This simpler description makes it blatantly obvious that

So, is a functor.

]]>One more remark. Every symplectic manifold is almost complex in a natural way, i.e. uniquely up to homotopy.

To see this, choose a Riemannian metric . This is a contractible set of choices and so harmless up to homotopy. Write the symplectic form as for a uniquely defined . Then so that . Hence is positive semi definite. In fact is positive definite because is non degenerate. Now define (e.g. using the functional calculus) . Since everything in sight commutes, you see that , and you have cooked yourself an almost complex structure.

]]>The tautological bundle absolutely does *not* restrict to the canonical bundle (except in special cases). In fact this is already wrong for P^n. For P^n he canonical bundle is K = O(-n -1). This is easy to see, as c_1(K) = c_1(T^* P^n) = -c_1(TP^n) and TP^n = Hom(S, Q) = Q(1) sits in the exact sequence

0 –> O –> C^{n+1}(1) –> Q(1) –> 0.

For a complete intersection M of degree d_1, d_2,…d_m in P^n the canonical bundle is O(\sum_i d_i – n -1) restricted to M. This follows from the adjunction formula which just follows by induction and noting that the normal bundle of the smooth zero of a polynomial of degree d is O(d) restricted to the zero. It shows in particular that not all projective varieties are complete intersections as there exist varieties with K^{\dim(M)} = 0.

(written in “asci art Tex” as I keep messing things up the latex/jax thingy).

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This offers me plenty to think about. For now I’ll just mention that 1) your main LaTeX typos are using the nonexistent commands \tensor and \dual and writing \matcal for \mathcal, but 2) I don’t mind fixing mistakes, since it gives me an excuse to really think about what you’re saying.

]]>Thanks! I’d taken Blau’s Postscript file, converted it to PDF, put it on my website, and given the wrong link. But I’ll take advantage of this mistake to point to the nLab version instead.

]]>Many thanks, John, and thanks also, Blake, for the corrected link. AFAICT, both the Blau and the math-ph/9904008 perpetuate the idea that the classical observables are only functions on phase space, discounting that one can as much ask in classical physics as one can in quantum physics whether a state is an eigenstate of a unary transformation operator (such as a rotation that is generated by the biderivational Poisson bracket). Keeping it simple, in the absence of constraints, with a trivial symplectic form, classical operators include all functions , which, given also a Liouville state over this operator algebra, allows us to GNS-construct a (real or complex) Hilbert space, from which observables would presumably be, as usual, self-adjoint or normal operators, to taste. I’m glad to be even more clear that my approach to this is *not* geometric quantization: it is, so to speak, Koopman-von Neumann-ization, so given that John wants to discuss only geometric quantization, if anyone wishes to follow my gravatar link to my Facebook page and comment there, I will be there and as welcoming as I can, but in any case I’m sorry that I misunderstood some of the comments in the OP to be closer to my approach than I had thought.

The link above to “Symplectic geometry and geometric quantization” appears to be broken; the one on the nLab points here.

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