Bergman kernel and Geometric quantization (joint with Weiping ...

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Quantization on symplectic manifolds Bergman kernel and qeometric quantization Index of D L Geometric quantization Quantization commutes with reduction Non-compact case : Vergne’s conjecture ◮ Ker DL L + , Ker D− are finite dimensional. ◮ Quantization space of L is the formal difference Q(L) := Ind(D L ) = Ker D L L + − Ker D− . It does not depend on the choice of J and the metric and connection on L. ◮ When (X, ω, J) is Kähler and L is holomorphic, then Q(L) = H 0,even (X, L) − H 0,odd (X, L). Xiaonan Ma Bergman kernel and Geometric quantization (jo
Quantization on symplectic manifolds Bergman kernel and qeometric quantization Index of D L Geometric quantization Quantization commutes with reduction Non-compact case : Vergne’s conjecture ◮ Ker DL L + , Ker D− are finite dimensional. ◮ Quantization space of L is the formal difference Q(L) := Ind(D L ) = Ker D L L + − Ker D− . It does not depend on the choice of J and the metric and connection on L. ◮ When (X, ω, J) is Kähler and L is holomorphic, then Q(L) = H 0,even (X, L) − H 0,odd (X, L). ◮ Atiyah-Singer : Q(L) = X Td(T (1,0) X) ch(L) e = det √ −1RT (1,0) X /2π e ω . X 1 − e −√ −1R T (1,0) X /2π Xiaonan Ma Bergman kernel and Geometric quantization (jo
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Bergman kernel and Geometric quantization (joint with Weiping ...

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