# 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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### Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
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### Why the origin of this complex is moving away from the origin (0,0)?

Why does the origin of the complex line z is moving away from the origin? $$Let\;z=x+i\cdot y \\\;z-1\;=\;\;(x-1)+i\cdot y\;$$ Following that I would say that the coordinates of the origin of z are ...
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### Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
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### How to see whether a tori is an abelian variety or not?

Given an explicit lattice $\Lambda \cong \mathbb{Z}^{2n}$ in $\mathbb{C}^n$, how can one check whether the complex torus $\mathbb{C}^n/\Lambda$ is a projective or not?
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### Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
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### Based on the Andreotti-Frankel theorem, what is the CW complex homotopy equivalent to $x^2 + y^2 - 1$?

I am referring to this theorem: http://en.wikipedia.org/wiki/Andreotti%E2%80%93Frankel_theorem I have no idea how to begin thinking about this.
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### What are the irreducible curves on the blow up of $\mathbb{P}^{2}$?

On the blow-up of $2$-dimensional complex projective space, $\mathbb{P}^2$, I know that $$Pic(\mathbb{\tilde{P}^2})=\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$$ where $\tilde{H}$ is the blow-up of the ...
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### Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
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### Higher dimensional analogue for Riemann Hurwitz formula

There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and ...
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### Global sections of holomorphic vector bundles

Let $X$ be a complex manifold, and $\mathbb{L}\rightarrow X$ a holomorphic line bundle over $X.$ Can we always find global sections of $\mathbb{L}$? (other from the one that's identically zero) On a ...
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### Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
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### Analytic hypersurface as union of irreducibles

Let $X$ be a complex manifold. Then any analytic subvariety $V$ of codimension 1 (that is, any analytic hypersurface) can be expressed uniquely as the union of irreducible analytic hypersurfaces ...
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### What is the linear series $|mL|$?

I am studying complex geometry and I am trying to find out what is the definition of the linear series $|mL|,$ where $L$ be a line bundle over a compact Kahler manifold $X^n.$ In particular, I know ...
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### Fundamental cycle $[Y]\in H_{2n-2}(X_{\mathbb{R}},\mathbb{Z})$ of irreducible analytic hypersurface on a complex manifold X.

I am studying about complex manifolds and I am trying to understand the following statement. Let $Y\subset X$ be an irreducible analytic hypersurface, where $X$ is an n-dimensional complex compact ...