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On the Hodge theory of the additive middle convolution

We compute the behaviour of Hodge data under additive middle convolution for irreducible variations of polarized complex Hodge structures on punctured complex affine lines.


Introduction
In previous work of Sabbah with one of the authors [4], the effect of the additive middle convolution MC χ (V ) = V ⋆ L χ of a complex polarized Hodge module V on A 1 with a Kummer module L χ on various local and global Hodge data was determined. This leads to an analog of Katz' algorithm for irreducible rigid local systems [7] in the context of Hodge modules. It is the aim of this work to extend these results to the case of the middle convolution V ⋆ L (cf. Section 1) of two irreducible and non-constant complex polarized Hodge modules on A 1 . It turns out that, to a large extent, the general case can be reduced to the middle convolution with Kummer modules treated in [4].
In Section 2, Theorem 2.1, the global Hodge numbers of tensor products V ⊗ L (the degrees of the associated Hodge bundles) are determined, generalizing [4], Prop. 2.3.2. We are indepted to Claude Sabbah for communicating the proof of Theorem 2.1 to us. This result is important in many applications where convolution is applied iteratively in combination with tensor operations (cf. [7], [4], [3]).
In Section 3 we determine the local Hodge data of the vanishing and nearby cycles (cf. Section 1 and [4] for these notions) at the finite singularities of a convolution V ⋆ L (Theorem 3.4). As in [4], the main tool for doing this is Saito's version of the Thom-Sebastiani theorem (cf. [4], Theorem 3.2.3, and its corrigendum [5], where a proof of the Thom-Sebastiani result is provided).
In Section 4 the global Hodge numbers of V ⋆ L are determined. The main observation is that the middle convolution V ⋆ L χ of an irreducible and nontrivial Hodge module V with a generic Kummer module L χ is parabolically rigid, meaning that the associated parabolic cohomology group vanishes (where RH (V ⋆ L χ ) is the perverse sheaf associated to V ⋆ L χ via Riemann-Hilbert correspondence and j is the projective embedding of A 1 ). Using the Riemann-Roch theorem, a formula for the Hodge numbers of H 1 par (V ⋆ L χ ) involving local and global data was given in [4], Proposition 2.3.3. Hence the vanishing of H 1 par (V ⋆ L χ ) gives a method to compute the global Hodge numbers of V ⋆ L.
The remaining local Hodge data at ∞ of V ⋆ L are determined in Section 5. For this, we make use of hypergeometric Hodge modules with prescribed local behaviour at ∞ and reduce the general case to the convolution of these. We believe that a more conceptual proof of these results may be given in the context of irregular Hodge filtrations on twistor modules and their behaviour under Fourier-Laplace transformation (cf. [6]).
In a forthcoming work, the authors prove similar results for the multiplicative convolution (also called Hadamard product). §1. Preliminary results Following [4], we review the basic notions of middle convolutions introduced by Katz [7], in the frame of holonomic D-modules. Let s : . It can also expressed as s † (M ⊠ N ), if s † := Ds + D denotes the adjoint by duality of s + , cf. [4] (under the Riemann Hilbert correspondence, the functor + corresponds to the derived * -functor and † corresponds to !, explaining the notion). Let us choose a projectivization s : X → A 1 of s, and let j : A 1 ×A 1 ֒→ X denote the open inclusion. Since s naturally commutes with duality, we have s † = s + and s † = s + • j † . Since there is a natural morphism ). Let P be the full subcategory of Mod hol (A 1 ) consisting of holonomic D(A 1 )-modules N such that for all holonomic D(A 1 )-modules M both types of convolutions N ⋆ * M and N ⋆ ! M are again holonomic. Definition 1.0.1. For N in P and M holonomic, the middle convolution M ⋆ mid N is defined as the image of M ⋆ ! N → M ⋆ * N in Mod hol (D(A 1 )). For simplicity we often set M ⋆ N := M ⋆ mid N. As explained in [4], Section 3.3, this notion extends to the category of complex polarized Hodge modules on A 1 , using the results of [11] and [12]. If M is smooth on A 1 x (x = {x 1 , . . . , x r } ∪ {∞}) and if N is smooth on A 1 y (y = {y 1 , . . . , y s } ∪ {∞}) then M ⋆ N as well as the other types of convolutions are smooth on A 1 x ⋆ y, where The following result follows from the Riemann-Hilbert correspondence and [7], Cor. 2.6.10 and Cor. 2.6.17: N is irreducible such that its isomorphism class is not translation invariant then N has the property P.
(ii) If N and M are in P then N ⋆ M again is in P.
Let W be a complex polarized Hodge module on the complex affine line A 1 which is smooth on A 1 {z 1 , . . . , z k }. The local system on A 1 {z 1 , . . . , z k } which is underlying W is denoted as W . The perverse sheaf on A 1 associated to W via the de Rham functor is denoted by RH (W ) and we view the i-th parabolic cohomology group to be equipped with its natural Hodge structure.
Throughout the article we will work with Hodge modules V, L, δ x , L χ which are as follows: . , x r , ∞} ⊂ P 1 (C)). In this situation we sometimes set x r+1 = ∞.
(ii) Let L = (L, F • L) be another Hodge module of the same kind which is the minimal extension of a variation of polarized complex Hodge structures on A 1 y (where y = {y 1 , . . . , y s , ∞}).
(iii) For a point x ∈ A 1 we write δ x for the Hodge module which corresponds to the rank-one skyscraper sheaf on A 1 supported in x, having trivial Hodge filtration (so that h 0 (H 0 (A 1 , δ x )) = 1).
For the following notions and stated results we refer to [4], Section 1.2 and Sections 2.2, 2.3: on the one hand, one has global Hodge data δ p (V ) given by the degrees of the Hodge bundles. On the other hand, one has local Hodge data: For each point x ∈ {x 1 , . . . , x r+1 } and each λ ∈ S 1 one has the notion ψ x,λ (V ) of the generalized λ-eigenspace of the nearby cycles ψ x (V ). We will also use the corresponding notion of vanishing cycles ϕ x,λ (V ). These spaces are mixed Hodge structures with associated nilpotent monodromy operator, derived from the local monodromy, which imposes an associated weight filtration W. One has the notion of l-primitive vectors P l ϕ x,λ (V ) with respect to the Lefschetz decomposition of ϕ x,λ (V ). For l ∈ N we define l-primitive local Hodge numbers as follows: where a ∈ R ∩ [0, 1) such that λ = exp(2πi(−a)). We set as well as One has corresponding notions for vanishing cycles These notions are related as follows (cf. loc.cit.): µ p x,a,l (V ) = ν p x,a,l (V ) if a = 0 and µ p x,0,l (V ) = ν p x,0,l+1 (V ).
Additionally to [4], we will use the following further local Hodge numbers, simplifying the computations below: cf. [4], (2.2.5*), and Let J p (a, l)(V ) denote a mixed C-Hodge structure which is associated to a nilpotent orbit belonging to a monodromy operator whose Jordan form is a single Jordan block of size l and having residue a ∈ [0, 1) such that ν p a,l−1 (V ) = 1.
(note that we use complex coefficients, so any pure Hodge structure decomposes into one-dimensional summands).
In the following, let j : A 1 x ֒→ P 1 be the natural inclusion. Using our above notion of ω i (V ) we obtain: Proof. By [4], Proposition 2.3.3, we have Remark. 1.6. The construction of nearby and vanishing cycles and their basic invariants is carried out for minimal extensions in [4], Section 2.2. The general case can be reduced with this at hand to the case of mixed Hodge-modules with punctual support. The Hodge invariants of these are as follows: Let V be a Hodge module supported on a closed point x i ֒→ A 1 (i.e., a complex polarized Hodge structure V placed at x) and let i + V its extension to A 1 . By the usual triangle which connects nearby and vanishing cycles, the nearby cycles of i + V are zero, while the vanishing cycles ϕ x (i + V ) can be identified with V. Note that the natural monodromy operation on ϕ x (i + V ) is trivial, hence µ p x,a (i + V ) = 0 for a = 0 and µ p x,0 (i + V ) = h p (V ). §2. Degrees of tensor products We will proceed using the notions of the previous section. The following theorem is a generalization of [4], Proposition 2.3.2. We are indepted to Claude Sabbah for communicating its proof to us.
The result depends on the following two lemmata. Let V 0 , L 0 , (V ⊗ L) 0 denote the Deligne extensions of V, L, V ⊗ L (resp.). There is also (V ⊗ L) 0 . We have the following Hodge filtrations: • Since V ⊗ L is a variation of Hodge structures on the punctured P 1 , with Hodge filtration equal to the tensor product filtration, we obtain the filtration F ℓ (V ⊗ L) 0 .
Let D = x ∪ y denote the reduced divisor away from which V and L are variations of Hodge structures. A local computation (without using Hodge theory) shows that there are F -filtered inclusions which are equalities away from D.
If this lemma is proved, we find that, for each ℓ, there is an injective morphism whose cokernel is supported on D and has dimension dim gr ℓ F (V ⊗ L) 0 /V 0 ⊗ L 0 . As a consequence, we find Proof of Lemma 2.2. The result is local, so the setting is on a small disc with coordinate t around one of the singularities of the variations of Hodge structures. The local computation mentioned above shows that there is an exact sequence giving rise to the exact sequence We also have the following Hodge filtration: • The tensor product filtration on any gr α V ⊗ gr β L considered above.
For the sake of simplicity, we will set Gr 0 V := V 0 /tV 0 (and similarly for L and V ⊗ L). This space is endowed with the induced filtration F • Gr 0 V . There is also a filtration E • Gr 0 V indexed by α ∈ [0, 1) induced by the decreasing V -filtration on Gr 0 V , so that gr α defined as follows: This morphism is compatible with the F -filtrations on each term. Grading with respect to E • gives a morphism α,β∈[0,1) The later morphism is also F -filtered, and is moreover a morphism of mixed Hodge structures. It is then F -strict. Therefore, (2.0.1) is also F -strict. Arguing similarly, we find that for any k 1 the natural morphism The previous result implies that the inclusion is strictly F -filtered, hence, regarding the previous morphism as an inclusion, By faithful flatness of O over O, we conclude that Proof of Lemma 2.3. We consider the composed F -filtered morphism After grading with respect to the E • filtration, it becomes and has image α,β∈(0,1) α+β 1 Being a morphism of mixed Hodge structures, it is also F -strict, and so is (2.0.2). Since the isomorphism t : V 0 → V 1 is F -strict (and similarly for L and V ⊗ L), the isomorphism t : is also F -strict. As a consequence, Transformation of local Hodge data away from ∞ under middle convolution Recall that in general, the convolution V ⋆ L is neither irreducible nor an intermediate extension anymore (cf. Assumption 1.2). The following definition gives the largest factor of V ⋆ L which is an intermediate extension: The following result clarifies the relation between V ⋆ L and V ⋆L : One has a short exact sequence of Hodge modules If H = 0 then p, c are uniquely determined.
Proof. As in the proof of [7], Proposition 2.6.9, one finds that H n (RH (V ⋆ † L)) vanishes outside n = −1, 0 and that if H 0 (RH (V ⋆ † L)) = 0, then there exists a unique point c ∈ A such that H 0 (RH (V ⋆ † L)) is a rank-one skyscraper sheaf with support at c such that the stalk at c is isomorphic to the Tate-twisted space of invariants Hom Since irreducible VPCHS are determined up to a Tate-twist by their local systems, there exists a unique p such that the Tate-twist V (p) becomes VPCHS-isomorphic to L (c − x) ∨ in this case. This implies that, taking Hodge structures into account, the stalk Hom The claim follows now from Proposition 1.5.
The following result determines the local Hodge data of the vanishing cycles: be a fixed element of the unit circle S 1 and let λ 1 , λ 2 be variable elements in where the expression ν p abbreviates dim gr p F .
Proof. By Saito's version of the Thom-Sebastiani theorem (cf. [4] Theorem 3.2.3 and its erratum) one knows that, for all (x i , t − y j ) as in the theorem, Moreover, the support of the vanishing cycles in the fibre over t is the union of these (x i , t − y j ). Since middle convolution is afterwards formed via higher direct image along the compactified (hence proper) pr 2 and since formation of vanishing cycles is compatible with higher direct images along projective morphisms the claim follows.
Corollary. 3.5. Let a ∈ [0, 1). Then the following holds: (ii) If L χ is generic with respect to L and V ⋆L, then Proof. Let us first treat the case where λ = 1, equivalent to a = 0 (note that inside Theorem 3.4 the residues a are contained in (0, 1], whereas in the rest of the paper a ∈ [0, 1), hence we have to adapt our notation to this situation). By Theorem 3.4 Note that in the above sum we switch from a i ∈ (0, 1] to a i ∈ [0, 1) so that the case a 1 + a 2 = 2 now corresponds to a 1 + a 2 = 0 (and l + k = p + 1). Analogously we get for 0 < a < 1 Hence the first claim follows. In the case where V ⋆ L = V ⋆L, the formula given in (ii) holds obviously true. By Theorem 3.2, if V ⋆ L = V ⋆L, then . This implies the first equation in (ii). Since L χ is generic, By associativity of the middle convolution (under the assumption that L, V, L χ are irreducible and nontrivial) where δ i,j denotes the usual Kronecker-delta. On the other hand, Rem. 1.6 and (3.0.2) imply that Theorem. 4.1. Let V ⋆ L χ be viewed as Hodge module on P 1 by taking the minimal extension using the canonical map A 1 t ֒→ P 1 . Let x 1 , . . . , x r ∈ A 1 (C) denote the finite singularities of V and let φ t : P 1 (C) → P 1 (C) be the Möbius transformation that exchanges 0 and ∞ by inverting the coordinate t of A 1 . If 0 is a smooth point of V then V ⋆ L χ can be obtained as Proof. We will view maps as φ x or φ t either as maps on P 1 or on G m , or even on smaller subsets of G m , depending on the context. Let U := A 1 (C) {0, x 1 , . . . , x r }, let W := U x × U t {x = t, 1 = xt} (with U x , U t denoting copies of U with coordinates x, t), let j : W ֒→ P 1 × U t denote the obvious embedding and let pr 2 : P 1 × U t → U t denote the second projection. Then, by construction, we have where V (x) is the Hodge module on W obtained by pulling back V | Ux along the x-projection and where, as above, the Hodge module L χ (t−x) is obtained by pulling back L χ along the map t−x. We use a similar notion below also for other sheaves and other polynomial maps, so that V (f (x, t)) denotes a pullback of V along a polynomial map given by f (x, t) ∈ C[x, t]. Then the following isomorphisms follow by looking at products of local sections (where the isomorphisms are seen as isomorphisms of the restrictions of Hodge modules to the respective smooth parts): where we use the following notions and arguments: Throughout we work over the largest smooth locus of the Hodge modules involved. Eq.
The following result was independently proven using different methods by Nicolas Martin in his Dissertation [8], Thm. 6.3.1:

Proposition. 4.3. (i) Let µ ∼ 1 (meaning that 1 − µ is chosen generically and small enough). Then
(ii) Let V ⋆ L χ = δ x for any x ∈ A 1 (equivalent to V being not isomorphic to a translate of the dual of L χ ). Then Moreover, the only other possibly non zero nearby cycle data at infinity of V ⋆ L χ are of the form ν i ∞,1−µ,0 (V ⋆ L χ ). If µ ∼ 1 then the formula simplifies to

Proof. By Theorem 3.3 and Theorem 2.1,
, which is the first formula in (i). The fixed space under the local monodromy at ∞ is trivial (otherwise, the last formula in the proof of Proposition 4.2 reads rk(H 1 par (V ⋆ L χ )) < 0, a contradiction). This implies ω i ∞ (V ⋆ L χ ) = h i (V ⋆ L χ ) which is the fourth equation in (i). By Proposition 4.2 V ⋆ L χ is parabolically rigid. Therefore, Therefore Using µ ∼ 1 and the second equality in Corollary 3.5(i) we obtain establishing the third equality in (i). Hence Summing up we get the second formula of (i): Claim (ii) follows by rewriting V ⋆ L χ via Theorem 4.1 as assuming via a translation that 0 is not a singularity of V . Thus the singularity ∞ of V becomes the finite singularity 0 of φ + x (V ⊗ L χ ) and we can apply Formula (3.0.1) in order to prove the first claim. Note that φ + x (V ⊗ L χ ) has scalar monodromy exp(−2πiµ) at ∞, which is called the standard situation for the middle convolution with the Kummer sheaf L χ in [4]. The first Formula in (ii) follows now from [4], Theorem 3.1.2(2). By Formula (4.0.9) and Theorem 3.4, the only other possibly non zero nearby cycle data at infinity are of the form ν i ∞,1−µ,0 (V ⋆ L χ ).
Proof. Let L χ be generic and µ ∼ 1. By Proposition 4.3(i) By the transformation of residues under convolution, described by Proposition 4.3(ii), the sum of a residue of L ⋆ L χ at ∞ and a residue of V at ∞ is not an integer. Hence the nearby cycles of V ⊗(L ⋆ L χ )(t − x) at ∞ coincide with the vanishing cycles. Therefore ).

One has
where we used the following arguments: the first equality uses the usual equality for tensor products, the second follows from basic properties of nearby cycles of tensor products, the third equality is a reorganisation of the sum and in the last equation we use the first and the third equation in Proposition 4.3(i). Summing up (4.0.10) yields where we use (4.0.11) for the first equality and Corollary 3.5(i) for the second. On the other hand, by the first equality in Proposition 4.3(i), Thus by Corollary 3.5(ii) and Eq. (4.0.12) By Theorem 2.1 and Proposition 4.3, Taking the previous two equations together one has as claimed. §5. Transformation of local Hodge data at ∞ under middle convolution In the following, the objects V, L, M satisfy the conditions in Assumption 1.2.
Theorem. 5.1. Let ε l := h l+1 (H 0 (H 0 (RH (V ⋆ L)))). Then Proof. By Theorem 1.5 Using subsequently Theorem 3.3, Theorem 2.1 and Theorem 1.5 we get By Theorem 4.4 Inserting the last two equations into the first equation we obtain By Corollary 3.5(ii), Moreover, by Corollary 3.5(i) Inserting this into the above equation yields as claimed.
The following result is also obtained in the Dissertation of Nicolas Martin [8], Thm. 6.3.1: Hence, since L χ is parabolically rigid, it follows from Theorem 5.1 that By definition of κ ∞ and the first formula in Proposition 4.3 (ii) Hence we obtain Remark. 5.3. The above theorem may also be derived from the well known formula )). In the last equation one uses the usual long exact cohomology sequence and The assumption that V ⊗L(t − x) has no unipotent Jordan block at ∞ implies that V is not dual to a translate of the form L(c − x). Hence V ⋆ L = V ⋆L by Theorem 3.2. Further V ⋆L is parabolically rigid without unipotent Jordan block at ∞ by Theorem 5.1, implying (i). By Theorem 5.1, Since by Remark 5.4 (ii), (V ⋆L) ⋆M = V ⋆(L ⋆M ) we deduce Remark. 5.6. Let m, n ∈ N >0 and let a m , b n ∈ R ∩ (0, 1). Proof. The tensor decomposition of J(n)⊗J(m) of the tensor product of two unipotent Jordan blocks of size n, resp. m, in characteristic zero is given by cf. Reference Chapter, Table 5, The stationary phase formula (cf. [10]), Theorem 5.1, implies that the Jordan blocks of M m ⋆N n at infinity are J(a m + b n , m + n − 1), J(a m + b n , m + n − 3), . . . , J(a m + b n , m + n + 1 − 2 min{m, n}) which are exactly the Jordan blocks of the tensor product J(a m , m)⊗J(b n , n) by Remark 5.6. The only possibility that the above derived Hodge numbers match this Jordan decomposition is given as follows: = J m+n−2 (a m + b n , m + n − 1) ⊕ · · · ⊕ J m+n−min{m,n} (a m + b n , m + n + 1 − 2 min{m, n}).
Proof. Assume first that V, L are parabolically rigid without unipotent Jordan block at ∞ such that V ⊗L(t − x) has also no unipotent Jordan block at ∞. By Corollary 5.8 there exists for each n, a n ∈ (0, 1) a parabolic rigid irreducible hypergeometric H m (a m ) such that ψ ∞ (H n (a n )) ≃ J n−1 (a n , n) and (V ⋆L) ⋆H n (a n ) = V ⋆(L ⋆H n (a n )).
Since these numbers determine uniquely the vanishing cycle structure of V ⋆L at infinity by Remark 5.6 we obtain using Corollary 5.8 In the general situation we proceed as follows: Let L χ1 , L χ2 be generic and µ 1 , µ 2 ∼ 1. Then V ⋆ L χ1 and L⋆L χ2 are parabolically rigid by Proposition 4.2 without unipotent Jordan block at ∞ by Proposition 4.3 Using commutativity and associativity of the middle convolution together with Theorem 3.2 one finds W ⋆ L χ1χ2 = (V ⋆ L)(−1).
By Proposition 4.3(ii) we deduce that a Jordan block J i (c, l) of ψ ∞ (W ) is transformed to a Jordan block of ψ ∞ (W ⋆ L χ1χ2 ) as follows: which implies the expression for ψ ∞ (V ⋆ L) = ψ ∞ (V ⋆L) in the theorem.