– Geometric Quantization of complex Monge-Ampère operator for certain diffusion flows – Julien Keller (Aix-Marseille University) 1 / 57
Kähler metrics Kähler metrics 1 Geometric flows 2 Quantum formalism and intrinsic geometric operators 3 Other related geometries 4 2 / 57
Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End ( TM ) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real ( 1 , 1 ) -form ω by ω ( X , Y ) = g ( JX , Y ) , for all tangent vectors X , Y . Locally √ − 1 n j dz i ∧ d ¯ ∑ ω = h i ¯ z j 2 i , j = 1 and ∀ p ∈ M , h i ¯ j ( p ) is positive definite hermitian matrix If d ω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric √ CP n = ⋃ n i = 0 U i , U i ≃ C n , ω FS ∣ U i = ∂ log ( ∑ l ≠ i ∣ z l ∣ 2 ∣ z i ∣ 2 ) 2 ∂ ¯ − 1 3 / 57
Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End ( TM ) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real ( 1 , 1 ) -form ω by ω ( X , Y ) = g ( JX , Y ) , for all tangent vectors X , Y . Locally √ − 1 n ω = j dz i ∧ d ¯ ∑ h i ¯ z j 2 i , j = 1 and ∀ p ∈ M , h i ¯ j ( p ) is positive definite hermitian matrix If d ω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CP n = ⋃ n √ i = 0 U i , U i ≃ C n , ω FS ∣ U i = ∂ log (∑ l ≠ i ∣ z i ∣ 2 ) ∣ z l ∣ 2 2 ∂ ¯ − 1 4 / 57
Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End ( TM ) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real ( 1 , 1 ) -form ω by ω ( X , Y ) = g ( JX , Y ) , for all tangent vectors X , Y . Locally √ − 1 n ω = j dz i ∧ d ¯ ∑ h i ¯ z j 2 i , j = 1 and ∀ p ∈ M , h i ¯ j ( p ) is positive definite hermitian matrix If d ω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CP n = ⋃ n √ i = 0 U i , U i ≃ C n , ω FS ∣ U i = ∂ log (∑ l ≠ i ∣ z i ∣ 2 ) ∣ z l ∣ 2 2 ∂ ¯ − 1 5 / 57
Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End ( TM ) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real ( 1 , 1 ) -form ω by ω ( X , Y ) = g ( JX , Y ) , for all tangent vectors X , Y . Locally √ − 1 n ω = j dz i ∧ d ¯ ∑ h i ¯ z j 2 i , j = 1 and ∀ p ∈ M , h i ¯ j ( p ) is positive definite hermitian matrix If d ω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CP n = ⋃ n √ i = 0 U i , U i ≃ C n , ω FS ∣ U i = ∂ log (∑ l ≠ i ∣ z i ∣ 2 ) ∣ z l ∣ 2 2 ∂ ¯ − 1 6 / 57
Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End ( TM ) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real ( 1 , 1 ) -form ω by ω ( X , Y ) = g ( JX , Y ) , for all tangent vectors X , Y . Locally √ − 1 n ω = j dz i ∧ d ¯ ∑ h i ¯ z j 2 i , j = 1 and ∀ p ∈ M , h i ¯ j ( p ) is positive definite hermitian matrix If d ω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CP n = ⋃ n √ i = 0 U i , U i ≃ C n , ω FS ∣ U i = ∂ log (∑ l ≠ i ∣ z i ∣ 2 ) ∣ z l ∣ 2 2 ∂ ¯ − 1 7 / 57
Kähler metrics M compact Kähler manifold, n = dim C M , ω Kähler form. √ Kähler class Ka ( ω ) = { φ ∈ C ∞ ( M , R ) ∶ ω + − 1 ∂ ¯ ∂φ > 0 } Theorem (Yau -1978) Let Ω a smooth volume form with ∫ M Ω = Vol ([ ω ]) . Then there exists a smooth solution φ to the Monge-Ampère equation √ ∂φ ) n = Ω ( ω + − 1 ∂ ¯ 8 / 57
Kähler metrics M compact Kähler manifold, n = dim C M , ω Kähler form. √ Kähler class Ka ( ω ) = { φ ∈ C ∞ ( M , R ) ∶ ω + − 1 ∂ ¯ ∂φ > 0 } Theorem (Yau -1978) Let Ω a smooth volume form with ∫ M Ω = Vol ([ ω ]) . Then there exists a smooth solution φ to the Monge-Ampère equation √ ∂φ ) n = Ω ( ω + − 1 ∂ ¯ ↪ Non constructive proof. Transcendental solution 9 / 57
Kähler metrics M compact Kähler manifold, n = dim C M , ω Kähler form. √ Kähler class Ka ( ω ) = { φ ∈ C ∞ ( M , R ) ∶ ω + − 1 ∂ ¯ ∂φ > 0 } Theorem (Yau -1978) Let Ω a smooth volume form with ∫ M Ω = Vol ([ ω ]) . Then there exists a smooth solution φ to the Monge-Ampère equation √ ∂φ ) n = Ω ( ω + − 1 ∂ ¯ ↪ Non constructive proof. Transcendental solution Ricci curvature of ω √ Ric ( ω ) = − − 1 ∂ ¯ ∂ log ( ω n ) “The Ricci Curvature as organizing principle” Scalar curvature scal ( ω ) = trace of the Ricci curvature 10 / 57
Kähler metrics Some consequences of Yau’s theorem A new physics (Supersymmetric String Theory) ↔ Ricci flat 3-folds 11 / 57
Kähler metrics Some consequences of Yau’s theorem Space Ka ( ω ) of Kähler metrics ← → Smooth probabilities den- sities on M compatible with the symplectic structure ω 12 / 57
Kähler metrics Some consequences of Yau’s theorem Space Ka ( ω ) of Kähler metrics ← → Smooth probabilities den- sities on M compatible with the symplectic structure ω ← → Rao-Fisher metric Calabi metric g F ( x , y ) ∣ µ = ∫ M g C ( α,β ) ∣ ω φ = ∫ M ∆ ω φ α ∆ ω φ β ω n y x φ µ µ µ n ! 13 / 57
Kähler metrics Some consequences of Yau’s theorem Space Ka ( ω ) of Kähler metrics ← → Smooth probabilities den- sities on M compatible with the symplectic structure ω ← → Rao-Fisher metric Calabi metric g F ( x , y ) ∣ µ = ∫ M g C ( α,β ) ∣ ω φ = ∫ M ∆ ω φ α ∆ ω φ β ω n y x φ µ µ µ n ! g C has constant > 0 sectional curvature. Geodesic equation wrt g C is an ODE ⇒ smoothness and uniqueness 14 / 57
Kähler metrics Some consequences of Yau’s theorem Space Ka ( ω ) of Kähler metrics ← → Smooth probabilities den- sities on M compatible with the symplectic structure ω ← → Rao-Fisher metric Calabi metric g F ( x , y ) ∣ µ = ∫ M g C ( α,β ) ∣ ω φ = ∫ M ∆ ω φ α ∆ ω φ β ω n y x φ µ µ µ n ! g C has constant > 0 sectional curvature. Geodesic equation wrt g C is an ODE ⇒ smoothness and uniqueness ↪ works in a more general setup (non compact, singular) 15 / 57
Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z 16 / 57
Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E [ ZZ ∗ ] that is (Toeplitz) hermitian > 0. √ − 1 ∂ ¯ ∂ logdet ( E [ ZZ ∗ ]) , studied by ↪ Kähler metric (of Bergman type) Burbea-Rao (1984) and more recently by F. Barbaresco. 17 / 57
Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E [ ZZ ∗ ] that is (Toeplitz) hermitian > 0. √ − 1 ∂ ¯ ∂ logdet ( E [ ZZ ∗ ]) , studied by ↪ Kähler metric (of Bergman type) Burbea-Rao (1984) and more recently by F. Barbaresco. ↪ Kähler metric with constant scalar curvature on bounded homogeneous domains 18 / 57
Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E [ ZZ ∗ ] that is (Toeplitz) hermitian > 0. √ − 1 ∂ ¯ ∂ logdet ( E [ ZZ ∗ ]) , studied by ↪ Kähler metric (of Bergman type) Burbea-Rao (1984) and more recently by F. Barbaresco. ↪ Kähler metric with constant scalar curvature on bounded homogeneous domains Open questions for target detection: – define a good distance between two Toeplitz covariance matrices ↔ geodesic distance on Kähler metrics – give a reasonable definition of the average of covariance matrices ↔ balancing/barycenter condition 19 / 57
Kähler metrics Quantum Field Theory Classical system: Phase space ( M ,ω ) , observables C ∞ ( M , R ) Quantized system: Hilbert space H ( M ,ω ) , hermitian operators on H ( M ,ω ) Quantum phase space P ( H ) : the Fubini-Study metric provides the means of measuring information in quantum mechanics 20 / 57
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