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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.

5 votes
0 answers
147 views

Toponogov comparison theorem for complex manifold

I would like to know some reference for the Toponogov comparison theorem for complex manifolds, in particular for complex manifolds with bounded holomorphic sectional curvature. As far as I know, the …
Daniel's user avatar
  • 303
3 votes
1 answer
183 views

Volume form under holomorphic automorphisms

$(M,\omega)$ is a compact Kaehler manifold and $f_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V_s$. My question is whether the function $\frac{f_{t,s}^* \omega^n}{\omega^n}$ i …
Daniel's user avatar
  • 303
2 votes
1 answer
270 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies $\ …
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  • 303
1 vote
1 answer
454 views

Properness of Ding-functional independent of the chosen Kahler metric

On Page 62 of "canonical metrics in Kahler geometry" written by Gang Tian, the author pointed out that properness of Ding-functional $F_\omega$ is independent of $\omega$, without proof. What I want t …
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  • 303
1 vote
0 answers
207 views

Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth symple …
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  • 303
0 votes
0 answers
308 views

Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. Furth …
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