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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On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
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Meromorphic function with bounded order of zeros and poles

The following problem has been bothering me for a long time; Let $X$ be a compact Riemann surface of genus $g$. Is there a non-zero meromorphic function on $X$ with all of its poles and zeros have ...
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Cartier divisors of schemes

In his notes, Ravi Vakil only defines the notion of an effective Cartier divisor. Furthermore, the Wikipedia page only defines the notion of an effective Cartier divisor for a general scheme. ...
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What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction. At the heart of this is the complex $$ V ...
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Two generating meromorphic functions seperate points on a compact Riemann surface?

Problem Suppose $z,f$ are two meromorphic functions on a compact Riemann surface $M$, whose meromorphic function field is $\mathbb C(M)=\mathbb C(z,f)$, where $\mathbb C(M)$ is a finite extension of ...
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Giving Holomorphic structure to Complex line bundles on projective space and torus

I am doing questions from [Huybrechts, Complex Geometry, An Introduction] page 143 questions 3.3.7 and 3.3.8. Basically the questions ask: For question 7, for any complex line bundles on the ...
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35 views

Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
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Homogeneous polynomials and line bundles

In Huybrechts' Complex Geometry text, he makes the following claim: (Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map ...
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Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
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34 views

Transcendence Degree of the Function field of $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense subset ...
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42 views

Basis for a line bundle

I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text: "Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, ...
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Undergraduate Complex Analysis: Use of Rouche's Theorem

We are asked to prove $ f = z^{3}e^{1-z} = 1 $ has exactly 2 roots inside $|z| = 1$ We've tried creating functions $p$ and $q$ where $p + q = f$, $p$ with 2 roots inside our boundary, and using ...
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explicit (holomorphic) map revealing blow-up as a connected sum with $\overline{\mathbb{CP}}^n$

I am trying to understand the statement that a blow-up of a complex manifold $M$ at a point $p$ is equivalent to the connected sum of $M$ with $\overline{\mathbb{CP^n}}$ and, being a physicist, I need ...
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32 views

Why is this function on the Riemann surface holomorphic?

Forster defines analytic continuation of a germ of a holomorphic function at a point on a Riemann surface as follows. Suppose $ X$ is a Riemann surface, $a\in X$ and $\phi\in\mathcal{O}_a$ is a ...
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1answer
285 views

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
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Hermitian manifold counterexample

I'm trying to come to come to grips with the notion of a hermitian manifold. Although I know some examples of hermitian manifolds, I am more interested in counterexamples: naturally occurring ...
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1answer
22 views

In the Riemann sphere 1 is not summe of holomorphics map vanishing on 0 and $\infty$

I want to prove (if it's right) that in the Riemann sphere one can not write the constant function 1 as a summe of two holomorphics map, one vanishing in 0 and one vanishing in $\infty$.
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Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
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Holomorphic vector fields on complex projective spaces

Question: is there a simple description of holomorphic vector fields on the complex projective space $\mathbb{C P}^n$ ? More precisely : for $n=1$, the holomorphic vector fields are of the form $P(z) ...
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antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
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Simpleminded example: flasque sheaves

Consider the sheaf $\mathcal{F}$ of polynomial functions on $\mathbb{R}^2$ endowed with the usual topology. A sheaf is said to be "flasque" (or "flabby") if, given $V \subset U$ both open sets, the ...
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1answer
19 views

Unitary transformation of Fubini-Study metric

I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation ...
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Why is $H^1(X, \mathcal{O}) \neq 0$ for $X = (B(0, 1)\times B(0, 2)) \cup (B(0, 2)\times B(0, 1))$?

In this MathOverflow answer, David Speyer says that \begin{align*} X &= \{ (z,w) \in \mathbb{C}^2 : (|z|, |w|) \in [0,1) \times [0,2) \cup [0,2) \times [0,1) \}\\ &= (B(0, 1)\times B(0, 2)) ...
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72 views

Almost complex structure which fails to be compatible

Let $V$ be a real vector space equipped with a scalar product $\langle, \rangle$ (i.e. a positive definite symmetric bilinear form). We say that an endomorphism $J: V \to V$ is an almost complex ...
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Working in complex number field

I have to draw the graphic of "group of points given by the equation $$(|z^2|-3|z|+2)(z^4+4)=0$$ I solved the first part by factoring and obtaining $|z|=1$ and $|z|=2$ so in the graphic I have the ...
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deformation space inside cohomology

For which smooth projective varieties $X$ is $H^1(X,T_X)$ (canonically ) contained in $H^\cdot(X,\mathbb C)$? If $K_X$ is trivial this is true. But are there other type of varieties?
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Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
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64 views

first chern class

If $M$ is a Fano manifold, and $K_M$ is the canonical line bundle of $M$. If $L$ is an ample line bundle over $M$, and $c_1(L)=\lambda c_1(M)$, for some positive number $\lambda$. What is the relation ...
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38 views

On the Chern connection

It is well kown that If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection ...
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Real part of a complex number divided: $\Re\frac{z+1}{z-1}=0$

$\Re\frac{z+1}{z-1}=0$ I've tried so many methods, they all end up with two variables $a, b$. I tried setting $z= a+ib$. This give me the equality $2\cdot\Re\frac{z+1}{z-1} = ...
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1answer
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Disconnecting a complex vector space

Can a (complex) dimension $n$ subspace disconnect a (complex) dimension $n+1$ vector space ? If the answer is no, what if we replace "vector space" by "manifold" ?
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Rational functions on varieties

To give a rational function $f$ in the function field $K(X)$ of a smooth projective curve $X$ is equivalent to giving a finite morphism $f:X\to \mathbb P^1$. What is the precise analogue of this ...
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A continuous map from $\mathbb S(\mathbb C^{n})$ to $U(n)$

Let $a$ in $\mathbb S(\mathbb C^{n})$, the unit sphere in $\mathbb C^n$. Does there exists a continuous map $x\mapsto u_x$, from $\mathbb S(\mathbb C^{n})$ to $U(n)$, the group of unitary ...
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3answers
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Finding an angle $\theta$ in a complex number

If we know that $z = \frac{1}{\sqrt2}(\cos\theta+i\cdot\sin\theta)$ and also that $z = \frac{(\sqrt3-1)+i(\sqrt3+1)}{4}$ How can I find $\cos\theta$ and $\sin\theta$? Using a calculator it gives me ...
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Is log-general type an intrinsic property of a variety

Let $X$ be a smooth quasi-projective variety over $\mathbb C$. Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor ...
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The first chern class of Fano manifold

If $M$ is a Fano manifold, $L$ is an ample line bundle over $M$. My question is that whether $c_1(L)=\alpha c_1(M)$ for some real number $\alpha$ always holds.
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
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extension of holomorphic functions on complex manifold

$M$ is a compact complex manifold of complex dimension $n$, and $D$ is a simple smooth divisor on $M$. My question is that whether a holomorphic function on $M-D$ can be extended to $M$. If not, what ...
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1answer
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Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
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2answers
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If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z $ is real, what is $z$?

Two questions from Grade Twelve class on complex numbers If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$? How do you solve graphically given the ...
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Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
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Why are these triangles formed by the product of two complex numbers similar?

I was trying to understand Eulers formula from this link and I came across this image on the second slide: I'm trying to understand why the specified triangles are similar. One intutive ...
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Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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Is $L^{*}L$ a real operator?

let $(M,h)$ be a compact complex manifold with a hermitian metric $h$. Let $L$ be a $\mathbb{C}$-linear differential operator with smooth coefficients \begin{equation} ...
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1answer
31 views

Vanishing of Nijenhuis tensor given complex linearity?

I believe this is a very simple question but I do get stuck here. Given the assertion that Lie bracket is complex linear for $v\to[v,w]$ (i.e. commutes with almost complex structure $J$), how can I ...
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1answer
88 views

Meaning of **Canonical metric** on complex manifolds

What is the meaning of Canonical metric on complex manifolds ?
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Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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35 views

Intersection product of submanifolds of complex manifolds - Selfintersection

I do not understand something about the intersection product. I'm kind of new to this topic, so please consider that. I write down everything we discussed in a lecture. We defined the intersection ...
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Number of vectors which are $\alpha$ angle apart

Let, $A\subseteq\{z=(z_1,z_2)\in\mathbb{C}^2:|z|^2=|z_1|^2+|z_2|^2=1\}$ such that any two vectors in $A$ have angle between them $\ge\alpha$ for some $0<\alpha<1$. I want to prove that ...
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Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...