Deligne's bigrading of a mixed Hodge structure #
Deligne's canonical bigrading of a mixed Hodge structure is the family of complex subspaces
I^{p,q} = (F^p ∩ W_{p+q}) ∩ (conj F^q ∩ W_{p+q} + ∑_{j ≥ 2} conj F^{q-j+1} ∩ W_{p+q-j}),
built from the Hodge filtration, its conjugate, and the complexified weight filtration. It is the working tool of the mixed theory: strictness of a morphism of mixed Hodge structures is proved through it, and it is what the theory of variations consumes. Accordingly it is defined here by Deligne's closed formula rather than obtained from an existence statement, so that lemmas can be stated about it.
This file constructs the bigrading and establishes the part of its theory that follows from the formula itself:
- the containments
I^{p,q} ≤ F^p,I^{p,q} ≤ W_{p+q}andI^{p,q} ≤ conj F^q ⊔ W_{p+q-2}— soI^{p,q}lies in the Hodge step and weight step of its bidegree, and moduloW_{p+q-2}inconj F^q; - the vanishing of
I^{p,q}outside the range fixed by the boundedness of the two filtrations; - the conjugate of
I^{p,q}, which is the same formula with the Hodge filtration and its conjugate exchanged; - functoriality: a morphism of mixed Hodge structures carries
I^{p,q}intoI^{p,q}; - the computation of the bigrading of a pure Hodge structure viewed as a mixed one: it is
concentrated on the antidiagonal
p + q = n, where it is the Hodge componentH^{p,q}.
That ⨁_{p,q} I^{p,q} is the whole space, and the recovery W_k = ⨆_{p+q ≤ k} I^{p,q} of the
weight filtration, are proved in TauCeti/Geometry/Hodge/Mixed/Decomposition.lean, which needs
the pure Hodge structure carried by the graded pieces and so cannot be part of this file. What
remains of Deligne's theorem after that module is the conjugation symmetry
I^{p,q} ≡ conj I^{q,p} modulo lower bidegree.
Main declarations #
TauCeti.Hodge.MixedHodgeStructure.deligneSplitting: the bigradingI^{p,q}.TauCeti.Hodge.MixedHodgeStructure.deligneSplitting_le_conjF_sup_WC: the pieceI^{p,q}lies inconj F^qmodulo the weight filtration two steps down.TauCeti.Hodge.MixedHodgeStructure.map_latticeConj_deligneSplitting: the conjugate of a bigrading piece.TauCeti.Hodge.MixedHodgeStructure.Hom.map_deligneSplitting_le: functoriality.TauCeti.Hodge.MixedHodgeStructure.ofPure_deligneSplitting_eq_piece_of_add_eqand…deligneSplitting_eq_bot_of_add_ne: the bigrading of a pure Hodge structure.
References #
Deligne, Théorie de Hodge II, 1.2.10 and 2.3.5; Peters–Steenbrink, Mixed Hodge Structures, Lemma-Definition 3.4 and Ch. 3. The formula and the conventions are those fixed for Layer L2 of the Hodge structures roadmap.
The bigrading #
Deligne's bigrading of a mixed Hodge structure:
I^{p,q} = (F^p ∩ W_{p+q}) ∩ (conj F^q ∩ W_{p+q} + ∑_{j ≥ 2} conj F^{q-j+1} ∩ W_{p+q-j}),
with the tail sum indexed here by j - 2 : ℕ.
Instances For
Deligne's closed formula for the bigrading.
Membership in Deligne's bigrading, expanded into its three defining conditions.
Vectors lying simultaneously in F^p, conj F^q, and W_{p+q} lie in the corresponding
Deligne bigrading piece.
Deligne's bigrading repackaged as a single family indexed by bidegrees, the form in which
DirectSum.IsInternal consumes it.
Instances For
The bidegree-indexed family is Deligne's bigrading.
A bigrading piece lies in the Hodge filtration step of its first index.
A bigrading piece lies in the weight filtration step of its total degree.
After shifting by m, a Deligne bigrading piece lies in conj F^{q-m} modulo the weight
filtration step W_{p+q-m-2}.
Modulo the weight filtration two steps below its total degree, a bigrading piece lies in the conjugate Hodge filtration step of its second index.
If the weight filtration vanishes two steps below the total degree, Deligne's formula collapses to the intersection of its three leading filtration steps.
If W_m = 0 with m ≤ p + q - 2, then
I^{p,q} ≤ F^p ∩ conj F^{m+2-p}.
Above a Hodge filtration index at which the filtration vanishes, the bigrading vanishes: the hypothesis bounds the first index.
Below a weight filtration index at which the filtration vanishes, the bigrading vanishes.
The conjugate of a bigrading piece is given by Deligne's formula with the Hodge filtration and its conjugate exchanged.
Functoriality #
Functoriality of Deligne's bigrading. A morphism of mixed Hodge structures carries the
bigrading piece I^{p,q} of its source into the bigrading piece I^{p,q} of its target.
Elementwise form of functoriality of Deligne's bigrading.
The bigrading of a pure Hodge structure #
On the antidiagonal the Deligne bigrading of a pure Hodge structure is its Hodge
decomposition: for p + q = n the piece I^{p,q} is the Hodge component H^{p,q}.
Off the antidiagonal the Deligne bigrading of a pure Hodge structure vanishes. Below it the weight filtration is zero; above it the defining intersection is cut out by two complementary filtration steps.