# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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### Manifolds whose tangent spaces have a special behavior

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let $$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$ be a local parametrization of $M$. ...
1 vote
80 views

### Maximal analytic continuation

I'm currently reading about the concept of a “maximal analytic continuation” from Forster's book Lectures on Riemann Surfaces (see Section 7). There are a bunch of definitions to unpack before I can ...
1 vote
131 views

### Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...
68 views

### Complete intersections in complex manifolds

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$. a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global ...
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### Riemann-Roch theorem for higher-dimensional complex manifolds

Does an analogue of the Riemann-Roch theorem hold for higher-dimensional complex manifolds? (Hirzebruch-Riemann-Roch theorem is for algebraic manifolds, but not for general complex manifolds, right?)
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### Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
187 views

### Example of usual Laplacian does not respect bidegree for general hermitian manifolds

We know that the Kähler identity $\Delta=2\Delta_{\partial}=2\Delta_{\bar{\partial}}$ on a Kähler manifold $(X,g)$ implies that the usual Laplacian $\Delta:=dd^*+d^*d$ respects the bidegree, i.e. for ... 1 vote
81 views

### Holomorphic mapping on a manifold approximating a constant map

Let $X,Y$ be complex manifold, $Y$ Stein. It sounds quite reasonable to formulate the following claim: given $y_0\in Y$, for every $\epsilon>0$ and $M\subset X$ compact, there exists an holomorphic ...
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### Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
1 vote
142 views