Kontsevich bundles via D-modules

In Part II, we use the setting and notation of §1.b. It will also be convenient to work algebraically with respect to P¹, in which case we will consider the DX[v]h∂vi-module E^vf(∗H) := OX(∗D)[v]·e^vf and the DX[u]h∂ui-module E^{f /u}(∗H) := OX(∗D)[u, u⁻¹]·

e^{f /u}, where e^vf and e^{f /u} are other notations for 1 which make clear the twist of the connection.

6.a. The Laplace Gauss-Manin bundles H^k(α). The bundles H^k(α) on P¹_vwill be obtained by gluing bundles on Cv and on Cu, that we describe below.

Over the chart Cv. Let us denote by H_v^k the restriction of H^k (see §1.c) to the v-chart. This is nothing but the Laplace transform of the (k − dim X)th Gauss-Manin system of f . It is known to have a regular singularity at v = 0 and no other singularity at finite distance (as follows by push-forward from the arguments recalled in §7.b, or by a general result about Laplace transform of regular holonomic D-modules in one variable). Moreover, H_v^k is equipped with the push-forward filtration F_•^irrH^k

v by O_C

v-coherent subsheaves, in a strict way according to Theorem 1.3. On C^∗_v we obtain in such a way a flat bundle (H_|C^k∗

v, ∇) equipped with a filtration F_•^irrH^k

|C^∗_v indexed by Q, which satisfies Griffiths transversality condition with respect to ∇ (see §7.f, see also Remark 6.3). This is the variation with respect to v of the irregular Hodge filtration of H^k(X, DR E^vf(∗H)).

We consider the limiting filtration (in the sense of Schmid) when v → 0. For α ∈ [0, 1), let us denote by VαH^k

v the αth term of the Kashiwara-Malgrange filtration of H_v^k at v = 0. Equivalently, due to the regularity property of the connection at v = 0, VαH^k

v is the Deligne extension of H_|C^k∗

v on which ∇ has a simple pole with residue Resv=0∇ having eigenvalues in [−α, −α + 1). We set gr^V_αH_v^k = VαH_v^k/V<αH_v^k.

Theorem 6.1. For each α ∈ [0, 1),

(1) the jumps β ∈ Q of the induced filtration

F_•^irrgr^V_αH_v^k:= F_•^irrH^k∩ VαH_v^k F_•^irrH^k∩ V<αH_v^k belong to α + Z,

(2) on each generalized eigenspace of Resv=0∇ acting on VαH_v^k/vVαH_v^k, the nilpotent part of the residue strictly shifts by one the filtration naturally induced by F_•^irrVαH^k

v .

Our proof in §7.i is obtained by showing (Proposition 7.19) that the conditions needed for applying M. Saito’s criterion [Sai88, Prop. 3.3.17] are fulfilled. More pre-cisely, we will work with a filtration F_•E^vf(∗H) easy to define, and we postpone to §9 the proof that this is indeed the irregular Hodge filtration of E^vf(∗H). It would also be possible, as observed by T. Mochizuki [Moc15], to directly refer to a similar property for twistor D-modules.

Over the chart Cu. Let us now consider the chart Cu. We denote by H_u^k the restric-tion of H^k to this chart. If j : Cur{0} ֒→ Cu denotes the open inclusion, we have H^k

u = j+H^k

|C^∗_v. There is a natural OC_u-lattice G0H^k

u of the free OC_u[u⁻¹]-module H^k

u called the Brieskorn lattice in analogy with the construction of Brieskorn in singu-larity theory [Bri70]. It can be defined in terms of the Hodge filtration of the Gauss-Manin system attached to f (see the appendix). It can also be defined (see §8.d) as the push-forward by q in a suitable sense of an OX×Cu(∗Pred)-module G0E^{f /u}(∗H) equipped with a u-connection ud + df : G0E^{f /u}(∗H) → G0E^{f /u}(∗H) ⊗ Ω¹_X and with a compatible action of u²∂u.

The connection on H^k has a pole of order at most two at u = 0 when restricted to G0H_u^k (see Remark 8.14). In the context of DX[u]h∂ui-modules E^{f /u}(∗H) corre-sponds to E^{f /u}(∗H) = OX(∗D)[u, u⁻¹]e^{f /u}.

Gluing. We can then glue G0H^k

u with VαH^k

v and obtain an OP¹-bundle H^k(α) with a connection having a pole of order one at v = 0 and of order two at u = 0.

6.b. The Kontsevich bundles K^k(α). We now consider the Kontsevich bundles introduced in §1.c. We can endow them with a natural meromorphic connection having a pole of order one at v = 0 and of order two at most at v = ∞, and no other pole.

In order to do so, we start⁽²⁾ by considering the morphism of complexes u²∂u− f : Ω^•_f(α)[u], ud + df

−→ Ω^•_f(α + 1)[u], ud + df
. Lemma 6.2. For α ∈ [0, 1), the natural inclusion of complexes

(Ω^•_f(α)[u], ud + df ) −→ (Ω^•_f(α + 1)[u], ud + df ) is a quasi-isomorphism.

This lemma allows us to define an action of u²∂u on each K^k(α)|Cu and therefore a meromorphic connection ∇ on K^k(α) with a pole of order at most two at u = 0.

We will show that ∇ has at most a simple pole at v = 0.

Remark 6.3 (due to T. Mochizuki). Let H be a vector bundle on P¹ equipped with a connection ∇ having a simple pole at v = 0 and a double pole (at most) at v = ∞. Then the Harder-Narasimhan filtration F^•H satisfies the Griffiths transver-sality property with respect to ∇.

Indeed, the property is obviously true with respect to the connection d on H coming from d on each summand in a Birkhoff-Grothendieck decomposition. We are thus reduced to proving a similar property for the O-linear morphism ∇ − d, and the result follows by noticing that (H /F^p−1H)⊗Ω¹P¹({v = 0}+2{u = 0}) has slopes < p while F^pH has slopes > p.

Proof of Lemma 6.2. We will show that the quotient complex has zero cohomology.

From [ESY15, Prop. 1.4.2] we know that the inclusion of complexes (Ω^•_f(α), df ) −→ (Ω^•_f(α + 1), df )

is a quasi-isomorphism, and thus the quotient complex (Q^•, df ) has zero cohomology.

Let ω = Pk

j=0ωju^j be a local section of Q^p[u] such that (ud + df )(ω) = 0. Then df ∧ω0= 0 and therefore there exists η0∈ Q^p−1such that ω = df ∧η0. By replacing ω with ω − (ud + df )η0 and iterating the process we can assume that ω = ωku^k and, dividing by u^k, that ω ∈ Q^p. It satisfies then dω = 0 and df ∧ω = 0, so ω = df ∧η for some η ∈ Q^p−1, and therefore df ∧ dη = 0. For any representativeη ∈ Ωe ^p−1_f (α + 1), we obtain

df ∧ dη ∈ Ωe ^p+1_f (α) ⊂ Ω^p+1_X (log D)([αP ]).

On the other hand, we note that

Ω^p−1_f (α + 1) = df ∧ Ω^p−2_X (log D)([αP ]) + Ω^p−1_X (log D)([αP ]),

so we can assume that eη ∈ Ω^p−1_X (log D)([αP ]). Then dη ∈ Ωe ^p_X(log D)([αP ]), and therefore dη ∈ Ωe ^p_f(α), that is, dη = 0, so ω = (ud + df )η.

(2)This was suggested to us by T. Mochizuki.

Proof of Theorem 1.11. We will compare the filtered complex σ^>p(Ω^•_f(α)[v], d + vdf ) with the filtered relative de Rham complex of E^vf(∗H) with respect to the projection to Cv. We introduce in §7.c a filtration F_α^•E^vf(∗H), which will be shown to coincide with F_α^irr,•E^vf(∗H) in Theorem 9.1.

Theorem 6.4 (see §7.g, modulo Th. 9.1). There is a natural quasi-isomorphism of filtered complexes

(OX×Cv ⊗O_X[v]Ω^•_f(α)[v], d + vdf, σ^>p) −→ (DRX×Cv/CvVαE^vf(∗H), F_α^irr,p) which is compatible with the meromorphic action of ∇∂v.

It follows from (1.9) that applying Rq∗ to the filtered complex on the right-hand side gives a strict complex (i.e., we have a similar injectivity statement).

We apply R^kq∗ to the quasi-isomorphism of Theorem 6.4. The non-filtered state-ment gives the first point of Theorem 1.11, since Vαis compatible with proper push-forward. The second point is then obtained by applying the second point of Theo-rem 6.1.

In a way similar to Theorem 6.4, but algebraically with respect to u, we in-troduce in §8.b a filtration F_α^•G0E^{f /u}(∗H), which will be shown to coincide with F_α^irr,•G0E^{f /u}(∗H) in Theorem 9.1, and we prove:

Theorem 6.5 (see §8.c, modulo Th. 9.1). There is a natural quasi-isomorphism of filtered complexes

(Ω^•_f(α)[u], ud + df, σ^>p) −→ (DRXG0E^{f /u}(∗H), F_α^irr,p) which is compatible with the action of ∇∂u.

As above, it follows from (1.9) that applying Rq∗ to the filtered complex on the right-hand side gives a strict complex.

By applying a degeneration statement similar to that of [Sai88, Prop. 3.3.17]

proved in the appendix, we obtain a concrete description of the irregular Hodge fil-tration of H^k.

Corollary 6.6. The isomorphism K^k(α)−→ H^{∼ k}(α) obtained by pushing forward the quasi-isomorphisms of Theorems 6.4 and 6.5 identifies the Harder-Narasimhan filtra-tion of K^k(α) (hence of H^k(α)) with the image on H^k(α) of the irregular Hodge filtration F^irrH^k.

Remark 6.7. Another proof of Theorem 1.11 has recently been given by T. Mochizuki [Moc15], by showing an analogue of Theorem 6.4 in the framework of mixed twistor D-modules, but not referring explicitly to the irregular Hodge filtration.