# 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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### Picard number of Kahler manifold

Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group. In ...
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### adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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### Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
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### Page 276 of Principles of Algebraic Geometry by Griffiths and Harris; wrong parameter count?

The following is taken from page $276$ of Principles of Algebraic Geometry by Griffiths and Harris: Now let $S$ be a Riemann surface of genus $g\ge 3$. By our last result, if $S$ has any ...
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### Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
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### Why is compactness needed for proving that Kahler forms are open

I am studying complex geometry from Huybrechts' Complex Geometry - An Introduction. In Corollary 3.1.8 he proves that: The set of all Kahler forms on a compact complex manifold $X$ is an open ...
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### Embedding Complex Tori in Projective Space

When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense ...
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### Chern class of ideal sheaf

Let $X$ be a smooth projective surface. Let $Z$ be a dimensional $0$ subscheme of length $l$. Suppose $I_Z$ is the ideal sheaf of $Z$. Then it claimed that $c_1(I_Z) = 0$ and $c_2(I_Z) = l$. (1)Why ...
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### Oka's coherence theorem and Cartan's theorem A and B

So reading up on some articles I've found that whilst attempting to characterize Oka's coherence theorem in sheaf-theoretic language, Cartan was ultimately led to formulating his theorem A and B in ...
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### Hyperelliptic Curves and Period Integrals

I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic ...
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### Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
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### Hilbert scheme of quasi-projective variety

Suppose $X$ is a projective scheme over an algebraically closed field $k$, denote its Hilbert scheme with Hilbert polynomial $p$ by $\text{Hilb}^p_X$, then from section 1.1 of Nakajima's book, ...
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### Prove L(T(γ))=L(γ) where L(γ) is the hyperbolic length

The first proof $\text{Im}(w)=(w-w^*)/2i$ I believe I have correct, but I need help on the second proof. $L(T(γ))=L(γ)$ I need to know if this is the same concept as, independent of choice of the ...
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### Symmetric Product of a Projective scheme

Following the question, Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): http://math....
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### Hilbert scheme of $n$ points on a smooth curve

If $C$ is a smooth curve over a field $k$, then from lots of references, e.g. Janos Kollar, Rational Curves on Algebraic Varieties, exercise 1.4.1, that the Hilbert scheme of $n$ points is \begin{...
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### The cylinder does not embed into $\Bbb C^n$

The cylinder $\Bbb R\times S^1$ can be viewed as a complex manifold with a flat metric by viewing it has the quotient $\Bbb R\times\Bbb R/\Bbb Z$, where $\Bbb R\times\Bbb R=\Bbb C$. (In fact it makes ...
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### Embedding of Kähler manifolds into $\Bbb C^n$

Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now ...
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### Asymptotics of Harmonic Functions

I am looking for some information (answers, references etc.) on existence of solutions to $$\Delta u = f$$ on the Euclidean unit ball $B \subset \mathbb{C}^n$ with ...
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### Is $X$ an algebraic subset? Analytic subset?

Suppose that $X$ is a subset of $\mathbb{C}^n$, and that every (complex) hyperplane section of $X$ is an algebraic subset (respectively analytic subset) of complex dimension at least one (or empty). ...
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### Locally finitely generated sheaf

Here is the image from the book So I don't quite understand the proof of lemma 3.9. Namely, I don't see why there exists $H_{jk}$ such that the formula is true on $U'$. I was wondering if someone ...
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### Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
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### Obstructions to putting a complex structure on a real vector bundle (other than, obviously, dimension)

A complex vector bundle is usually described as one with structure group $GL(n,\mathbb{C})$. If I take a real $2n$ when is it not the underlying real bundle of some complex bundle?
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### Example of a complex manifold with certain curvature properties?

Are there (nice) examples of a complex manifold such that the sectional curvatures through all the complex planes are non-positive but the sectional curvatures through the real planes are mixed?
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### Diffeomorphism from Riemannian metric to Hermitian metric on a complex manifold.

It is known that any complex manifold, $M$ admits a Hermitian metric, i.e., a Riemannian metric, $g$, which satisfies $$g_p(J_pX,J_pY)=g_p(X,Y)$$ at each point $p\in M$, ...
A complex $n$-dimensional vector space $V$ can be thought of as a real $2n$-dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$. Complex-linear maps are then linear maps $V \to V$ ...
### Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?
Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...