Hodge-Deligne polynomials of character varieties of free abelian groups

Proposition 2.1

Let X and Y be complex quasi-projective varieties. Then,

1. For all k , p , q , we have H k , q , p ( X ) ≅ H k , p , q ( X ) ¯ ;

2. The weight and Hodge filtrations are preserved by algebraic maps. Therefore, so are MHSs;

3. The Hodge and weight filtrations are preserved by the Künneth isomorphism. Therefore, so are MHSs;

4. The MHSs are compatible with the cup product:

   H k , p , q ( X ) ⌣ H k ′ , p ′ , q ′ ( X ) ↪ H k + k ′ , p + p ′ , q + q ′ ( X ) ;

5. If X is smooth of complex dimension n , MHSs are compatible with Poincaré duality:

   H k , p , q ( X ) ≅ ( H c 2 n − k , n − p , n − q ( X ) ) ∗ .

Proof

All of these statements are standard. For convenience, we point to appropriate references. The proof of (1) follows from the purity of the Hodge structure on the graded pieces of the weight filtration.

The other proofs are found in chapters 4 to 6 of the book [27]. Specifically, for (2) see [27, Proposition 4.18]; (3) and (4) appear in [27, Theorem 5.44, Corollary 5.45] and (5) in [27, Proposition 6.19].□

Definition 2.2

Let X be a complex quasi-projective variety of complex dimension d . The mixed Hodge polynomial of X (also called Hodge-Deligne polynomial) is the three-variable polynomial of degree ≤ 2 d

Its specialization for t = − 1

is called the E -polynomial of X .

Remark 2.3

1. The specialization of μ X for u = v = 1 gives the Poincaré polynomial of X :

   P X ( t ) ≔ ∑ k ≥ 0 b k ( X ) t k ,

   with b k ( X ) ≔ dim C H k ( X , C ) being the Betti numbers of X . Note that the coefficients of μ X and of P X are non-negative integers, whereas E X lives in the ring Z [ u , v ] .

2. As mentioned earlier, there is an entirely parallel theory for the compactly supported cohomology. Here, the associated Hodge numbers are denoted by h c k , p , q ≔ dim C H c k , p , q ( X ) . If P X stands for one of the polynomials in the aforementioned definition, we will distinguish its compactly supported version by writing P X c .

3. Comment on terminology: there are inconsistencies in the literature on the terminology used for these polynomials. Since h k , p , q ( X ) are generally called Hodge-Deligne (or mixed Hodge) numbers, we refer to μ X as Hodge-Deligne or mixed Hodge polynomial. To emphasize the distinction, the compactly supported E -polynomial E X c will also be called the Serre polynomial of X , since its crucial behavior, as a generalized Euler characteristic, was first used by Serre in connection with the Weil conjectures (see [30]).

4. Many specializations of the E -polynomial have been studied in the literature. There is, for example, the weight polynomial W X ( y ) ≔ ∑ k , p ( − 1 ) k w k , p ( X ) y p , using the graded pieces of the weight filtration w k , p ( X ) ≔ dim C G r p W C H k ( X , C ) (see [21]). This is a specialization of the E -polynomial since W X ( y ) = E X ( y , y ) . Also, Hirzebruch’s χ y -genus and the signature σ of a complex manifold X are given, in terms of E X ( u , v ) , as: χ y ( X ) = E X ( − y , 1 ) and σ ( X ) = E X ( − 1 , 1 ) , respectively (see Hirzebruch [31]).

Proposition 2.4

For a quasi-projective variety X , we have:

1. The polynomials μ X and E X are symmetric in the variables u and v ; in particular, if h k , p , q ( X ) ≠ 0 then h k , q , p ( X ) ≠ 0 .

2. Let h k , p , q ( X ) ≠ 0 . Then p , q ≤ k . Moreover, if X is smooth, then p + q ≥ k ; if X is projective, then p + q ≤ k . In particular, if X is a compact Kähler manifold p + q = k .

3. The (topological) Euler characteristic χ ( X ) is given by χ ( X ) = E X ( 1 , 1 ) .

4. The Serre polynomial (compactly supported E -polynomial) E X c is additive for stratifications of X by locally closed subsets, and its degree is equal to 2 dim C X .

5. All polynomials μ X , P X and E X are multiplicative under Cartesian products.

Proof

(1) Follows from item (1) of Proposition 2.1. Item (2) is proved in [27, Proposition 4.20] and [27, Theorem 5.39]. Item (3) is immediate from the definition. The proof of (4) can be found in [27, Corollary 5.57] and (5) follows directly from 2.1(4).□

Definition 2.5

A quasi-projective variety X is said to be balanced or of Hodge-Tate type if for every non-negative integer k ∈ N 0 , and all p ≠ q , h k , p , q ( X ) = 0 . In other words, if h k , p , q ( X ) ≠ 0 , then q = p . We call p + q the total weight of H k , p , q ( X ) .

Example 2.6

1. If X is connected, H 0 ( X , C ) ≅ C has always a pure Hodge structure, with trivial decomposition H 0 ( X , C ) = H 0 , 0 , 0 ( X ) . Dually, when X is also smooth, the compactly supported cohomology is also a trivial decomposition H c 2 n ( X , C ) = H c 2 n , n , n ( X ) .

2. X = C n is a (non-compact) Kähler manifold with cohomology only in degree zero. By the above, it has trivial pure Hodge structure: H ∗ ( X , C ) = H 0 ( X , C ) = H 0 , 0 , 0 ( X ) and so

   μ C n ( t , u , v ) = 1 , μ C n c ( t , u , v ) = t 2 n u n v n ,

   where the compactly supported version follows from Poincaré duality.

3. Let X = C ∗ = C ⧹ { 0 } . Although Kähler, its cohomology has no pure Hodge structure, since dim C H 1 ( X , C ) = 1 . Being smooth, using Proposition 2.1(1)–(2), the only non-zero h 1 , p , q is h 1 , 1 , 1 , so h 0 , 0 , 0 = h 1 , 1 , 1 = 1 and all other Hodge numbers vanish. We then get

   μ C ∗ ( t , u , v ) = 1 + t u v ,

   and the only non-zero h c k , p , q are h c 2 , 1 , 1 = h c 1 , 0 , 0 = 1 , by Poincaré duality. Thus, μ C ∗ c ( t , u , v ) = t 2 u v + t and E C ∗ c ( u , v ) = u v − 1 . Observe that this is compatible with the decomposition into locally closed subsets C 1 = C ∗ ⊔ C 0 as in Proposition 2.4(4). Hence, C n and ( C ∗ ) n are examples of balanced varieties. For a simple example of a non-balanced variety we can take an elliptic curve or any compact Riemann surface of positive genus.

4. Consider the total space X of the trivial line bundle over an elliptic curve X ≅ ( C / Λ ) × C , where Λ is a rank two lattice in ( C , + ) . It is easy to see that X is real analytically isomorphic to ( C ∗ ) 2 (but not complex analytically or algebraically isomorphic). From the Künneth isomorphism and considerations analogous to Example 2.6, we get:

   μ ( C ∗ ) 2 ( t , u , v ) = ( 1 + t u v ) 2 = 1 + 2 t u v + t 2 u 2 v 2 , μ X ( t , u , v ) = ( 1 + t u ) ( 1 + t v ) = 1 + t ( u + v ) + t 2 u v .

   Indeed, ( C ∗ ) 2 is balanced, whereas the cohomology of X is pure.

Remark 2.7

1. The last example is a very special case (the genus 1, rank 1 case) of the non-abelian Hodge correspondence mentioned in Section 1, which produces diffeomorphisms between (Zariski open subsets of) moduli spaces of flat connections and certain moduli spaces of Higgs bundles over a given Riemann surface. The fact that one diffeomorphism type is balanced (the flat connection side of the correspondence) and the other is pure is a general feature (see [3,4]).

2. If X is balanced, its E -polynomial depends only on the product u v , so it is common to adopt the change of variables x ≡ u v . When written in this variable, the degree of E X c ( x ) is now equal to dim C X , instead of 2 dim C X .

Definition 3.1

Let X be a quasi-projective variety. X (or its cohomology) is called elementary if its MHSs are trivial decompositions of the cohomology, so that for every k ∈ N there is only one p ∈ N such that h k , p , p ( X ) ≠ 0 (and h k , p , q ( X ) ≠ 0 for q ≠ p ).

X is said to be separably pure if the MHS on each H k ( X , C ) is in fact pure of total weight w k , and such that w j ≠ w k for every j ≠ k .

Remark 3.2

1. Note that X is elementary if it is balanced and there is a weight function k ↦ p k (defined only for those k ∈ N 0 with H k ( X , C ) ≠ 0 ) such that h k , p , q = 0 for every pair ( p , q ) not equal to ( p k , p k ) . In this case,

   (2) G r 2 p k W C H k ( X , C ) = H k , p k , p k ( X ) = H k ( X , C ) .

   A general weight function is not enough to recover μ X from the weight or the E -polynomials (different degrees of cohomology may have equal total weights). However, this can be done (see Theorem 3.6) if the weight function k ↦ p k is injective, in which case the equality (2) takes the stronger form: ⊕ m G r 2 p k W C H m ( X , C ) = H k , p k , p k ( X ) = H k ( X , C ) .

2. In a pure Hodge structure of total weight k on H k ( X , Q ) the only non-zero weight summand is G r 2 k W H k ( X , Q ) . So, a pure total cohomology is separably pure, but not conversely, as the case C ∗ shows (Example 2.6).

3. When X is separably pure, instead of the weight function, one can define a degree function ( p , q ) ↦ k = k ( p , q ) (defined only on pairs ( p , q ) such that h k , p , q ( X ) ≠ 0 ). Noting that, in fact, the degree k only depends on the total weight p + q (being separably pure) we can write this as ( p , q ) ↦ k p + q .

Lemma 3.3

A quasi-projective variety X is separably pure and balanced if and only if it is elementary and its weight function k ↦ p k is injective.

Proof

If X is separably pure, the total weight in each H k ( X , C ) has to be constant. But if X is also balanced, given k , all h k , p , q ( X ) vanish except for a unique pair ( p , q ) = ( p k , p k ) , so we have an assignment k ↦ p k proving that X is elementary. Moreover, since the total weights are different for distinct k , the weight function is injective. The converse statement is easy since an elementary variety is Hodge-Tate and an injective weight function implies injectivity for total weights.□

Example 3.4

A family of balanced and pure varieties are the smooth projective toric varieties (hence these are elementary). Indeed, every such toric variety X , where the C ∗ action has d j orbits in (complex) dimension j = 0 , … , dim C X , has (see [27, Example 5.58]):

and the weight function is 2 j ↦ j (the weights being ( j , j ) ). For example, the Hodge-Deligne polynomials of projective spaces are μ P C n ( t , u , v ) = ∑ j = 0 n t 2 j u j v j .

Example 3.5

The Poincaré polynomial of G L ( n , C ) is well known, given by P G L ( n , C ) ( t ) = ∏ j = 1 n ( 1 + t 2 j − 1 ) . Also, by [29, Theorem 9.1.5], we have:

For example, μ G L ( 3 , C ) ( t , x ) = 1 + t x + t 3 x 2 + t 4 x 3 + t 5 x 3 + t 6 x 4 + t 9 x 6 (writing x = u v , see Remark 2.7(2)). So, G L ( 3 , C ) is elementary (hence balanced) but not separably pure: both degrees 4 and 5 have associated total weight 6 (the terms with x 3 ), so G L ( n , C ) is not separably pure, for n ≥ 3 . Moreover, the same argument readily shows that G L ( n , C ) is not elementary for n ≥ 5 .

Theorem 3.6

Let X be a quasi-projective variety of dimension n . Then:

1. If X is elementary, with known weight function, its Poincaré polynomial determines its Hodge-Deligne polynomial.

2. If X is separably pure, with known degree function, its E -polynomial determines its Hodge-Deligne polynomial.

Proof

1. Suppose the Poincaré polynomial of X is P X ( t ) = ∑ k b k t k and the weight function is k ↦ ( p k , p k ) . Then, since the only non-trivial mixed Hodge pieces are H k , p k , p k ( X ) , we get μ X ( t , u , v ) = ∑ k b k t k u p k v p k .

2. Similarly, writing E X ( u , v ) = ∑ p , q a p , q u p v q , and the degree function as ( p , q ) ↦ k p + q , since the total weights are in one-to-one correspondence with the degrees of cohomology, we obtain μ X ( t , u , v ) = ∑ k , p , q a p , q ( − t ) k p + q u p v q .□

Definition 3.7

Let X be a quasi-projective variety. If the only non-zero Hodge numbers are of type h k , k , k ( X ) , k ∈ { 0 , … , 2 dim C X } , we say that X is round.

Remark 3.8

In general, Cartesian products of elementary varieties are not elementary, and similarly for separably pure varieties. For instance, using Example 3.5 with n = 2 , we see that G L ( 2 , C ) × G L ( 2 , C ) is not separably pure. On the other hand, the following result holds for round varieties.

Proposition 3.9

Let X and Y be round varieties. Then:

1. The Hodge-Deligne polynomial of X reduces to a one-variable polynomial, and can be reconstructed from either the E or the Poincaré polynomial:

   μ X ( t , u , v ) = P X ( t u v ) = E X ( − t u , v ) .

2. The Cartesian product X × Y is round.

Proof

1. By definition, if X is round, we can write:

   μ X ( t , u , v ) = ∑ k ≥ 0 h k , k , k ( X ) t k u k v k ,

   so that μ X is a polynomial in t u v . Moreover, its Betti numbers are b k ( X ) = h k , k , k ( X ) giving the first equality. The second follows from E X ( x ) = ∑ k ≥ 0 ( − 1 ) k h k , k , k ( X ) x k .

2. This follows at once from (1) and from Proposition 2.4(5).□

Remark 3.10

1. If X satisfies Poincaré duality on MHS, and dim C X = n , one has

   μ X c ( t , u , v ) = ( t 2 u v ) n μ X 1 t , 1 u , 1 v , P X c ( t ) = t 2 n P X 1 t , E X c ( u , v ) = ( u v ) n E X 1 u , 1 v .

   In particular, χ ( X ) = E X c ( 1 , 1 ) . If X is additionally round, analogously to Proposition 3.9, μ X c can be reconstructed from P X c and E X c as:

   μ X c ( t , u , v ) = ( u v ) − n P X c ( t u v ) = ( − t ) n E X c ( − t u , v ) .

2. A sufficient condition for roundness is the following: if X is balanced and separably pure and its cohomology has no gaps, in the sense that for every k ∈ N , the condition H k ( X , C ) ≠ 0 implies H k − 1 ( X , C ) ≠ 0 , then X is round. This is easy to see from Lemma 3.3 and the restrictions on weights (Proposition 2.4(2)).

Definition 4.1

The equivariant mixed Hodge polynomial is defined as:

Evaluating at t = − 1 , gives us the equivariant E-polynomial:

Proposition 4.2

Let X be a quasi-projective F -variety, for a finite group F , and let P X F be one of the polynomials in Definition 4.1. Then

1. P X is obtained by replacing each representation in P X F by its dimension;

2. The Künneth formula and Poincaré duality, for X smooth, are compatible with equivariant MHS:

   P X × Y F = P X F ⊗ P Y F , [ G r p + q W C H k ( X , C ) ] = [ ( G r 2 n − ( p + q ) W C H c 2 n − k ( X , C ) ) ∗ ] , [ H k , p , q ( X ) ] = [ ( H c 2 n − k , n − p , n − q ( X ) ) ∗ ] ,

   where ⊗ means that we take tensor products of graded F -representations.

Proof

(1) This follows immediately from the definition of dimension of representation. For (2), it suffices to see that the Künneth and Poincaré maps are also morphisms in the category of F -modules, which is easily checked.□

Proposition 4.3

Let F be a finite group and X a smooth quasi-projective F -variety. Then, the pullback of the quotient map π ∗ : H ∗ ( X / F , Q ) → H ∗ ( X , Q ) is injective and has H ∗ ( X , Q ) F as its image.

Proof

Assume first that F acts freely on X . Then, X / F has a well-defined manifold structure, and one can realize the pullback in cohomology by the pullback in differential forms. In particular, this shows that the image of the pullback π ∗ : H ∗ ( X / F , Q ) → H ∗ ( X , Q ) is given by H ∗ ( X , Q ) F . Using (5), this means that the pullback map is bijective onto H ∗ ( X , Q ) F .

If F does not act freely, the same argument can be reproduced for the de Rham orbifold cohomology, in which representatives of orbifold cohomology classes are sections of exterior powers of the orbifold cotangent bundle (see [35]). The result then follows because, for manifolds such as X , the de Rham orbifold cohomology reduces to the usual de Rham cohomology.□