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▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣s ✐♥ ♠✉❧t✐s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②

▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ ❙❡♣t❡♠❜❡r ✷✺✱ ✷✵✷✵

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❖✉t❧✐♥❡

✶ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ✷ ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C ∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

✸ ❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s

❉❡✜♥✐t✐♦♥ ❆ ♣❛✐r (M, ω) ✐s ❛♥ ♥✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ ✐❢ ω ✐s ❛ ❝❧♦s❡❞ ♥♦♥❞❡❣❡♥❡r❛t❡ n + ✶ ❢♦r♠✿ ddRω = ✵ ∀m ∈ M✱ ∀v ∈ TmM ✇❡ ❤❛✈❡ ιvω = ✵ ⇒ v = ✵

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊①❛♠♣❧❡s

❙②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞s ❢♦r n = ✶ ❆♥ ♦r✐❡♥t❛❜❧❡ ♠❛♥✐❢♦❧❞ M t♦❣❡t❤❡r ✇✐t❤ ❛ ✈♦❧✉♠❡ ❢♦r♠✳ ∧nT ∗M ✇✐t❤ ω = −dθ✱ ✇❤❡r❡ θ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ n✲❢♦r♠ ❞❡✜♥❡❞ ❜②✿ θ|(m,α)(v✶, ..., vn) = α(π∗v✶, ..., π∗vn). ❈♦♠♣❛❝t s❡♠✐✲s✐♠♣❧❡ ▲✐❡ ❣r♦✉♣ ● ✇✐t❤ ω = θ, [θ, θ], ✇❤❡r❡ , ✐s ❛♥ Ad✲✐♥✈❛r✐❛♥t ✐♥♥❡r ♣r♦❞✉❝t ♦♥ g✱ ❛♥❞ θ ✐s t❤❡ ▼❛✉r❡r✲❈❛rt❛♥ ❢♦r♠✿ θL

g : TgG → TeG, v → Lg−✶∗v✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞s ❛♥❞ (n − ✶)✲❢♦r♠s

❉❡✜♥✐t✐♦♥ ▲❡t (M, ω) ❜❡ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✳ ❆♥ (n − ✶)✲❢♦r♠ α ∈ Ωn−✶(M) ✐s ❍❛♠✐❧t♦♥✐❛♥ ✐✛ t❤❡r❡ ❡①✐sts ❛ ✈❡❝t♦r ✜❡❧❞ vα ∈ X(M) s✉❝❤ t❤❛t dα = −ιvαω. ❚❤❡ ✈❡❝t♦r ✜❡❧❞ vα ✐s t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ α✳ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ (n − ✶)✲❢♦r♠s ❜② Ωn−✶

Ham(M)✳

❊①❛♠♣❧❡ ❋♦r n = ✶ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ (n − ✶)✲❢♦r♠s ❛r❡ t❤❡ s♠♦♦t❤ ❢✉♥❝t✐♦♥s ♦♥ M✳ ❆♥② f ∈ C ∞(M) ❤❛s ❛ ✉♥✐q✉❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ vf ✿

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C ∞(M)

❉❡✜♥✐t✐♦♥ ▲❡t (M, ω) ❜❡ ❛ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✳ ❚❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ ♦❢ ♦❜s❡r✈❛❜❧❡s ♦♥ M ✐s C ∞(M) ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜r❛❝❦❡t {f , g} = ω(vf , vg), ✇❤❡r❡ vf ❛♥❞ vg ❛r❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ f ❛♥❞ g✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ♦❜s❡r✈❛❜❧❡s❄❄

❈❛♥❞✐❞❛t❡✿ ❍❛♠✐❧t♦♥✐❛♥ (n − ✶)✲❢♦r♠s ❈❛♥ tr②✿ ❢♦r α, β ∈ Ωn−✶

Ham(M)

{α, β} = ιvβιvαω. ❲❤❛t ✇♦r❦s✿ d{α, β} = −ι[vα,vβ]ω s❦❡✇✲s②♠♠❡tr② ❲❤❛t ❞♦❡s ♥♦t ✇♦r❦✿ ❏❛❝♦❜✐ ✐❞❡♥t✐t②✦ {α, {β, γ}} + {β, {γ, α}} + {γ, {α, β}} = −dιvγιvβιvαω ❲❤❛t t♦ ❞♦❄ ❚❡❛s❡r✿ L∞✲❛❧❣❡❜r❛s✦

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

L∞✲❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭▲❛❞❛✱ ❙t❛s❤❡✛ ❬✹❪✮ ❆♥ L∞✲❛❧❣❡❜r❛ ✐s ❛ ❣r❛❞❡❞ ✈❡❝t♦r s♣❛❝❡ L ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❝♦❧❧❡❝t✐♦♥ {[ , ..., ]k : L⊗k → L|✶ ≤ k < ∞} ♦❢ ❣r❛❞❡❞ s❦❡✇✲s②♠♠❡tr✐❝ ❧✐♥❡❛r ♠❛♣s ✭❛❧s♦ ❝❛❧❧❡❞ ♠✉❧t✐❜r❛❝❦❡ts✮ ♦❢ ❞❡❣r❡❡ |[ , ..., ]k| = ✷ − k s❛t✐s❢②✐♥❣ t❤❡ ❤✐❣❤❡r ❏❛❝♦❜✐ ✐❞❡♥t✐t✐❡s✳

[ ]✶ sq✉❛r❡s t♦ ✵ ❛♥❞ ✐s ♦❢ ❞❡❣r❡❡ ✶✱ ✐✳❡✳✱ ✐s ❛ ❞✐✛❡r❡♥t✐❛❧✱ ❛♥❞ ❛♥ L∞✲❛❧❣❡❜r❛ ✐s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❛ ❝♦❝❤❛✐♥ ❝♦♠♣❧❡①✳ ❲❡ ❞❡♥♦t❡ [ ]✶ ❜② d✳ d ✐s ❛ ❣r❛❞❡❞ ❞❡r✐✈❛t✐♦♥ ♦❢ [ , ]✷✳ [ , , ]✸ s❛t✐s✜❡s✿ [[x, y]✷, z]✷ ± [[x, z]✷, y]✷ ± [[y, z]✷, x]✷ = ± d([x, y, z]✸) ± [d(x), y, z]✸ ± [d(y), x, z]✸ ± [d(z), x, y]✸,

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊①❛♠♣❧❡s

❆ ❝♦❝❤❛✐♥ ❝♦♠♣❧❡① (L, d) · · · d − → Li−✶

d

− → Li

d

− → Li+✶ · ··

❆ ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ▲✐❡ ❛❧❣❡❜r❛ (L, d, [ , ]✷, [ , , ]✸ = ✵) · · ·

d

− → Li−✶

d

− → Li

d

− → Li+✶ · ·· s✉❝❤ t❤❛t d[x, y] = [d(x), y] − (−✶)|x||y|[dy, x] ❛♥❞ (−✶)|x||z|[x, [y, z]]+(−✶)|y||x|[y, [z, x]]+(−✶)|z||y|[z, [x, y]] = ✵. ◆♦t❡✿ ✇❤❡♥ L ✐s ❝♦♥❝❡♥tr❛t❡❞ ✐♥ ❞❡❣r❡❡ ✵✱ ❛♥❞ l✶ = ✵✱ t❤✐s ❜❡❝♦♠❡s ❛ ▲✐❡ ❛❧❣❡❜r❛✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

L∞✲❛❧❣❡❜r❛s ❛s ❞✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❝♦✲❛❧❣❡❜r❛s

❚❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞❡♥❝❡ {L∞ − ❛❧❣❡❜r❛s} − → {❉✐✛❡r❡♥t✐❛❧ ❣r❛❞❡❞ ❝♦✲❛❧❣❡❜r❛s} (L, [, ..., ]k) − → (C(L), D) ❚❤❡♥ {❚❤❡ ❤✐❣❤❡r ❏❛❝♦❜✐ ✐❞❡♥t✐t✐❡s} ⇔ {D✷ = ✵}. ❉❡✜♥✐t✐♦♥ ❆♥ L∞✲♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ (L, [, ..., ]k) ❛♥❞ (L′, [, ..., ]′

k) ✐s ❛ ❝♦✲❛❧❣❡❜r❛

♠♦r♣❤✐s♠ F : C(L) → C(L′) ♦❢ ❣r❛❞❡❞ ❝♦✲❛❧❣❡❜r❛s s✉❝❤ t❤❛t F ◦ D = D′ ◦ F. ❚❤✐s tr❛♥s❧❛t❡s t♦✿ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✭❣r❛❞❡❞✮ s❦❡✇✲s②♠♠❡tr✐❝ ♠❛♣s fk : L⊗k → L′, k ≥ ✶ ♦❢ ❞❡❣r❡❡ ✶ − k✱ t❤❛t ❛r❡ ✧❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❜r❛❝❦❡ts✧✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

L∞✲❛❧❣❡❜r❛ ♦❢ ♦❜s❡r✈❛❜❧❡s ♦❢ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞

❚❤❡♦r❡♠ ✭❘♦❣❡rs✱ ❬✻❪✮

• ✐✈❡♥ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ t❤❡r❡ ✐s ❛ ❝♦rr❡s♣♦♥❞✐♥❣ L∞✲❛❧❣❡❜r❛

(L, {[ , ... , ]k}) ✇✐t❤ t❤❡ ✉♥❞❡r❧②✐♥❣ ❝♦❝❤❛✐♥ ❝♦♠♣❧❡① C ∞(M)

d

− → Ω✶(M)

d

− → · · ·

d

− → Ωn−✷(M)

d

− → Ωn−✶

Ham(M)

✇✐t❤ Ωn−✶

Ham ✐♥ ❞❡❣r❡❡ ✵ ❛♥❞ C ∞(M) ✐♥ ❞❡❣r❡❡ ✶ − n✱

❛♥❞ ♠❛♣s {[ , ... , ]k : Ωn−✶

Ham(M)⊗k → Ωn+✶−k(M)} ❢♦r k > ✶✱

[α✶, ... , αk]k = −(−✶)

k(k+✶) ✷

ι(vα✶ ∧ ... ∧ vαk)ω ✇❤❡r❡ vαi ✐s t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞ ❛ss♦❝✐❛t❡❞ t♦ αi✱ ❛♥❞ i(. . . ) ❞❡♥♦t❡s ❝♦♥tr❛❝t✐♦♥ ✇✐t❤ ❛ ♠✉❧t✐✈❡❝t♦r ✜❡❧❞✿ ι(vα✶ ∧ ... ∧ vαk)ω = ιvαk ...ιvα✶ω✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊①❛♠♣❧❡✿ ❛ ✶✲♣❧❡❝t✐❝ ✭s②♠♣❧❡❝t✐❝✮ ♠❛♥✐❢♦❧❞

■❢ (M, ω) ✐s ❛ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ L∞(M, ω) ❤❛s C ∞(M) ❛s t❤❡ ✉♥❞❡r❧②✐♥❣ ✈❡❝t♦r s♣❛❝❡✱ ❝♦♥❝❡♥tr❛t❡❞ ✐♥ ❞❡❣r❡❡ ✵✳ ❚❤❡ ♠✉❧t✐❜r❛❝❦❡t [ , ] ✐s ❣✐✈❡♥ ❜② [α✶, α✷] = ω(vα✶, vα✷).

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊①❛♠♣❧❡✿ ❛ ✷✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞

■❢ (M, ω) ✐s ❛ ✷✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ L∞(M, ω) ❤❛s C ∞(M) d − → Ω✶

Ham(M)

❛s t❤❡ ❝♦❝❤❛✐♥ ❝♦♠♣❧❡①✱ ✇✐t❤ C ∞(M) ✐♥ ❞❡❣r❡❡ ✲✶✱ ❛♥❞ Ω✶

Ham(M) ✐♥ ❞❡❣r❡❡ ✵✳

❚❤❡ ♠✉❧t✐❜r❛❝❦❡ts [ , ]✱ [ , , ] ❛r❡ ❣✐✈❡♥ ❜② [α✶, α✷] = ι(vα✶ ∧ vα✷)ω [α✶, α✷, α✸] = ω(vα✶, vα✷, vα✸).

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣✿ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②

❉❡✜♥✐t✐♦♥ ▲❡t ❛ ▲✐❡ ❛❧❣❡❜r❛ g ❛❝t ♦♥ (M, ω)✱ ❛♥❞ ❧❡t vx ❜❡ t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ ❛❝t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ x ∈ g. ❆ ✭❝♦✮♠♦♠❡♥t ♠❛♣ ❢♦r g ✐s ❛ ▲✐❡ ❛❧❣❡❜r❛ ♠♦r♣❤✐s♠ µ : g → C ∞(M) s✉❝❤ t❤❛t d(µ(x)) = −ivxω. C ∞(M) g XHam(M)

µ

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s

❚❤❡ ♠♦♠❡♥t ♠❛♣ ✧❡♥❛❜❧❡s ♦♥❡ t♦ r❡❧❛t❡ t❤❡ ❣❡♦♠❡tr② ♦❢ ❛ s②♠♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ ✇✐t❤ s②♠♠❡tr② t♦ t❤❡ str✉❝t✉r❡ ♦❢ ✐ts s②♠♠❡tr② ❣r♦✉♣✧ ✭❬✽❪✮✳ ❆♣♣❧✐❝❛t✐♦♥s✿ ◆♦❡t❤❡r✬s t❤❡♦r❡♠ ✭❍❛♠✐❧t♦♥✐❛♥ ✈❡rs✐♦♥✮ ❙②♠♣❧❡❝t✐❝ r❡❞✉❝t✐♦♥ ✭❡✳❣✳✱ ♠♦❞✉❧✐ s♣❛❝❡s ♦❢ ❣❛✉❣❡ t❤❡♦r✐❡s✮ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ s②♠♣❧❡❝t✐❝ t♦r✐❝ ♠❛♥✐❢♦❧❞s✳ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❍♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣

❉❡✜♥✐t✐♦♥ ✭❈❛❧❧✐❡s✱ ❋r❡❣✐❡r✱ ❘♦❣❡rs✱ ❩❛♠❜♦♥✱ ❬✶❪✮ ▲❡t g → X(M), x → vx ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ❛❝t✐♦♥ ♦♥ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ (M, ω) ❜② ❍❛♠✐❧t♦♥✐❛♥ ✈❡❝t♦r ✜❡❧❞s✳ ❆ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ ❢♦r t❤✐s ❛❝t✐♦♥ ✐s ❛♥ L∞✲♠♦r♣❤✐s♠ {fk} : g → L∞(M, ω) s✉❝❤ t❤❛t −ivxω = d(f✶(x)) ∀x ∈ g.

L∞(M, ω) g XHam(M)

{fk}

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❍♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣

■♥ ♦t❤❡r ✇♦r❞s✱ ❧❡t δk : ∧kg → ∧k−✶g ❜❡ t❤❡ k✲t❤ ▲✐❡ ❛❧❣❡❜r❛ ❤♦♠♦❧♦❣② ❞✐✛❡r❡♥t✐❛❧ ❣✐✈❡♥ ❜② δk : x✶∧...∧xk →

• ✶≤i<j≤k

(−✶)i+j[xi, xj]∧x✶∧... xi ∧...∧ xj ∧...xk.

❉❡✜♥✐t✐♦♥ ❆ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ ❢♦r t❤❡ ❛❝t✐♦♥ ♦❢ g ♦♥ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ (M, ω) ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧✐♥❡❛r ♠❛♣s fk : ∧kg → Ωn−k(M)✱ s✉❝❤ t❤❛t ❢♦r ✶ ≤ k ≤ n + ✶ ❛♥❞ ❛❧❧ p ∈ ∧kg✿ −fk−✶(δk(p)) = dfk(p) + ζ(k)ιvpω ✇❤❡r❡ vp ✐s t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✈❡❝t♦r ✜❡❧❞ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ p✱ ❛♥❞ f✵ ❛♥❞ fn+✶ ❛r❡ ❞❡✜♥❡❞ t♦ ❜❡ ③❡r♦✿ f✵ = fn+✶ = ✵✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❲❡❛❦ ✭❤♦♠♦t♦♣②✮ ♠♦♠❡♥t ♠❛♣

❉❡✜♥✐t✐♦♥ ✭❏✳ ❍❡r♠❛♥✱ ❬✸❪✮ ▲❡t g → X(M), x → vx ❜❡ ❛ ▲✐❡ ❛❧❣❡❜r❛ ❛❝t✐♦♥ ♦♥ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞ (M, ω)✳ ❆ ✇❡❛❦ ✭❤♦♠♦t♦♣②✮ ♠♦♠❡♥t ♠❛♣ ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧✐♥❡❛r ♠❛♣s fk : Pk,g → Ωn−k(M)✱ ✇❤❡r❡ ✶ ≤ k ≤ n✱ s❛t✐s❢②✐♥❣ dfk(p) = −ζ(k)ιvpω ❢♦r k ∈ ✶, ..., n ❛♥❞ ❛❧❧ p ∈ Pk,g✱ ✇❤❡r❡ Pk,g ⊂ ∧kg ✐s t❤❡ k✲t❤ ▲✐❡ ❦❡r♥❡❧ ♦❢ g✱ ✐✳❡✳✱ t❤❡ ❦❡r♥❡❧ ♦❢ δk : ∧kg → ∧k−✶g✳ ❆♣♣❧✐❝❛t✐♦♥s✿ n✲♣❧❡❝t✐❝ ◆♦❡t❤❡r✬s t❤❡♦r❡♠✱ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ♣♦s✐t✐♦♥ ❛♥❞ ♠♦♠❡♥t✉♠ ❢✉♥❝t✐♦♥s t♦ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✱ ❡t❝ ✭s❡❡ ❬✸❪✮✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❈♦♠♣❛r✐♥❣ t❤❡ t✇♦ ♠❛♣s

❇② ❝♦♠♣❛r✐♥❣ t❤❡ t✇♦ ❞❡✜♥✐t✐♦♥s✱ ✐t ✐s ❝❧❡❛r t❤❛t ❛ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣✱ ✇❤❡♥ ✐t ❡①✐sts✱ ❣✐✈❡s ❛ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣ ❜② r❡str✐❝t✐♥❣ t❤❡ fk t♦ Pk,g✱ ✐✳❡✳✱ ❊①✐st❡♥❝❡ ♦❢ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ ⇒ ❊①✐st❡♥❝❡ ♦❢ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣ ◗✉❡st✐♦♥ ■s t❤❡ ❝♦♥✈❡rs❡ tr✉❡❄

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❞♦✉❜❧❡ ❝♦♠♣❧❡①❡s

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡①❡s✿ ❚❤❡ t♦t❛❧ ❝♦♠♣❧❡① ( C, dtot) ♦❢ t❤❡ ❞♦✉❜❧❡ ❝♦♠♣❧❡① (∧≥✶g∗ ⊗ Ω(M), dg, d), ✇✐t❤ t❤❡ ❈❤❡✈❛❧❧❡②✲❊✐❧❡♥❜❡r❣ ❞✐✛❡r❡♥t✐❛❧ dg ♦♥ ∧≥✶g∗ :=

k=✶ ∧kg∗ ❛♥❞ t❤❡ ❞❡ ❘❤❛♠

❞✐✛❡r❡♥t✐❛❧ ♦♥ Ω(M)✳ ❍❡r❡ dtot := dg ⊗ ✶ + ✶ ⊗ d✳ ❚❤❡ t♦t❛❧ ❝♦♠♣❧❡① ( C, dtot) ♦❢ t❤❡ ❞♦✉❜❧❡ ❝♦♠♣❧❡① (P∗

≥✶,g ⊗ Ω(M), ✵, d) ✇✐t❤ ③❡r♦ ❞✐✛❡r❡♥t✐❛❧ ♦♥

P∗

≥✶,g := k=✶ P∗ k,g ❛♥❞ t❤❡ ❞❡ ❘❤❛♠ ❞✐✛❡r❡♥t✐❛❧ ♦♥

Ω(M)✳ ❍❡r❡ dtot := ✶ ⊗ d✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✐♥ t❡r♠s ♦❢ ❞♦✉❜❧❡ ❝♦♠♣❧❡①❡s

❚❤❡♦r❡♠ ✭❋r❡❣✐❡r✱ ▲❛✉r❡♥t✲●❡♥❣♦✉①✱ ❩❛♠❜♦♥❬✷❪✱ ❘②✈❦✐♥✱ ❲✉r③❜❛❝❤❡r ❬✼❪✮ ❚❤❡r❡ ❡①✐sts ❛ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ ❢♦r t❤❡ ❛❝t✐♦♥ ♦❢ g ♦♥ (M, ω) ✐❢ ❛♥❞ ♦♥❧② ✐❢ [ ω] = ✵ ∈ Hn+✶( C)✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣ ❢♦r t❤❡ ❛❝t✐♦♥ ♦❢ g ♦♥ (M, ω) ✐❢ ❛♥❞ ♦♥❧② ✐❢ [ ω] = ✵ ∈ Hn+✶( C)✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊①✐st❡♥❝❡ r❡s✉❧t ❢♦r ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣s

❚❤❡♦r❡♠ ✭▼✳✱ ❘②✈❦✐♥✱ ❬✺❪✮ ▲❡t (M, ω) ❜❡ ❛♥ n✲♣❧❡❝t✐❝ ♠❛♥✐❢♦❧❞✱ ❛♥❞ ❧❡t g ❛❝t ♦♥ (M, ω) ❜② ♣r❡s❡r✈✐♥❣ ω✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ❚❤❡ ❛❝t✐♦♥ ♦❢ g ♦♥ (M, ω) ❛❞♠✐ts ❛ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ ❚❤❡ ❛❝t✐♦♥ ♦❢ g ♦♥ (M, ω) ❛❞♠✐ts ❛ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣ ❛♥❞ φ ∈ P∗

n+✶,g ⊗ C ∞(M) ❞❡✜♥❡❞ ❜②

φ : Pn+✶,g → C ∞(M) p → ιvpω ✈❛♥✐s❤❡s ✐❞❡♥t✐❝❛❧❧②✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊❧❡♠❡♥ts ♦❢ t❤❡ ♣r♦♦❢

❈♦♥str✉❝t ❛ ♠❛♣ Hn+✶( C) → Hn+✶( C) t❤❛t ♠❛♣s [ ω] t♦ [ ω]✳ ■❢ t❤✐s ♠❛♣ ✐s ✐♥❥❡❝t✐✈❡✱ t❤❡♥ [ ω] = ✵ ⇒ [ ω] = ✵ ❈♦♥s✐❞❡r t❤❡ ❡①❛❝t s❡q✉❡♥❝❡✳ ✵ → Pk,g

i

− → ∧kg

δk

− → ∧k−✶g ◆♦t❡ t❤❡ ❞✉❛❧ s❡q✉❡♥❝❡ ✐s ❛❧s♦ ❡①❛❝t✳ ✵ ← P∗

k,g π

← − ∧kg∗

dk−✶

g

← − − − ∧k−✶g∗ ❚❤✉s✱ P∗

k,g = ∧kg∗/im(dk−✶ g

) ← ֓ ker(dk

g)/im(dk−✶ g

) = Hk(g),

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❊❧❡♠❡♥ts ♦❢ t❤❡ ♣r♦♦❢

❆♣♣❧② t❤❡ ❑ü♥♥❡t❤ ❢♦r♠✉❧❛✿ Hn+✶( C) = Hn+✶(∧≥✶g∗⊗Ω(M)) =

• k≥✶

Hkg⊗Hn+✶−k(M) ❛♥❞ Hn+✶( C) = Hn+✶(P∗

≥✶,g⊗Ω(M)) =

• k≥✶

P∗

k,g⊗Hn+✶−k(M)

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

❙tr✐❝t ❡①t❡♥s✐♦♥s

◗✉❡st✐♦♥✿ ●✐✈❡♥ ❛ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣ ❛♥❞ ❛ss✉♠✐♥❣ φ ≡ ✵✱ ❞♦❡s t❤❡r❡ ❛❧✇❛②s ❡①✐st ❛ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ t❤❛t r❡str✐❝ts t♦ t❤❡ ❣✐✈❡♥ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣❄ Pr♦♣♦s✐t✐♦♥ ✭✹✳✹✳✶✮ ▲❡t f ❜❡ ❛ ✇❡❛❦ ♠♦♠❡♥t ♠❛♣✱ ❛♥❞ φ = ✵✳ ❚❤❡r❡ ❡①✐sts ❛ ✇❡❧❧✲❞❡✜♥❡❞ ❝❧❛ss [γ]

dtot ∈ Hn+✶(

C) s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ [γ]

dtot = ✵ ❛♥❞ γ ❛❞♠✐ts ❛

dtot✲♣r✐♠✐t✐✈❡ ✐♥ n

k=✶ dg(Λkg∗) ⊗ Ωn−k−✶(M)

❚❤❡r❡ ❡①✐sts ❛ ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣ f ✱ s✉❝❤ t❤❛t

• f |Pg =

f ✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

▼✳ ❈❛❧❧✐❡s✱ ❨✳ ❋r❡❣✐❡r✱ ❈✳ ▲✳ ❘♦❣❡rs✱ ❛♥❞ ▼✳ ❩❛♠❜♦♥✳ ❍♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣s✳ ❆❞✈❛♥❝❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✸✵✸✿✾✺✹✕✶✵✹✸✱ ✷✵✶✻✳ ❨✳ ❋ré❣✐❡r✱ ❈✳ ▲❛✉r❡♥t✲●❡♥❣♦✉①✱ ❛♥❞ ▼✳ ❩❛♠❜♦♥✳ ❆ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ❢r❛♠❡✇♦r❦ ❢♦r ❤♦♠♦t♦♣② ♠♦♠❡♥t ♠❛♣s✳ ❏✳ ●❡♦♠✳ P❤②s✳✱ ✾✼✿✶✶✾✕✶✸✷✱ ✷✵✶✺✳ ❏✳ ❍❡r♠❛♥✳ ◆♦❡t❤❡r✬s t❤❡♦r❡♠ ✐♥ ♠✉❧t✐s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✳ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr② ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✺✻✿✷✻✵✕✷✾✹✱ ✷✵✶✽✳ ❚✳ ▲❛❞❛ ❛♥❞ ❏✳ ❙t❛s❤❡✛✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❙❍ ▲✐❡ ❛❧❣❡❜r❛s ❢♦r ♣❤②s✐❝✐sts✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s✱ ✸✷✭✼✮✿✶✵✽✼✕✶✶✵✸✱ ✶✾✾✸✳

▼♦♠❡♥t ♠❛♣s ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▲❡②❧✐ ▼❛♠♠❛❞♦✈❛ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❖❜s❡r✈❛❜❧❡s

❙②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ P♦✐ss♦♥ ❛❧❣❡❜r❛ C∞(M) n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr②✿ t❤❡ L∞✲❛❧❣❡❜r❛ L∞(M, ω)

❚❤❡ ♠♦♠❡♥t ♠❛♣s

▼♦♠❡♥t ♠❛♣ ✐♥ s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr② ▼♦♠❡♥t ♠❛♣ ✐♥ n✲♣❧❡❝t✐❝ ❣❡♦♠❡tr② ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ t✇♦ ♥♦t✐♦♥s ♦❢ n✲♣❧❡❝t✐❝ ♠♦♠❡♥t ♠❛♣s

▲✳ ▼❛♠♠❛❞♦✈❛ ❛♥❞ ▲✳ ❘②✈❦✐♥✳ ❖♥ t❤❡ ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ✇❡❛❦ ♠♦♠❡♥t ♠❛♣s✳ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✷✵✵✶✳✵✵✷✻✹✱ ✷✵✷✵✳ ❈✳ ▲✳ ❘♦❣❡rs✳ L∞✲❛❧❣❡❜r❛s ❢r♦♠ ♠✉❧t✐s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✳ ▲❡tt✳ ▼❛t❤✳ P❤②s✳✱ ✶✵✵✭✶✮✿✷✾✕✺✵✱ ✷✵✶✷✳ ▲✳ ❘②✈❦✐♥ ❛♥❞ ❚✳ ❲✉r③❜❛❝❤❡r✳ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐❝✐t② ♦❢ ❝♦✲♠♦♠❡♥ts ✐♥ ♠✉❧t✐s②♠♣❧❡❝t✐❝ ❣❡♦♠❡tr②✳ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠✳ ❆♣♣❧✳✱ ✹✶✿✶✕✶✶✱ ✷✵✶✺✳ ◆✳ ▼✳ ❏✳ ❲♦♦❞❤♦✉s❡✳

• ❡♦♠❡tr✐❝ q✉❛♥t✐③❛t✐♦♥✳

❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✼✳