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Results tagged with complex-geometry
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user 13268
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.
6
votes
Accepted
Locality of Kähler-Ricci flow
The standard heat equation on the real number line $u_t=u_{xx}$ has solution
$$
u(x,t)=\int K(t,x,y)u(y,0)\,dy
$$
where
$$
K(x,y,t)= \frac{1}{\left(4\pi t\right)^{d/2}} e^{-\|x - y\|^2 / 4t}
$$
So if …
- 26.3k
1
vote
Accepted
Common holomorphic forms for two distinct complex structures
Any $C^1$ function $g$ on a connected Riemann surface is holomorphic if and only if $dg\wedge \omega=0$, for $\omega$ holomorphic and not the zero form. Where $\omega\ne 0$ this is clear, expanding ou …
- 26.3k
5
votes
How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negat...
On the complex torus, all Chern numbers vanish, but the same is true on the compact complex manifold $G/\Gamma$, given by quotienting a complex Lie group by a cocompact lattice. Such lattices exist in …
- 26.3k
3
votes
first order quasilinear partial differential equations
Vekua, I. N. Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962 xxix+668: starting at page 334, Vekua discusses the existe …
- 26.3k
3
votes
Accepted
Torsion free (1,0)-connections on the holomorphic tangent bundle?
I will write in terms of the holomorphic frame bundle, i.e. the bundle of choices of complex linear bases of tangent spaces, a holomorphic principal $\operatorname{GL}_n$-bundle.
Any sum $\gamma=\sum …
- 26.3k
4
votes
Accepted
Holomorphic vector fields with a non-degenerate isolated zero
A relevant result, but not a complete answer.
Vladimir Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, p. 181:
A zero of a vector field is in the Poincare domain if the …
- 26.3k
3
votes
Reconstructing the metric on $CP^2$ with special one forms
My calculation has the exterior derivative of the left hand side being
$$\frac{(1+|z_2|^2)d\bar{z}_1\wedge dz_1 + (1+|z_1|^2)d\bar{z}_2\wedge dz_2 - z_1\bar{z}_2 d\bar{z}_1 \wedge dz_2 -z_2\bar{z}_1 d …
- 26.3k
3
votes
English reference for Douady/Grauert construction of versal deformations of compact complex ...
D. Barlet and J. Magnusson, Complex Analytic Cycles II. Unfortunately, the book is not in press at the moment, so you have to wait. While you wait, you can read Complex Analytic Cycles I.
- 26.3k
1
vote
1
vote
Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex struc...
You change complex linearity to conjugate linearity, and vice versa, by replacing $I$ by $-I$, but only on the domain or the range independently. If you want to change them both, as the same vector sp …
- 26.3k
1
vote
Chart in $1$-parameter family of Lagrangians in a Kähler manifold
In any holomorphic chart, real analytic submanifolds remain real analytic. If your Lagrangian manifolds are not real analytic, they cannot become real analytic in holomorphic coordinates. In fact, you …
- 26.3k
4
votes
Accepted
Do non-compact Fano manifolds exist?
By the Bonnet Myers theorem, bounded positive Ricci curvature and complete Riemannian metric forces compact. David Wraith once explained to me that if the Ricci decays more slowly than quadratically i …
- 26.3k
1
vote
Regarding definition of Kobayashi length
The vector $\xi$ is tangent to the complex line spanned by $\xi$. The significance of "infinitesimal" is that this definition defines a length of a tangent vector (in the sense of differential geometr …
- 26.3k
2
votes
3
votes