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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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
21 votes
5 answers
7k views

References for complex analytic geometry?

I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc. A textbo …
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
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
6 votes
1 answer
1k views

Harmonic forms on Ricci-flat Kahler manifolds

Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian. …
Gunnar Þór Magnússon's user avatar
6 votes
0 answers
223 views

Complex manifold with non-finitely generated canonical ring

P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have t …
Gunnar Þór Magnússon's user avatar
3 votes
0 answers
131 views

Slicing the fibres of a meromorphic function with the zero set of a section of an ample line...

I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall: I've got a complex projective manifold $X$, …
Gunnar Þór Magnússon's user avatar
2 votes
1 answer
281 views

Is a certain composition of harmonic forms again harmonic?

Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an a …
Gunnar Þór Magnússon's user avatar
1 vote
0 answers
717 views

Does the tangent bundle of this fiber product split?

Let $\mathcal X \to S$ be the local universal family of an elliptic curve, and let $E \to S$ be a vector bundle over $S$. Then we can form the fiber product $\mathcal Y = \mathcal X \times_S E$, which …
Gunnar Þór Magnússon's user avatar