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

33 votes
2 answers
6k views

Which almost complex manifolds admit a complex structure?

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau …
Gunnar Þór Magnússon's user avatar
18 votes
2 answers
4k views

What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of …
Gunnar Þór Magnússon's user avatar
16 votes
4 answers
1k views

Wanted: an example of a natural non-Kähler metric on a Kähler manifold

Let $X$ be a Kähler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kähler form of $h$. One of several equivalent conditions …
Gunnar Þór Magnússon's user avatar
16 votes
3 answers
3k views

References for holomorphic foliations

I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds. Any little helps, but I'm particularily interested in problems of the type where we have a hermitian man …
Gunnar Þór Magnússon's user avatar
13 votes
4 answers
1k views

"Simple" Kahler manifolds

I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X …
Gunnar Þór Magnússon's user avatar
13 votes
1 answer
2k views

Surgery in complex geometry

I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something …
Gunnar Þór Magnússon's user avatar
12 votes
1 answer
540 views

Which complex manifolds embed into tori?

If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to em …
Gunnar Þór Magnússon's user avatar
11 votes
3 answers
1k views

Can a metric conformal to a Kahler metric be Kahler?

Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on $X …
Gunnar Þór Magnússon's user avatar
11 votes
2 answers
2k views

Complex analytic vs algebraic families of manifolds

I'm studying the deformation theory of compact complex manifolds as developed by Kodaira and Spencer. On the side I'm reading as much about deformation theory in general as I can get my hands on (and …
Gunnar Þór Magnússon's user avatar
9 votes
2 answers
1k views

Is the deformation limit of Ricci-flat Kahler manifolds Kahler?

Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are compac …
Gunnar Þór Magnússon's user avatar
9 votes
2 answers
2k views

Hodge theory on complex spaces

If $X$ is a compact Kahler manifold, then Hodge theory says that its cohomology decomposes as a direct sum $$ H^{p+q}(X,\mathbb C) = \bigoplus_{p,q} H^{p,q}(X,\mathbb C) $$ where $H^{p,q}(X,\mathbb …
Gunnar Þór Magnússon's user avatar
9 votes
1 answer
1k views

Calculating a second fundamental form in the space of hermitian metrics

Let $X$ be a compact Kahler manifold and let $\mathcal M$ denote the space of hermitian metrics on $X$. We'll identify a hermitian metric with a smooth, real and positive $(1,1)$-form $\omega$. Let $\ …
Gunnar Þór Magnússon's user avatar
8 votes
2 answers
556 views

Inequality on Kähler classes

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. Th …
Gunnar Þór Magnússon's user avatar
8 votes
1 answer
1k views

Calculating a curvature tensor by polarization

I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson me …
Gunnar Þór Magnússon's user avatar
6 votes
0 answers
157 views

The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes

Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an alge …
Gunnar Þór Magnússon's user avatar

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