Recall that a DG-module over a DG Lie algebra is a module over a Lie algebra in the category of complexes rather then vector spaces. So a DG module over a DG Lie algebra (L•, ∂) is a complex M• plus an action of the graded Lie algebra L• on the graded object M• given by a map of complexes µ : L•⊗ M•→ M•.
Given two DG-modules M• and N•, Hom•L•(M•, N•) is a complex, consisting of elements of the complex Hom•(M•, N•) commuting with the action of L•. Taking H0’s of these Hom-complexes we arrive at the homotopy category of DG-modules. Here the morphisms are mor-phisms of L•-modules M• → N• commuting with the differentials in M• and N•, up to homo-topies given by morphisms of L•-modules M•→ N•[−1]. Localizing by the quasiisomorphisms, we get the derived category D(L•). One easily translates this to the world of DG Lie coalgebras.
Recall the DG Lie coalgebraL∗H;X in the pure category HodX defined in Section 1.9.
Definition 5.20. The DG Hodge category GH∗(X) of a regular complex projective variety X is the derived category D(L∗H;X) of DG comodules over the DG coalgebra L∗H;X in the pure category HodX.
The precise form of Conjecture 1.11 is
Conjecture 5.21. The category of smooth complexes of real Hodge sheaves on X is equivalent to the derived category D(L∗H;X).
Let us describe DG comodules over the DG Lie coalgebraL∗H;X. Consider the following data:
• A bounded complex V = {. . . −→ V1 −→ V∂ 2 −→ V∂ 3 −→ V∂ 4−→ . . .} in Hod∂ X.
• A Maurer-Cartan element:
G ∈ CH1(EndV) such that δG + G ∧ G = 0, (169) defined modulo elements of type δB + [B,G], where B ∈ CH0(EndV).
Here EndV = V∨⊗ V is a complex in HodX. The complexCH∗(EndV) is the total complex of the bicomplex CH•(EndV). So an element (169) includes a function on X with values in End0V, a 1-form with values in End−1V, etc. Thanks to Theorem 5.7 it provides the cohomology H∗(V) with a variation of R-MHS’s data.
Lemma 5.22. (i) DG comodules over the DG Lie coalgebra CH∗(EndV) are the same thing as Maurer-Cartan elements (169).
(ii) Homotopy equivalent DG comodules correspond to the equivalent Maurer-Cartan ele-ments.
Proof. (i) Neglecting the differential, the graded Lie coalgebra L∗H;X is a free Lie coalgebra cogenerated by the graded object DX[1] in the category HodX. Therefore a comodule over L∗H;X is determined by the coaction of the cogenerators. The latter is given by a graded object V plus a linear map V −→ V ⊗R DX[1]. It can be interpreted as a degree zero element in DX[1]⊗REndV, which is the same as an element G ∈ CH1(EndV).
ii) Similarly a homotopy is given by an element V −→ V ⊗RDX, which can be interpreted as an element B∈ CH0(EndV).
Assume that Lk = 0 for k > 0. Then subcategory of DG-modules concentrated in the degree zero is canonically identified with the category of modules over the Lie algebra H0(L•). Indeed, given such a module M0, Lk where k 6= 0 acts on it trivially since M0 a graded module over a graded Lie algebra. Further, since the action map µ is a map of complexes, ∂(L−1) acts on M0 trivially.
There is a similar picture for DG comodules over a DG Lie coalgebra concentrated in degrees
≥ 0, which we use below.
Proof of Lemma 5.6. Follows immediately from Lemma 5.22.
A DG-connection on X is a first order linear differential operator d +A where A is an EndV-valued differential form on X of total degree 1. A DG-connection is flat if (d + A)2 = 0.
Definition 5.23. Let V be a variation of real Hodge structures on X, and G ∈ CH1(EndV). The twistor DG-connection of G is a DG-connection on X given by ∇G := d + γ(G).
Proposition 5.24. A form G ∈ CH1(EndV) satisfies the Maurer-Cartan equation δG + G ∧ G = 0 if and only if the twistor DG-connection ∇G is flat.
Proof. Follows immediately from Theorem 1.9.
6 DG Lie algebra C
H,Sand L
∞-algebra of plane decorated trees
Let H∨ be a finite dimensional symplectic vector space, and S a finite set. Set
V∨H,S := H∨⊕ C[S]. (170)
Let CH,S∨ := CT(V∨H,S) be the cyclic envelope of the tensor algebra of V∨H,S. In Section 6.1 we define a Lie coalgebra structure on CH,S∨ . The coproduct generalizes the coproduct for the dihedral Lie coalgebras from [G4] – the latter corresponds to the H∨= 0 case. The Lie coalgebra structure on C∨H,S provides the dual vector space CH,S with a Lie algebra structure. The Lie algebra CH,S plays a key role in our story. We give a different interpretation of the Lie algebra CH,S in Section 8.
In Section 6.2 we define an L∞-coalgebra (TH,S∨,•[1], ∂) of the plane V∨H,S-decorated trees.
Its dual is an L∞-algebra TH,S• [−1] of plane VH,S-decorated trees.
The standard cochain complex of the L∞-algebraTH,S• [−1] is identified with a commutative differential graded algebra FH,S∨,• of plane V∨H,S-decorated forests:
S•(TH,S∨,•) =FH,S∨,•.
The L∞-coalgebra structure on TH,S∨,•[1] generalises both the Kontsevich-Boardman differ-ential ∂∆ in the graph complex, and the differential defined in [G5]. The differential ∂∆ is indentified with the differential on the TH,S∨,•[1]. Our definition was suggested by computation of the differential of the Hodge correlator in Theorem 7.11.
The sum over all plane trivalent trees with a given decoration gives rise to an injective map F :C∨H,S֒→ TH,S∨,1[1].
The L∞-algebra TH,S• [−1] with the differential ∂∆ is a resolution of the Lie algebraCH,S. So CH,S = H0(TH,S• [−1]).
6.1 A Lie algebra structure on CH,S
Let us define a cobracket δ :CH,S∨ −→ Λ2CH,S∨ . It is a sum of two maps:
δ = δCas+ δS.
Let us define these maps. We picture a cyclic word W on an oriented circle.
(i) The map δCas. Cut two different arcs of the circle. We get two semicircles. Make an oriented circle out of each of them. Each of them comes with a special point on it, obtained by gluing the ends of the semicircle. Decorate the special point of the first circle by αk, and the special point of the second circle by α∨k, take the sum over k, and put the sign. This is illustrated on Fig 8. The map δCas is obtained by taking the sum over all possible cuts.
Observe that the wedge product comes with the plus sign, and changing the order of the cuts (and hence the order of the terms in the wedge product) we do not change it, since if{αk}, {α∨k} is a basis and its symplectic dual, then the same is true for {−α∨k}, {αk}.
(ii) The map δS. Cut the circle at an arc and at an S-decorated point, which is not at the end of the arc, and make two circles as before. Their special points inherit an S-decoration from the S-decorated point, see Fig 9. Make the sum over all possible cuts.
This time the order of the terms in the wedge product matters. We put the semicircle which, going according to the orientation of the circle, has the S-cut at the end as the first term.
.
Figure 8: The component δCas of the differential.
.
Figure 9: The component δS of the differential.
Examples. One has
δC({s0} ⊗ {s1}) = C({s0} ⊗ αk)∧ C(α∨k ⊗ {s1}). (171) δC({s0} ⊗ {s1} ⊗ {s2}) =
Cycle0,1,2 C({s0} ⊗ {s1} ⊗ αk)∧ C(α∨k ⊗ {s2}) + C({s0} ⊗ {s1}) ∧ C({s1} ⊗ {s2}
. (172) where Cycle0,1,2 means the cyclic sum. See Fig 10 illustrating the last formula.
s1
Proposition 6.1 is deduced from the construction of the L∞-coalgebra of plane decorated trees, see the end of Section 6.
6.2 A commutative differential graded algebra of plane decorated forests