Diffgeometer1 on Nostr: some additional detail on what the article is calling a Kahler structure (as I ...

some additional detail on what the article is calling a Kahler structure (as I understand it)

First, any hermitian manifold M can be regarded as a real manifold with extra data, namely a Riemannian metric \(g\) and an almost complex structure \(J\) where \(g\) is invariant under \(J\). The fundamental 2-form is \(\omega:=g(J\cdot,\cdot)\) which is a symplectic form when M is Kahler. If we regard TM as a complex vector bundle by defining \(\sqrt{-1}X:=JX\) and define\[h(𝑋,𝑌):=𝑔(𝑋,Y)-\sqrt{-1}\omega(X,Y)\] we get the traditional notion of a Hermitian metric on TM (where the latter is viewed as a complex vector bundle with complex scalar multiplication induced by \(J\). In other words, \(h(JX,Y)=\sqrt{-1}h(X,Y)\), \(h(Y,X)=\overline{h(X,Y)}\). So the Kahler structure is \(h\). I think most people working in complex geometry would call the triple \((g,J,\omega)\), the Kahler structure (where \(\omega\) is a closed 2-form which in turn is equivalent to the statement that \(J\) is parallel to the Levi-Civita connection.

The word “quantum” is overused in pop science articles to make things sound fancier than it really is IMO.