Let $M$ and $N$ be real manifolds of dimension $n$ which happen to admit complex structures (so that necessarily $n=2k$ and both are orientable). Then does their connected sum $M\# N$ also admit a complex structure? This is true for $n=2$, because every oriented $2$-dimensional topological manifold admits a complex structure. However, this operation doesn’t […]

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 […]

On this page, the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor vanish ($g_{\alpha\beta}=g_{\bar{\alpha}\bar{\beta}}=0$). My question is why do those vanish if the metric is Hermitian?

I want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure, i.e., a bundle isomorphism $J\colon TM\to TM$ such that $J^2=-I$, where $I$ is the identity. For every tangent space $T_pM$, there exists tangent vectors $v_1,\cdots, v_n$ […]

What are holomorphic and anti-holomorphic components? Why don’t we call them complex components and their conjugates? What is holomorphic coordinate transformation?

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