The Weil operator of a pure Hodge structure #
The Weil operator C of a pure Hodge structure of weight n is the complex-linear automorphism
acting on the Hodge component H^{p,q} by the scalar i^{p-q}. Since q = n - p on the
component piece p, that scalar is i^{2p-n}, and the Hodge decomposition
TauCeti.Hodge.HodgeStructureOn.isInternal_piece extends it to the whole ambient space.
Two identities make C the carrier of the Hermitian theory. It squares to the weight sign,
C ∘ C = (-1)^n, so C is an automorphism, and is a complex structure in odd weight.
And it commutes with the conjugation, so it is defined over the real form: this is what allows a
symmetric bilinear form Q to be turned into the Hermitian form h(u, v) = Q(C u, conj v) on all
of the ambient space, rather than only on homogeneous vectors where i^{p-q} makes sense.
Main declarations #
TauCeti.Hodge.HodgeStructureOn.weilOperator: the Weil operatorC.TauCeti.Hodge.HodgeStructureOn.weilOperator_apply_of_mem:Cacts onH^{p,n-p}byi^{2p-n}.TauCeti.Hodge.HodgeStructureOn.weilOperator_unique: that action characterizesC.TauCeti.Hodge.HodgeStructureOn.weilOperator_piece,…_F,…_conjF:Cpreserves the Hodge components and both filtrations.TauCeti.Hodge.HodgeStructureOn.eigenspace_weilOperator: theμ-eigenspace ofCis spanned by the Hodge components on whichi^{2p-n} = μ.TauCeti.Hodge.HodgeStructureOn.weilOperator_comp_weilOperator:C ∘ C = (-1)^n.TauCeti.Hodge.HodgeStructureOn.weilOperatorEquiv:Cbundled as a linear automorphism.TauCeti.Hodge.HodgeStructureOn.weilOperator_comap:Cis compatible with transport of Hodge structures along equivalences.TauCeti.Hodge.HodgeStructureOn.conj_weilOperator:Ccommutes with the conjugation.TauCeti.Hodge.HodgeStructureOn.realAlmostComplexStructure: in odd weight,Crestricted to the real form as an almost complex structure, together with its scalar-extension comparison.TauCeti.Hodge.HodgeStructureOn.almostComplexStructure: on a literal complexification in odd weight, the almost complex structure whose scalar extension isC.TauCeti.Hodge.IsPolarization.isOrthogonal_weilOperator:Cis an isometry of a complexified polarizing form.TauCeti.Hodge.HodgeStructure.Hom.commutes_weilOperator: morphisms commute withC.TauCeti.Hodge.tate_weilOperator: the Weil operator ofℤ(m)is the identity.
This supplies the Weil operator targeted in Layer L1 of
TauCetiRoadmap/HodgeStructures/README.md. The sign conventions are those of Voisin, Hodge
Theory and Complex Algebraic Geometry I, §7.1.2, and Peters–Steenbrink, Mixed Hodge
Structures, §2.
The Weil operator of a pure Hodge structure of weight n: the complex-linear map acting
on the Hodge component H^{p,q} by i^{p-q} = i^{2p-n}.
Instances For
The Weil operator acts on the Hodge component H^{p,n-p} by the scalar i^{2p-n}.
The Weil operator is the only complex-linear map acting on each Hodge component H^{p,n-p}
by i^{2p-n}.
The Weil operator preserves every Hodge component.
The Weil operator restricts to an automorphism of every Hodge component.
The Weil operator preserves every step of the Hodge filtration.
The Weil operator preserves every step of the conjugate Hodge filtration.
Elementwise form of preservation of the Hodge filtration.
Elementwise form of preservation of the conjugate Hodge filtration.
The Hodge projections intertwine the Weil operator with multiplication by i^{2p-n}.
The eigenspaces of the Weil operator. The μ-eigenspace of the Weil operator is spanned
by the Hodge components H^{p,n-p} on which it acts by i^{2p-n} = μ.
The Weil operator squares to the weight sign: C ∘ C = (-1)^n.
The Weil operator squares to -1 in odd weight, where it is therefore a complex structure.
The Weil operator is an involution in even weight.
The Weil operator is inverted by (-1)^n times itself, on the right.
The Weil operator is inverted by (-1)^n times itself, on the left.
The Weil operator, bundled as a complex-linear automorphism of the ambient space.
Instances For
The bundled Weil operator acts as the Weil operator.
The inverse of the bundled Weil operator is (-1)^n times the Weil operator.
The Weil operator is bijective.
The Weil operator of a Hodge structure transported along an equivalence intertwining the conjugations is the transported Weil operator.
In weight one the Weil operator acts on H^{1,0} by i. Together with the next lemma this
identifies it with the complex structure of an effective weight-one Hodge structure.
In weight one the Weil operator acts on H^{0,1} by -i.
The Weil operator is real: it commutes with the conjugation of the Hodge structure. This
is what lets the Hermitian form Q(C u, conj v) be assembled from the scalars i^{p-q}.
The Weil operator of an odd-weight Hodge structure, restricted to the real form fixed by its
conjugation. It squares to -1, so it is an almost complex structure on that real vector space.
Instances For
The almost complex structure on the real form acts by the ambient Weil operator.
The real almost complex structure complexifies to the Weil operator. Under the canonical
equivalence ℂ ⊗[ℝ] V_ℝ ≃ₗ[ℂ] W, extending J from the real form sends the same vectors to the
same place as the Weil operator on W.
The complexification comparison as an identity of complex-linear maps.
Odd-weight Hodge structures on a complexification #
The almost complex structure on a real vector space U determined by a Hodge structure of odd
weight on its complexification ℂ ⊗[ℝ] U. The Weil operator commutes with conjugation, so it
preserves the real vectors 1 ⊗ₜ u, and it squares to -1 in odd weight.
Instances For
On a real vector 1 ⊗ₜ u the Weil operator acts through the induced almost complex
structure.
The induced almost complex structure complexifies to the Weil operator.
Compatibility with polarizing forms #
The Weil operator is an isometry of the polarizing form. Two Hodge components pair to
zero unless their degrees add up to the weight, and on a pair of components that do, the two
scalars i^(2p-n) and i^(2p'-n) by which the Weil operator acts are inverse to each other.
The Weil operator is an isometry of the complex form of a polarization.
A morphism of pure Hodge structures commutes with the Weil operators: it preserves every Hodge component, on which both operators are the same scalar.
The Weil operator of the Tate structure ℤ(m) is the identity: its single Hodge component
has p = q = -m, where the scalar i^{p-q} is one.