This simpler description makes it blatantly obvious that

So, is a functor.

]]>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.

]]>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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