I’ve been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I’m not missing something. Let’s look at this from a purely linear algebra perspective: let $h$ be a Hermitian inner product on a complex vector space. Should […]

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.

Recall the following definitions: A Hermitian manifold is a smooth manifold $M$ endowed with a tensor field $J\in\mathcal{T}^1_1(M)$ and a Riemannian metric $g$ such that $$\forall x\in M,\ (J_x)^2 = -\mathrm{id}_{\mathrm{T}_x}$$ and such that the tensor field $\omega\in\mathcal{T}^2_0(M)$ defined by $\omega(-,-):=g(-,J(-))$ is a nondegenerate differential 2-form. Additionally, a Hermitian manifold $(M,J,g,\omega)$ is said to be […]

I was woundering if anyone knows any good references about Kähler and complex manifolds? I’m studying supergravity theories and for the simpelest N=1 supergravity we’ll get these. Now in the course-notes the’re quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite complete book) about the subject […]

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on topological solitons. I have been trying to read a paper that uses Kähler Manifolds. My guide just expects me to […]

Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now I am wondering about the converse: Question: Suppose that $X$ is a Kähler manifold that is also a smooth affine variety over $\Bbb C$. Can […]

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