2 implies that the isomorphism of complex vector bundles is actually R an,exp -definable.Since the morphism of trivial bundles is obviously R an,exp -definable, we get the desired result.□

5.5.
We are ready to prove a first definability result for self-dual vectors in a single Γ -orbit.Suppose that we have an integral vector a ∈ H Z such that Ca = a.We are interested in self-dual classes in the orbit Γ a ⊆ H Z .As noted after Lemma 3.1, points of the symmetric space G(R)/K correspond to Weil operators for (H Q , Q); we identity a coset gK with the Weil operator C g = gCg −1 .
Proposition 5.2.Let a ∈ H Z be a nonzero integral vector with Ca = a.The set Proof.We introduce the additional subgroups For the same reason as before, K a is a maximal compact subgroup of G a (R), and Γ a is an arithmetic subgroup of G a (Q).Lemma 3.5 shows that the image of consists of all cosets gK whose corresponding Weil operator C g = gCg −1 satisfies C g a = a.According to Proposition 3.6 and [BKT20, Theorem 1.2], the morphism of arithmetic quotients is R alg -definable.This morphism has a well-defined lifting which is R alg -definable for the same reason.In more detail, let S be an arbitrary Siegel set in G a (R) with respect to the maximal compact subgroup K a .The formula gives us an R alg -definable local lifting of i.Because of Proposition 3.6, the composition of ĩ with the projection to Γ \(G(R)/K × H C ) is definable; it follows that the mapping i is itself R alg -definable.Now the image of i is exactly the set we are interested in.Indeed, suppose that Γ (gK, v) = Γ (hK, a) for some g ∈ G(R), h ∈ G a (R), and v ∈ H C .Then there is an element γ ∈ Γ such that g = γh and v = γa, and one easily deduces that v ∈ Γ a and C g v = v.In fact, the mapping i is an embedding: if Γ (hK, a) = Γ (h ′ K, a) for two elements h, h ′ ∈ G a (R), then there is some γ ∈ Γ and some k ∈ K such that h ′ = γhk and γa = a; but then γ ∈ Γ a , and therefore k ∈ K a , and so the double cosets Γ a h ′ K a = Γ a hK a are equal.The locus of self-dual classes in our given Γ -orbit is therefore an R alg -definable subset that is isomorphic to the smaller arithmetic quotient Γ a \G a (R)/K a .□ 5.6.We now extend the above result to integral vectors v ∈ H Z with a fixed self-intersection number Proposition 5.3.Let q ∈ N be a positive integer.Then the set Proof.The crucial point is that Γ acts on the set v ∈ H Z Q(v, v) = q with only finitely many orbits; this makes the result a direct consequence of Proposition

5.7.
Finally, we assemble all the pieces and prove Theorem 1.1.
Proof of Theorem 1.1.The polarized integral variation of Hodge structure H on X gives rise to a kind of period mapping that, up to the action by Γ , associates to every point x ∈ X the Weil operator C x of the corresponding Hodge structure.We already know that Φ is R an,exp -definable [BKT20, Theorem 1.2].We also have a morphism of complex vector bundles from the algebraic vector bundle p : E → X to the "universal" vector bundle on the right.We also know that Φ E is R an,exp -definable (by Proposition 5.1).The two morphisms fit into the following commutative diagram: Fix a positive integer q ∈ N. By Proposition 5.3, the set is R alg -definable, and so its preimage under Φ E is an R an,exp -definable subset of E. Since it is easy to see that a point (x, v) ∈ E lies in the preimage exactly when v ∈ E x is integral and satisfies Q x (v, v) = q and C x v = v, we get the result.□

Additional results
6.1.In this section, we prove the two variants of the main theorem stated in the introduction.The idea is simple enough: we tensor a given integral variation of Hodge structure by an auxiliary Hodge structure of weight 1 or 2, and then apply Theorem 1.1.The first Hodge structure that we need is the following.
Example 6.1.Consider the Hodge structure on the first cohomology of the elliptic curve C/(Z ⊕ Zi).
Concretely, this is an integral Hodge structure of weight 1 on the free Z-module Z ⊕2 , whose Hodge decomposition is given by The Hodge structure is polarized by the skew-symmetric bilinear form and the Weil operator is easily seen to be the operator (a 1 , a 2 ) → (a 2 , −a 1 ); note that it happens to preserve the integral structure in this case.
Now suppose that H is a polarized integral Hodge structure of odd weight 2k − 1.Let Q : H Z ⊗ Z H Z → Z be the skew-symmetric bilinear form giving the polarization, and let C ∈ End(H R ) be the Weil operator (which now satisfies C 2 = − id).After taking the tensor product with the above Hodge structure of weight 1, we obtain a polarized integral Hodge structure H of weight 2k on , and with Weil operator C(a 1 , a 2 ) = (Ca 2 , −Ca 1 ).Evidently, and Q(a 1 , a 2 ) = q ; for polarized integral variations of Hodge structure of odd weight, Corollary 1.3 is therefore an immediate consequence of Theorem 1.1.

6.2.
The remaining assertions concern variations of Hodge structure of even weight.We can deal with them by the same method, using the following Hodge structure.Example 6.2.Consider now the symmetric square of the Hodge structure in Example 6.1.Concretely, we get an integral Hodge structure of weight 2 on the free Z-module Z ⊕3 , whose Hodge decomposition is The Hodge structure is polarized by the symmetric bilinear form and the Weil operator is easily seen to be the operator (a 1 , a 2 , a 3 ) → (a 3 , −a 2 , a 1 ), which again preserves the integral structure.
Suppose that H is a polarized integral Hodge structure of even weight 2k, with polarization Q : H Z ⊗ Z H Z → Z and Weil operator C ∈ End(H R ).After taking the tensor product with the Hodge structure in Example 6.2, we obtain a polarized integral Hodge structure H of weight 2k + 2 on polarized by the symmetric bilinear form and with Weil operator C(a 1 , a 2 , a 3 ) = (Ca 3 , −Ca 2 , Ca 1 ).This time around, , and 2Q(a 1 , a 3 ) − Q(a 2 , a 2 ) = q .We now obtain Corollary 1.3 for polarized integral variations of Hodge structure of even weight by looking at triples of the form (a 1 , 0, a 3 ), and Corollary 1.2 by looking at triples of the form (0, a 2 , 0).String theory is a candidate theory of quantum gravity that unifies Einstein's theory of general relativity and quantum field theory.Quantum consistency forces the string to travel through a higher-dimensional space-time manifold, extending beyond the four space-time dimensions that we currently observe in our universe.In prominent variants of string theory this implies that either six or eight extra dimensions need to be present.These extra dimensions are often considered to be on a tiny compact manifold.Particularly well-studied choices are Calabi-Yau manifolds, which are defined to be Kähler manifolds that admit a Ricci-flat metric.While it is not known which Calabi-Yau manifold one should pick, it has been studied intensively how the physical four-dimensional theory can be determined after making a choice.7.2.We now describe one example from physics that originally suggested the result in Theorem 1.1.Let Y be a compact polarized Calabi-Yau manifold of complex dimension D, with D = 3, 4 being the cases most relevant in the string theory application.One can associate a family of manifolds Y t to Y that is obtained by deforming its complex structure.It is shown by the Bogomolov-Tian-Todorov theorem [Tia87,Tod89] that the Kuranishi space of Y is unobstructed.Hence Y t varies over a finite-dimensional moduli space M if one demands that all Y t are Calabi-Yau manifolds.For polarized Calabi-Yau manifolds of complex dimension D the moduli space M is quasi-projective [Vie95] and of complex dimension h D−1,1 = dim H D−1,1 (Y ).The existence of such a moduli space leads to several physical problems when using such Y t as backgrounds of string theory.In particular, one finds modifications of Newton's law or Einstein's equations that are in contradiction with observations.To avoid this immediate conclusion further ingredients known as background fluxes can be introduced.These fluxes are integral classes in the cohomology of Y .Compared with the general considerations above, we thus identify X = M, where M is the resolution of M [Hir64].Furthermore, we set H Z = H D (Y , Z) and Q = Y v ∧ w.Introducing a Weil operator C acting on H p,q (Y t ) with i p−q , we define a norm ∥w∥ 2 = Q( w, Cw).

7.3.
The best understood string theory settings with integral fluxes are obtained from Type IIB string theory [GVW00,GKP02].Let us consider this ten-dimensional theory on a Calabi-Yau manifold Y of complex dimension three.In this string theory setting one is also free to chose in addition to Y two integral three-forms F, H ∈ H 3 (Y , Z), which set the flux background.They naturally combine to a complex three-form G = F − τH with τ ∈ C. The fluxes F, H are constrained by a consistency condition (7.1) where ℓ is a fixed positive rational number that can be derived for a given setting.This condition is known as a tadpole cancellation condition and plays a crucial role in finding consistent solutions of string theory.Furthermore, the presence of G impacts the physical four-dimensional theory by giving rise to an energy potential V (G), which generally changes for different choices Y t within the family.Concretely, it takes the form [GKP02] (7.2) where CG − = −iG − and κ is τ-dependent but constant over M. The loci in M that minimize this energy potential with G − = 0 have been shown to be consistent background solutions of Type IIB string theory [GKP02].It has been a long-standing question of whether or not the number of distinct H, F with G − = 0 and (7.1) is finite.

7.4.
A more general setting that leads to a similar question arises in a geometric higher-dimensional version of Type IIB string theory known as F-theory [Vaf96,Den08].In F-theory the extra dimensions are constrained to reside on an eight-dimensional compact manifold to extract a four-dimensional physical theory.The consistency equations for such twelve-dimensional string backgrounds admit solutions that are (conformal) Calabi-Yau manifolds Y of complex dimension D = 4 that admit a four-form flux background.Let us consider v ∈ H 4 (Y , Z) and assume that v is primitive with respect to the Kähler form J of Y .The condition (7.1) generalizes to Q(v, v) = ℓ, which is the consistency condition every solution to F-theory with a compact Y has to satisfy.A non-trivial v induces again an energy potential where Cv − = −v − and λ is a constant.V (v) changes in the family Y t and hence is a function on the moduli space M. The self-dual loci in M are by definition those that satisfy Cv = v and hence minimize V (v).They comprise consistent solutions to F-theory and are of central interest to some of the most prominent scenarios on realizing our four-dimensional universe in string theory.Each choice of v satisfying these consistency conditions can imply different values for physical observables.It is thus of profound importance to know if there are infinitely many choices for v.
centrally based on the definability of the period mapping which suffices to exclude the pathological examples discussed in [AD06].In dimension 1, it is possible to prove Theorem 1.1 along the lines of [CDK95], by using more details about the SL(2)-orbit theorem [Sch20], see also [Gri21] for a sketch of the argument.In higher dimensions, this kind of argument looks completely infeasible.
descends to a polarized integral variation of Hodge structure on ∆ * , whose monodromy transformation is T = e N .Let us describe the locus of points in H where a given integral class v ∈ H Z is self-dual.Write z = x + iy.Let C ♯ ∈ End(H R ) denote the Weil operator of the Hodge structure F ♯ .From the identity e zN F = e xN e iyN F = e xN e − 1 2 log y Y e iN e 1 2 log y Y F = e xN e − 1 2 log y Y F ♯ and the fact that both exponential factors are elements of the real Lie group G(R), it follows that the Weil operator of the Hodge structure e zN F is e xN e − 1 2 log y Y C ♯ e 1 2 log y Y e −xN .The set of points z ∈ H where our integral class v ∈ H Z is self-dual is therefore defined by the simple equation (1.3) C ♯ e 1 2 log y Y e −xN v = e 1 2 log y Y e −xN v.At each point, the Hodge norm of v is of course equal to Q(v, v).If the set contains points with y = Im z arbitrarily large, then necessarily v ∈ W 0 (N ).Since W 0 (N ) = ℓ≤0 E ℓ (Y ), we have a decomposition v = v 0 + v −1 + • • • , where Y v ℓ = ℓv ℓ .Now the Weil operator C ♯ interchanges the two weight spaces E ±ℓ (Y ), because Y C ♯ + C ♯ Y = 0, for example by [CKS86, formulas (6.35)].Since e 1 2 log y Y e −xN v ∈ W 0 (N ), the identity in (1.3) implies that e 1 2 log y Y e −xN v ∈ E 0 (Y ), and hence that v = e xN e − 1 2 log y Y v 0 = e xN v 0 .Now there are two possibilities.Either N v = 0 and v = v 0 , or N v 0. In the first case, v is self-dual at every point z ∈ H; in the second case, the equation v = e xN v 0 uniquely determines the value of x ∈ R, and v is self-dual along the vertical ray Re z = x.The connected components of the locus of self-dual classes are therefore of two different kinds: one kind projects isomorphically to the entire punctured disk ∆ * ; the other to a single angular ray in ∆ * .
The result is a more precise version of a general theorem by Orr [Orr18, Theorem 1.2], with a small correction contained in[OS21].Proposition 3.2.Any Siegel set in G(R) for the maximal compact subgroup K is contained in at most two G(Q)-translates of a Siegel set in G(R) (for K).

( a )
The eigenvalues of the local monodromy transformations around the components of D are complex numbers of absolute value 1.(b) For each irreducible component D j of the divisor D, the pointwise eigenvalues of the residue operators Res D j ∇ are real numbers.

7. 1 .
A motivation for studying the locus of self-dual integral Hodge classes stems from string theory.
Our second example is self-dual classes in certain nilpotent orbits.This is less geometric, but provides us with a large family of examples.For the general theory, see [CKS86, §3] and the survey paper [CK89, §3] by Cattani and Kaplan.
Example 1.2.Let H Z be a free Z-module of finite rank, and let Q : H Z ⊗ Z H Z → Z be a nondegenerate symmetric bilinear pairing.Suppose that we have a representation ρ : sl 2 (C) → End(H C ) of the Lie algebra sl 2 (C), such that and such that e N ∈ End(H Z ).Let W • (N ) denote the weight filtration of the nilpotent operator N .Let F be a decreasing filtration of H C such that The first assertion follows immediately from Proposition 3.2 and the discussion above.To prove the second assertion, observe that any set of the form S(v 1 , . . ., v n , t) is contained in a Siegel set in GL(H R ) (for K); moreover, the intersection of such a Siegel set with the subgroup G(R) is contained in finitely many G(Q)-translates of a Siegel set in G(R) by Proposition 3.4 below.We then get the desired result by recalling that any G(Q)-translate of a Siegel set for K is again a Siegel set for K (with a possibly different minimal parabolic Q-subgroup).In this section, we prove a proposition concerning the intersection of a Siegel set with a reductive subgroup.The result is similar to [BHC62, Lemma 7.5], except that we are working with Siegel sets for Q-algebraic groups (instead of with Siegel domains for R-algebraic groups), and that we are making a different set of assumptions about the subgroup.To simplify the notation, let G be an arbitrary reductive Q-algebraic group, and H ⊆ G a reductive Q-algebraic subgroup.Further, let K G ⊆ G(R) and K H ⊆ H(R) be maximal compact subgroups such that K H = K G ∩ H(R).Note that the Cartan involutions on G R and H R are not necessarily compatible with each other.
reduced relative to a given basis v 1 , . .., v n ∈ H Q .Being defined by a collection of inequalities, this is clearly a subalgebraic subset of G(R).It is easy to see that gS(v 1 , . .., v n , t) = S(gv 1 , . .., gv n , t) for any g ∈ G(Q), which means that the collection of these sets is stable under translation by elements in G(Q).The following theorem gives a useful criterion for checking whether a given subset of G(R) is contained in finitely many G(Q)-translates of a Siegel set for K. Theorem 3.3.Any Siegel set in G(R) for the maximal compact subgroup K is contained in a finite union of sets of the form S(v 1 , . .., v n , t); conversely, any set of the form S(v 1 , . .., v n , t) is contained in a finite union of Siegel sets (for K).Proof.□ 3.12.Proposition 3.4.Suppose that every Siegel set in H(R) for the maximal compact subgroup K H is contained in finitely many G(Q)-translates of a Siegel set in G(R) (for K G ). Let S G ⊆ G(R) be any Siegel set for K G .Then there is a Siegel set S H ⊆ H(R) for K H , and a finite set F ⊆ H(Q), such that 5.2.The finiteness of the number of Γ -orbits follows from [Kne02, Satz 30.2]; since Kneser works in much greater generality, let us briefly explain how to deduce the statement we need.The quadratic form v → Q(v, v) makes H Z into a lattice in the Q-vector space H Q , and an integral vector v ∈ H Z with Q(v, v) = q defines an isometry v : [q] → H Z , where [q] means the lattice Z with the quadratic form n → qn 2 .(Kneser calls this a "Darstellung" of [q] in H Z .)Now [ℓ] is nondegenerate because ℓ ≥ 1, and [Kne02, Satz 30.2] guarantees that there are only finitely many equivalence classes of such isometries.But since Γ = O(H Z , Q), two vectors v, v ′ ∈ H Z are in the same equivalence class, in the sense of [Kne02, Definition 30.1], exactly when there is an element γ ∈ Γ such that v ′ = γv.□