Compat: small MForm helpers needed by residue-theorem (candidates for canonical-forms) #
Unit: residue-theorem (docs/design/residue-theorem.md). Per CONVENTIONS.md's Compat rule and
the task brief's file discipline, these three one-step corollaries of canonical-forms' frozen
quotient API live in a clearly-marked NEW file rather than editing Jacobian/CanonicalForms/:
RS.MForm.resAt_eq_zero_of_ord_nonneg— residues vanish where the order is nonnegative (the §1.2 gap flagged in the residue-theorem design;RS.resAt_of_order_nonnegcomposed with the definition ofMForm.resAt).RS.MForm.finite_setOf_ord_neg— the pole set of a meromorphic 1-form on a compact surface is finite (via the divisor's finite support; no connectedness needed).RS.MForm.finite_support_resAt— henceresAthas finite support.
Request filed in docs/requests/canonical-forms.md-spirit: these belong upstream eventually.
theorem
RS.MForm.resAt_eq_zero_of_ord_nonneg
{X : Type u_1}
[TopologicalSpace X]
[ChartedSpace ℂ X]
[IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X]
{Θ : MForm X}
{x : X}
(h : 0 ≤ Θ.ord x)
:
The residue of a meromorphic 1-form vanishes at any point of nonnegative order.
theorem
RS.MForm.finite_setOf_ord_neg
{X : Type u_1}
[TopologicalSpace X]
[ChartedSpace ℂ X]
[IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X]
[T1Space X]
[T2Space X]
[CompactSpace X]
(Θ : MForm X)
:
The pole set of a meromorphic 1-form on a compact surface is finite (via the divisor's finite support; connectedness-free).
theorem
RS.MForm.finite_support_resAt
{X : Type u_1}
[TopologicalSpace X]
[ChartedSpace ℂ X]
[IsManifold (modelWithCornersSelf ℂ ℂ) ⊤ X]
[T1Space X]
[T2Space X]
[CompactSpace X]
(Θ : MForm X)
:
(Function.support fun (x : X) => Θ.resAt x).Finite
MForm.resAt has finite support on a compact surface.