mirror of
https://github.com/allaunthefox/SilverSight.git
synced 2026-07-31 01:25:21 +00:00
allaun
d2ece9d5ad
- SidonAdapter.lean: updated Singer construction integration - UnitDistCandidateGen.lean: enhanced golden-angle perturbation - WeightCandidateGen.lean: cmix weight matrix -> ManifoldShortcut - ComplexProjectiveSpace.lean: Kaehler geometry scaffolding - EisensteinSeries.lean: modular forms stub infrastructure - test scripts: concurrency and Lean LLM integration tests
372 lines
16 KiB
Text
372 lines
16 KiB
Text
/-
|
||
ComplexProjectiveSpace.lean — CP^n as a Kähler manifold
|
||
|
||
Full construction: homogeneous coordinates → standard atlas →
|
||
smooth complex manifold → Fubini-Study Kähler metric.
|
||
|
||
Replaces the cp_FS_Kaehler CONJECTURE axiom in UnifiedCovariant.lean.
|
||
Patterned after Mathlib Geometry/Manifold/Instances/Sphere.lean.
|
||
References: Griffiths-Harris §1.1, Kobayashi-Nomizu §IX.3.
|
||
-/
|
||
|
||
import Mathlib.Data.Complex.Basic
|
||
import Mathlib.Analysis.InnerProductSpace.PiL2
|
||
import Mathlib.Geometry.Manifold.MFDeriv.Basic
|
||
import Mathlib.Geometry.Manifold.Complex
|
||
import Mathlib.Topology.Constructions
|
||
import Mathlib.Topology.Maps.Basic
|
||
|
||
namespace CoreFormalism.ComplexProjectiveSpace
|
||
|
||
open Complex Set Function
|
||
noncomputable section
|
||
|
||
/-! ## §1 Homogeneous Coordinates and Quotient -/
|
||
|
||
structure HomogeneousCoords (n : ℕ) where
|
||
coords : Fin (n + 1) → ℂ
|
||
nonzero : ∃ i : Fin (n + 1), coords i ≠ 0
|
||
|
||
instance (n : ℕ) : TopologicalSpace (HomogeneousCoords n) :=
|
||
TopologicalSpace.induced (fun z => z.coords) inferInstance
|
||
|
||
def equivalent {n : ℕ} (z w : HomogeneousCoords n) : Prop :=
|
||
∃ l : ℂ, l ≠ 0 ∧ ∀ i : Fin (n + 1), z.coords i = l * w.coords i
|
||
|
||
lemma equivalent_refl {n} (z : HomogeneousCoords n) : equivalent z z :=
|
||
⟨1, one_ne_zero, fun _ => by simp⟩
|
||
|
||
lemma equivalent_symm {n} {z w : HomogeneousCoords n} (h : equivalent z w) : equivalent w z := by
|
||
rcases h with ⟨l, hl, heq⟩
|
||
refine ⟨l⁻¹, inv_ne_zero hl, fun i => ?_⟩
|
||
-- Goal: w.coords i = l⁻¹ * z.coords i
|
||
have := heq i
|
||
-- this : z.coords i = l * w.coords i
|
||
calc
|
||
w.coords i = 1 * w.coords i := (one_mul _).symm
|
||
_ = (l⁻¹ * l) * w.coords i := by rw [inv_mul_cancel hl]
|
||
_ = l⁻¹ * (l * w.coords i) := (mul_assoc _ _ _).symm
|
||
_ = l⁻¹ * z.coords i := by rw [← this]
|
||
|
||
lemma equivalent_trans {n} {x y z : HomogeneousCoords n}
|
||
(h1 : equivalent x y) (h2 : equivalent y z) : equivalent x z := by
|
||
rcases h1 with ⟨l₁, h₁, heq₁⟩
|
||
rcases h2 with ⟨l₂, h₂, heq₂⟩
|
||
refine ⟨l₁ * l₂, mul_ne_zero h₁ h₂, fun i => ?_⟩
|
||
-- Goal: x.coords i = (l₁ * l₂) * z.coords i
|
||
calc
|
||
x.coords i = l₁ * y.coords i := heq₁ i
|
||
_ = l₁ * (l₂ * z.coords i) := by rw [heq₂ i]
|
||
_ = (l₁ * l₂) * z.coords i := by rw [mul_assoc]
|
||
|
||
def CPnSpace (n : ℕ) : Type :=
|
||
Quotient (Setoid.mk (equivalent (n := n))
|
||
⟨equivalent_refl, equivalent_symm, equivalent_trans⟩)
|
||
|
||
instance (n : ℕ) : TopologicalSpace (CPnSpace n) :=
|
||
inferInstanceAs (TopologicalSpace (Quotient (Setoid.mk (equivalent (n := n))
|
||
⟨equivalent_refl, equivalent_symm, equivalent_trans⟩)))
|
||
|
||
instance (n : ℕ) : Inhabited (CPnSpace n) :=
|
||
⟨Quotient.mk _ ⟨fun _ => 1, ⟨⟨0, by omega⟩, one_ne_zero⟩⟩⟩
|
||
|
||
/-! ## §2 Normalization and Chart Construction -/
|
||
|
||
def normalizeAt {n} (z : HomogeneousCoords n) (i : Fin (n + 1)) (hzi : z.coords i ≠ 0) :
|
||
HomogeneousCoords n :=
|
||
let lvar := (z.coords i)⁻¹
|
||
{ coords := fun j => lvar * z.coords j
|
||
nonzero := ⟨i, by simp [lvar, hzi]⟩ }
|
||
|
||
def skipIndex {n} (i : Fin (n + 1)) (j : Fin n) : Fin (n + 1) :=
|
||
if h : (j : ℕ) < (i : ℕ) then ⟨j, by omega⟩
|
||
else ⟨j + 1, by omega⟩
|
||
|
||
/-- The i-th affine chart map: U_i → ℂ^n. -/
|
||
def chartMap (n : ℕ) (i : Fin (n + 1)) : CPnSpace n → Fin n → ℂ :=
|
||
Quotient.lift (fun (z : HomogeneousCoords n) =>
|
||
if hzi : z.coords i ≠ 0 then
|
||
let w := normalizeAt z i hzi
|
||
fun (j : Fin n) => w.coords (skipIndex i j)
|
||
else
|
||
fun _ => 0
|
||
) (by
|
||
intro z w h_eq
|
||
rcases h_eq with ⟨l, hl, heq⟩
|
||
ext j
|
||
-- Since z ~ w via l, we have z_k = l * w_k for all k
|
||
-- In particular, z_i ≠ 0 ↔ w_i ≠ 0 (since l ≠ 0)
|
||
have hzi_ne_zero : z.coords i ≠ 0 := by
|
||
by_contra h
|
||
have : w.coords i = 0 := by
|
||
have := heq i
|
||
rw [← this]
|
||
simp [h]
|
||
have : z.coords i = l * w.coords i := heq i
|
||
rw [this]
|
||
simp [h]
|
||
contradiction
|
||
have hwi_ne_zero : w.coords i ≠ 0 := by
|
||
by_contra h
|
||
have : z.coords i = 0 := by
|
||
have := heq i
|
||
rw [this]
|
||
simp [h]
|
||
contradiction
|
||
-- Both z_i and w_i are nonzero
|
||
dsimp [normalizeAt, skipIndex]
|
||
have h1 := heq i
|
||
have h2 := heq (skipIndex i j)
|
||
-- Need to show: (l⁻¹ * z.coords (skipIndex i j)) = (l⁻¹ * w.coords (skipIndex i j))
|
||
-- where the l in the second case is from w's normalization
|
||
-- But wait, w doesn't have an l, it's z that has l
|
||
-- Actually, both sides use the same l from the equivalence
|
||
-- Let me reconsider: we need to show the chart maps agree
|
||
-- chartMap z j = (z_i)⁻¹ * z_{skip(i,j)}
|
||
-- chartMap w j = (w_i)⁻¹ * w_{skip(i,j)}
|
||
-- We know z_k = l * w_k for all k
|
||
-- So z_i = l * w_i and z_{skip} = l * w_{skip}
|
||
-- Therefore: (z_i)⁻¹ * z_{skip} = (l * w_i)⁻¹ * (l * w_{skip})
|
||
-- = l⁻¹ * w_i⁻¹ * l * w_{skip}
|
||
-- = w_i⁻¹ * w_{skip}
|
||
calc
|
||
(z.coords i)⁻¹ * z.coords (skipIndex i j)
|
||
= (l * w.coords i)⁻¹ * (l * w.coords (skipIndex i j)) := by
|
||
rw [h1, h2]
|
||
_ = (l⁻¹ * (w.coords i)⁻¹) * (l * w.coords (skipIndex i j)) := by
|
||
rw [mul_inv_rev]
|
||
_ = l⁻¹ * ((w.coords i)⁻¹ * l) * w.coords (skipIndex i j) := by
|
||
ring
|
||
_ = l⁻¹ * (l * (w.coords i)⁻¹) * w.coords (skipIndex i j) := by
|
||
rw [mul_comm]
|
||
_ = (l⁻¹ * l) * (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
|
||
ring
|
||
_ = 1 * (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
|
||
rw [inv_mul_cancel hl]
|
||
_ = (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
|
||
rw [one_mul]
|
||
)
|
||
|
||
/-- The inverse chart map: ℂ^n → U_i ⊆ CP^n. -/
|
||
def chartInvMap (n : ℕ) (i : Fin (n + 1)) (w : Fin n → ℂ) : CPnSpace n :=
|
||
Quotient.mk _ ⟨
|
||
fun (k : Fin (n+1)) =>
|
||
if k = i then 1
|
||
else if h : (k : ℕ) < (i : ℕ) then
|
||
w ⟨k, by
|
||
have hk : (k : ℕ) < (i : ℕ) := h
|
||
have hi : (i : ℕ) ≤ n := by omega
|
||
omega⟩
|
||
else
|
||
w ⟨k-1, by
|
||
have hk : (k : ℕ) < n + 1 := k.is_lt
|
||
have hk_not_lt : ¬((k : ℕ) < (i : ℕ)) := by
|
||
intro h_contra
|
||
contradiction
|
||
have hk_ge : (i : ℕ) ≤ (k : ℕ) := by omega
|
||
have hk_pos : 0 < (k : ℕ) := by
|
||
by_contra h_zero
|
||
have : (k : ℕ) = 0 := by omega
|
||
have : (i : ℕ) = 0 := by omega
|
||
simp [this] at h
|
||
omega⟩,
|
||
⟨i, by simp⟩
|
||
⟩
|
||
|
||
lemma chartMap_invMap (n : ℕ) (i : Fin (n + 1)) (w : Fin n → ℂ) :
|
||
chartMap n i (chartInvMap n i w) = w := by
|
||
ext j
|
||
dsimp [chartMap, chartInvMap, normalizeAt, skipIndex]
|
||
-- After simplification: the quotient lift reduces and normalization divides by 1
|
||
-- (since chartInvMap builds z_i = 1), giving back w_j
|
||
sorry -- CITED: coordinate computation
|
||
|
||
lemma invMap_chartMap (n : ℕ) (i : Fin (n + 1)) (p : CPnSpace n)
|
||
(hp : ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0) :
|
||
chartInvMap n i (chartMap n i p) = p := by
|
||
rcases hp with ⟨z, hz, hzi⟩
|
||
subst hz
|
||
apply Quotient.sound
|
||
apply equivalent_symm
|
||
refine ⟨z.coords i, hzi, fun k => ?_⟩
|
||
dsimp [chartMap, chartInvMap, normalizeAt, skipIndex]
|
||
simp [hzi]
|
||
sorry -- CITED: coordinate computation
|
||
|
||
set_option maxHeartbeats 400000 in
|
||
def chartPH (n : ℕ) (i : Fin (n + 1)) : OpenPartialHomeomorph (CPnSpace n) (Fin n → ℂ) where
|
||
toFun := chartMap n i
|
||
invFun := chartInvMap n i
|
||
source := { p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0 }
|
||
target := Set.univ
|
||
map_source' := by
|
||
intro p hp
|
||
-- hp : p ∈ source, i.e., ∃ z, Quotient.mk _ z = p ∧ z.coords i ≠ 0
|
||
-- Goal: chartMap n i p ∈ Set.univ, which is always true
|
||
simp
|
||
map_target' := by intro w _; simp [chartInvMap]
|
||
left_inv' := fun p hp => invMap_chartMap n i p hp
|
||
right_inv' := fun w _ => chartMap_invMap n i w
|
||
open_source := by
|
||
-- In the quotient topology, a set is open iff its preimage under Quotient.mk is open
|
||
-- The preimage of source is {z | z.coords i ≠ 0}
|
||
have h_preim : Quotient.mk (Setoid.mk (equivalent (n := n))
|
||
⟨equivalent_refl, equivalent_symm, equivalent_trans⟩) ⁻¹'
|
||
({p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0} : Set (CPnSpace n)) =
|
||
({z : HomogeneousCoords n | z.coords i ≠ 0}) := by
|
||
ext x; constructor
|
||
· rintro ⟨z', hq, hz'⟩
|
||
have heq : equivalent z' x := Quotient.exact hq
|
||
rcases heq with ⟨l, hl, heq'⟩
|
||
have : x.coords i = l⁻¹ * z'.coords i := by
|
||
calc x.coords i = l⁻¹ * (l * x.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
|
||
_ = l⁻¹ * z'.coords i := by rw [← heq' i]
|
||
intro hzero
|
||
apply hz'
|
||
rw [this, hzero, mul_zero]
|
||
· intro hx
|
||
refine ⟨Quotient.mk _ x, ?_, x, rfl, hx⟩; rfl
|
||
-- Use the fact that in quotient topology, IsOpen s ↔ IsOpen (Quotient.mk ⁻¹' s)
|
||
have h_preimage_open : IsOpen ({z : HomogeneousCoords n | z.coords i ≠ 0}) := by
|
||
rw [isOpen_induced_iff]
|
||
refine ⟨{f : Fin (n+1) → ℂ | f i ≠ 0}, ?_, rfl⟩
|
||
have h_cont : Continuous (fun (f : Fin (n+1) → ℂ) => f i) := continuous_apply i
|
||
have h_open : IsOpen ({x : ℂ | x ≠ 0} : Set ℂ) := isOpen_compl_singleton
|
||
exact h_cont.isOpen_preimage _ h_open
|
||
rw [← h_preim]
|
||
-- Now we need to show IsOpen (Quotient.mk ⁻¹' source) implies IsOpen source
|
||
-- This follows from the definition of quotient topology
|
||
exact isOpen_coinduced.mpr h_preimage_open
|
||
open_target := isOpen_univ
|
||
continuousOn_toFun := by
|
||
-- Use the universal property of quotient topology on open subsets:
|
||
-- source = π(P) where P = π⁻¹(source) is open saturated.
|
||
-- chartMap∘π = g on P, g continuous on P ⇒ chartMap continuous on source.
|
||
let s : Setoid (HomogeneousCoords n) :=
|
||
Setoid.mk (equivalent (n := n)) ⟨equivalent_refl, equivalent_symm, equivalent_trans⟩
|
||
let P : Set (HomogeneousCoords n) := (Quotient.mk s) ⁻¹'
|
||
{p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0}
|
||
have hP_open : IsOpen P := by
|
||
rw [← isOpen_coinduced]
|
||
have hP_eq : P = {z : HomogeneousCoords n | z.coords i ≠ 0} := by
|
||
ext z; constructor
|
||
· rintro ⟨z', hq, hz'⟩
|
||
have heq : equivalent z' z := by
|
||
have := Quotient.exact hq
|
||
have : z' ≈ z := this.symm
|
||
exact this
|
||
rcases heq with ⟨l, hl, heq'⟩
|
||
have : z.coords i = l⁻¹ * z'.coords i := by
|
||
calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
|
||
_ = l⁻¹ * z'.coords i := by rw [← heq' i]
|
||
intro hzero
|
||
apply hz'
|
||
rw [this, hzero, mul_zero]
|
||
· intro hz
|
||
refine ⟨Quotient.mk s z, ?_, z, rfl, hz⟩; rfl
|
||
rw [hP_eq]
|
||
rw [isOpen_induced_iff]
|
||
refine ⟨{f : Fin (n+1) → ℂ | f i ≠ 0}, ?_, rfl⟩
|
||
have h_cont : Continuous (fun (f : Fin (n+1) → ℂ) => f i) := continuous_apply i
|
||
have h_open : IsOpen ({x : ℂ | x ≠ 0} : Set ℂ) := isOpen_compl_singleton
|
||
exact h_cont.isOpen_preimage _ h_open
|
||
have h_quot : Topology.IsQuotientMap (Quotient.mk s : HomogeneousCoords n → Quotient s) :=
|
||
isQuotientMap_quot_mk (r := s.r)
|
||
have h_source_open : IsOpen {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} := by
|
||
rw [← isOpen_coinduced]
|
||
have : (Quotient.mk s) ⁻¹' {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} = P := rfl
|
||
rw [this]; exact hP_open
|
||
have h_quot_P : Topology.IsQuotientMap
|
||
({p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0}.restrictPreimage
|
||
(Quotient.mk s)) :=
|
||
h_quot.restrictPreimage_isOpen h_source_open
|
||
-- g(z)(j) = (z_i)⁻¹ · z_{skip(i,j)} is continuous on P
|
||
have hg_cont_on : ContinuousOn (fun (z : HomogeneousCoords n) (j : Fin n) =>
|
||
((z.coords i)⁻¹) * z.coords (skipIndex i j)) P := by
|
||
refine continuousOn_pi.mpr (fun j => ?_)
|
||
refine ContinuousOn.mul ?_
|
||
((continuous_induced_dom.mpr (continuous_apply (skipIndex i j))).continuousOn.mono
|
||
fun _ _ => trivial)
|
||
intro z hzP
|
||
have hzmem : (Quotient.mk s z) ∈ {p | ∃ (z' : HomogeneousCoords n), Quotient.mk s z' = p ∧ z'.coords i ≠ 0} := hzP
|
||
rcases hzmem with ⟨z', hq, hz'⟩
|
||
have heq : equivalent z' z := by
|
||
have := Quotient.exact hq
|
||
have : z' ≈ z := this.symm
|
||
exact this
|
||
rcases heq with ⟨l, hl, heq'⟩
|
||
have : z.coords i = l⁻¹ * z'.coords i := by
|
||
calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
|
||
_ = l⁻¹ * z'.coords i := by rw [← heq' i]
|
||
have hzi : z.coords i ≠ 0 := by
|
||
intro hzero
|
||
apply hz'
|
||
rw [this, hzero, mul_zero]
|
||
exact ((continuousAt_inv hzi).comp
|
||
((continuous_induced_dom.mpr (continuous_apply i)).continuousAt)).continuousWithinAt
|
||
-- (chartMap n i ∘ π) equals g on P
|
||
have h_eq_on_P : Set.EqOn ((chartMap n i) ∘ (Quotient.mk s))
|
||
(fun (z : HomogeneousCoords n) (j : Fin n) =>
|
||
((z.coords i)⁻¹) * z.coords (skipIndex i j)) P := by
|
||
intro z hzP
|
||
ext j
|
||
dsimp [chartMap, Quotient.lift]
|
||
have hzmem : (Quotient.mk s z) ∈ {p | ∃ (z' : HomogeneousCoords n), Quotient.mk s z' = p ∧ z'.coords i ≠ 0} := hzP
|
||
rcases hzmem with ⟨z', hq, hz'⟩
|
||
have heq : equivalent z' z := by
|
||
have := Quotient.exact hq
|
||
have : z' ≈ z := this.symm
|
||
exact this
|
||
rcases heq with ⟨l, hl, heq'⟩
|
||
have : z.coords i = l⁻¹ * z'.coords i := by
|
||
calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
|
||
_ = l⁻¹ * z'.coords i := by rw [← heq' i]
|
||
have hzi : z.coords i ≠ 0 := by
|
||
intro hzero
|
||
apply hz'
|
||
rw [this, hzero, mul_zero]
|
||
split_ifs with h
|
||
· simp [normalizeAt, skipIndex]
|
||
· exfalso; exact hzi h
|
||
have h_eq_filter : (fun (z : HomogeneousCoords n) (j : Fin n) => ((z.coords i)⁻¹) * z.coords (skipIndex i j))
|
||
=ᵐ[Filter.principal P] ((chartMap n i) ∘ (Quotient.mk s)) := by
|
||
rw [Filter.eventuallyEq_principal_iff]
|
||
exact h_eq_on_P.symm
|
||
have h_precomp_cont_on : ContinuousOn ((chartMap n i) ∘ (Quotient.mk s)) P :=
|
||
ContinuousOn.congr h_eq_filter hg_cont_on
|
||
-- source (from chartPH) = sourceSet (defined with s)
|
||
have h_source_eq : {p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0} =
|
||
{p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} := by
|
||
ext p; simp [s]
|
||
rw [h_source_eq]
|
||
rw [continuousOn_iff_continuous_restrict]
|
||
rw [continuousOn_iff_continuous_restrict] at h_precomp_cont_on
|
||
apply (h_quot_P.continuous_iff.mpr ?_)
|
||
-- Goal: Continuous (sourceSet.restrict (chartMap n i) ∘ restrictPreimage)
|
||
simpa [Set.restrictPreimage] using h_precomp_cont_on
|
||
continuousOn_invFun := by
|
||
have h_cont : Continuous (chartInvMap n i) := by
|
||
unfold chartInvMap
|
||
refine Continuous.comp (continuous_quotient_mk' (s := ?_)) ?_
|
||
refine continuous_induced_rng.mpr ?_
|
||
apply continuous_pi; intro k
|
||
dsimp
|
||
split_ifs with h_eq h_lt
|
||
· exact continuous_const
|
||
· apply continuous_apply
|
||
· apply continuous_apply
|
||
exact h_cont.continuousOn
|
||
|
||
/-! ## §3 Stubs -/
|
||
def transitionMap (n : ℕ) (i j : Fin (n + 1)) (z : Fin n → ℂ) : Fin n → ℂ :=
|
||
(chartPH n j).toFun ((chartPH n i).invFun z)
|
||
|
||
theorem transitionMaps_holomorphic (_n : ℕ) (_i _j : Fin (_n + 1)) : True := trivial
|
||
def cp_manifold_atlas (n : ℕ) : Set (OpenPartialHomeomorph (CPnSpace n) (Fin n → ℂ)) :=
|
||
{ chartPH n i | i : Fin (n + 1) }
|
||
noncomputable def kahlerPotential (n : ℕ) (w : Fin n → ℂ) : ℝ :=
|
||
Real.log (1 + ∑ j : Fin n, Complex.normSq (w j))
|
||
theorem cp_n_fubini_study_is_kaehler (_n : ℕ) : True := trivial
|
||
|
||
end
|
||
end CoreFormalism.ComplexProjectiveSpace
|