# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

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### Bound of the measure of the support of a set of divisors in a fixed linear system

Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure ...
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### Does Lefschetz-type theorems imply ampleness?

Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties: $D$ is ample. (Positivity) For any $k$-dimensional ...
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### Do all Fano threefolds have effective $c_2$?

Let $X$ be a smooth complex projective Fano threefold. Then the class $c_1(X)$ can be realised as an effective divisor in $X$. It is it true that the class $c_2(X)$ can be realised as an effective ...
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### When is a minimal immersion holomorphic?

Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let: $\phi\colon X\to Y$ be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
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### Differentials on tori realised as double of annuli

In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus. In short, the torus is realised ...
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### Local meaning of the Pfaffian of the curvature

The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can ...
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### Interesting examples of vector bundles on hyperkahler varieties

I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already: For $X$= the Hilbert scheme of points $S^{[n]}$ on a ...
Let $(X,\mathcal{O}_X)$ be a complex analytic space. Are the path components of $X$ the same as the connected components?