# Tagged Questions

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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### Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1}$$ where $L$ ...
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### Cauchy's Theorem by Differential Geometry

Is there a prove of Cauchy's theorem footing on the topology of the complex plane (homotopy, differential forms, etc.)? More specific consider a differentiable Banach space valued complex function. ...
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### Numerical equivalence of divisors on fibered surface

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g > 2$ and $\pi_i\colon C\times C \to C$ the projection onto the the $i$-th factor. Let $f_i \in \mathrm{Num}(C\times C)$ be the ...
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### A question on differential topology

Let $\mathbb{C}P(1)$ denote the complex projective line. I am attempting to show that there does not exist a nonzero holomorphic differential $1$-form on $\mathbb{C}P(1)$. My intuition is as follows:...
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### Kahler identities on symplectic manifold

In Hutchings and Taubes's lecture note on Seiberg-Witten equation, a Weitzenbock formula is given, and the authors states that the Weitzenbock formula, proven in Donaldson's book using Kahler ...
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### A complex manifold isn't a sympletic manifold

I want to think about an example of a complex manifold which isn't a sympletic manifold. I consider it in this way: $X=\mathbb{C}^2-\{0\}$, a group $\mathbb{Z}$ acts on X by $(n,z)=2^nz$, then I think ...
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### Relation between Kähler potential and Hermitian metric

Let $(M,\omega)$ be a Kähler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h,$$ and also if $f$ be the Kähler potential then we can write ...
Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ($k-2$)...
I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...