Irregular Hodge numbers of confluent hypergeometric differential equations

We give a formula computing the irregular Hodge numbers for a confluent hypergeometric differential equation.


Introduction
Differential equations with irregular singularities occur in various branches of Algebraic geometry, like mirror symmetry or the theory of exponential periods. They are also of interest as providing a complex analogue of -adic sheaves with wild ramification in positive characteristic. Irregular Hodge theory, as initiated by Deligne (see [Del07]), gives, for a large class of such equations, a convenient analogue to Hodge theory for Picard-Fuchs equations appearing more classically in complex Algebraic geometry. It is proved in [Sab18] that any rigid irreducible differential equation on the Riemann sphere, having regular singularities or not, and having real formal exponents at each singular point, underlies a variation of irregular Hodge structures away from its singular points. In this article, we consider the first and most classical example of such irregular differential equations, namely that of confluent hypergeometric differential equations, and we aim at determining the ranks of the irregular Hodge bundles. In the non-confluent case, the Hodge numbers of this variation, as well as the limiting Hodge numbers at the singularities, have been computed by R. Fedorov [Fed17], relying on [DS13].
We consider the possibly confluent hypergeometric differential equation with the usual convention that a product indexed by the empty set is equal to 1. We know that the associated meromorphic flat bundle H (α, β) on P 1 is irreducible (see [Kat90,Cor. 3 . If n = m, it has singularities at 0, 1, ∞, and they are regular. If n > m, it has an irregular singularity at t = ∞, a regular singularity at t = 0, and no other singularity. If n < m, the roles of 0 and ∞ are exchanged. Since the α i 's and β j 's are real, the local monodromy of H (α, β) at its regular singularities is unitary and the formal monodromy at its irregular singular point is also unitary. If n = m (regular singularities), there exists a variation of polarizable Hodge structure on C * that H (α, β) underlies, which is unique up to a shift of the Hodge filtration.
In this article, we consider the confluent case n > m (the case n < m can be obtained by a change of variable t → 1/t), so we fix two integers n > m 0 and we set µ = n − m > 0. Then H (α, β) has a regular singularity at t = 0, an irregular singularity of pure slope 1/µ at t = ∞, and no other singularity. By [Sab18, Th. 0.7], the minimal extension H min (α, β) at t = 0 of H (α, β) underlies a unique pure object T min (α, β) of the category IrrMHM(P 1 t ) of irregular mixed Hodge modules on P 1 , and it comes equipped with a irregular Hodge filtration. In contrast with the non-confluent case, this filtration is indexed by a set A + Z, where A is a finite set in R, and this filtration is unique up to a shift of A by a real number. We determine these numbers and their multiplicities in term of the pair (α, β).

Theorem. The jumps of the irregular Hodge filtration
and for any p ∈ R we have rk gr Remarks.
(i) Recall that the irregular Hodge filtration is unique up to a shift by a real number, so the formula (3) above is understood up to an R-shift of the filtration.
(ii) The statement and proof of the theorem hold under the assumption that n > m. However, the formula remains meaningful when m = n, and it gives back the formula of R. Fedorov

Fourier transforms of Kummer pullbacks of hypergeometrics
We recall here a useful result of N. Katz (see [Kat90, Th. 6.2.1]) which reduces the study of confluent hypergeometric differential equations to that of regular ones. Recall that we set µ := n − m. We denote by β the sequence of length n obtained by concatenating and reordering the sequences β and 0, 1/µ, . . . , (µ − 1)/µ. We also have 0 β 1 · · · β n < 1.
In the remaining part of this section, we will make the following assumptions, the first one being mainly for convenience.
The theorem of N. Katz relates the confluent H (α, β) to the non-confluent H (β, α). We recall below this correspondence.

Reduction of the proof of the theorem to the case where Assumption B is fulfilled
Recall that we assume that the pair (α, β) is non-resonant (Assumption A). Then for γ > 0 small enough, setting α i = γ + α i and β j = γ + β j , the sequences α , β are in (0, 1), increasing, and remains nonresonant. Moreover, one can choose γ such that the pair (α , β ) is non-resonant, i.e., Assumption B is fulfilled for it. Moreover, since the irregular Hodge filtration is defined up to an R-shift, we can add µγ to Formula (2). In order to reduce the proof of the theorem to the case where Assumption B is fulfilled, we apply the following general lemma to the case of the rank-one local system L on C * with monodromy exp(− 2πi γ) around 0. • a proper morphism f : X → P 1 , with X smooth projective, • a normal crossing divisor D in X and a subdivisor H ⊂ D, • a regular holonomic D X -module N underlying a mixed Hodge module, • a meromorphic function ϕ with poles in H, . By a suitable change of data as above, we can assume that the pole and zero divisors of ϕ + ψ • f do not intersect, and that D ∪ D 1 is a normal crossing divisor. Then (see loc. cit.) M is obtained as the image of The irregular Hodge filtration on M is obtained by pushing forward by f 0 † the irregular Hodge filtrations on E ϕ ⊗ Γ [!H] N and E ϕ ⊗ Γ [ * H] N, and by considering the image of it by the morphism above, and similarly for M . We notice that, away from Σ, both constructions possibly differ only because the irregular Hodge filtrations on N and f + L reg ⊗ N possibly differ. Since the only choice involved is that of the jumping index of the irregular Hodge filtration of L reg , which can be an arbitrary real number, they actually do not differ, and we obtain the desired result.

Nearby cycles for the Kummer pullback
We take up here the notation as in [DS13]. Let (V , F • V , ∇) be a filtered flat vector bundle on the punctured disc ∆ * x with coordinate x underlying a variation of polarized complex Hodge structure. We denote by V a the Deligne extension of V on ∆ x on which the residue of ∇ has eigenvalues in [a, a + 1), and we set 5. The filtered inverse stationary phase formula and irregular Hodge numbers 6 5. The filtered inverse stationary phase formula and irregular Hodge numbers V −∞ = a V a , which is a free O ∆ ( * 0)-module of finite rank. For any p ∈ Z we set F p V a = j * F p V ∩ V a , where j : ∆ * → ∆ denotes the inclusion. This is a locally free O ∆ -module and multiplication by x induces an isomorphism F p V a ∼ −→ F p V a+1 , so that for fixed p and a, gr p F gr a+k (V ) := F p V a+k /(F p+1 V a+k + F p V >a+k ) has dimension independent of k ∈ Z. For χ = exp − 2πi a with a ∈ R, we set For µ ∈ N * , let ρ : ∆ y → ∆ x be the cyclic ramification y → x = y µ of order µ. The data (ρ * V , ρ * F • V , ρ * ∇) underlies a variation of polarized complex Hodge structure.
Proof. Considering the germs at the origin, we have a natural identification with a natural structure on the right-hand side, which leads to and similarly The lemma follows.
Let us apply this formula to H ( * 0) defined by (4) and its pullback ι * H µ ( * 0). Assumption B is supposed to hold. We consider nearby cycles at x = 0 and τ = 0. We set χ k = exp(− 2πi α k ) and λ k = exp(− 2πi µα k ). The formula of [Fed17,Th. 3(b)] reads, since the nilpotent part for each eigenvalue of the monodromy of H ( * 0) at x = 0 consists of one Jordan block, otherwise.
Let us denote by {µα j } ∈ [0, 1) the fractional part of µα j (it belongs to (0, 1), due to Assumption B). Since ρ(j) = {µα j } + p j , the previous formula can be rewritten as and, after applying ι, it reads

The filtered inverse stationary phase formula and irregular Hodge numbers
By [Sab18,Prop. 2.61], the general fibre of the Laplace transform of a mixed Hodge module on A 1 carries a canonical irregular Hodge structure. We will give a formula for the irregular Hodge numbers in terms of the limit mixed Hodge structure at infinity of the mixed Hodge module. We start by recalling some of the results in [Sab10], since the way we formulate them is implicit in loc. cit.
Let (M, F • M) be a well-filtered regular holonomic C[τ] ∂ τ -module underlying a polarizable pure complex Hodge module. For the sake of simplicity, and since we will only use the result in this setting, we assume that the monodromy of M around τ = ∞ does not have 1 as an eigenvalue. We associate with (M, F • M) • the Rees module R F M, • the localized Laplace transform G, that we regard as a C[v, v −1 ]-module, and which is free of finite rank as such, • the Brieskorn lattice G 0 = G 0 (F) associated to the filtration, which is a free C[u]-module (u = v −1 ) with an action of u 2 ∂ u (see e.g. [SY15, App.]), • the Rees module R G (F) (G) attached to the decreasing filtration G p (F) = u p G 0 .
Let M be the D P 1 -module such that M = M( * ∞) and M = Γ (P 1 , M). Our assumption above implies that M is equal to its minimal extension at τ = ∞. We denote by τ = 1/τ the coordinate centered at ∞ and by V • M the V -filtration of M with respect to τ . For α ∈ R and λ = exp(− 2πi α), we set ψ τ ,λ M = gr α V M. The Hodge filtration F • M extends naturally to V α M, as indicated in Section 4. In such a way, (M, F • M) is strictly specializable at τ = 0. The space ψ τ ,λ M comes equipped with the induced filtration given by F • ψ τ ,λ M = F • gr α V M, from which we construct the Rees module R F ψ τ ,λ M. On the other hand, let V • G be the V -filtration of G with respect to the function v. We set similarly ψ v,λ G = gr α V G, and the filtration G • (F) induces on it the filtration G • (F) ψ v,λ G, form which we construct the Rees module R G (F) ψ v,λ G.