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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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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25 views

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 ...
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30 views

relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
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51 views

Using the Riemann Hurwitz Formula

I am working with the function $f(z)=\frac{z^3}{1-z^2}$ from the Riemann Sphere to itself. I'm trying to show that this satisfies the Riemann-Hurwitz formula given ...
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57 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
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43 views

Holomorphic line bundle over complex torus.

Let $X$ be a complex torus, given by $X = \mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$. How to specify a holomorphic line bundle over $X$? One standard way is to glue it together from trivial bundles ...
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1answer
34 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
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21 views

If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
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24 views

Rank of a Holomorphism

Let $f \colon M \rightarrow N$ be a holomorphism of complex manifolds. Let $p \in M$. Let $(U,\phi)$ and $(V,\psi)$ be coordinate charts on $M,N$, respectively, satisfying $U \ni p$ and $V \ni f(p)$. ...
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23 views

computing the components of $f^*g_N$

Let $M$ and $N$ are to compact complex manifolds of dimensions $m$ and $k$ respectively, and $f:M\to N$ is a holomorphic map then how can we compute the components of $f^*g_N$
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51 views

Question about a paragraph in the book complex analysis by Ahlfors.

By $C_1$, we denote family of circles passing through $a,b$ and by $C_2$ we denote family of Appolonius circles with limit point $a,b$. In section $3.5$ entitled Families of circles, in one paragraph ...
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1answer
27 views

What is the Weil-Petersson metric of the moduli space of elliptic curves?

One can define the Weil-Petersson metric on the moduli space of Riemann surfaces. I would like to know an explicit example of such a metric. What is the Weil-Petersson metric of the moduli space of ...
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1answer
76 views

Analog of holomorphic Lefschetz fixed point theorem for smooth algebraic varieties

If $X$ is a compact complex manifold and $f: X \to X$ is a holomorphic map with isolated nondegenerate zeroes. Then there is a version of Lefschetz fixed point formula with traces on Dolbeaut ...
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1answer
56 views

set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
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53 views

use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
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21 views

Left and right module on the cohomology of a sheaf

Let $X$ a topological space, say a complex variety, and $\mathbb{C}_X$ its constant sheaf. $\mathcal{D}(X)$ is the derived category of sheaves of $\mathbb{C}_X$-modules. Let $F^\bullet\in ...
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71 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
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37 views

Example of irreducible analytic set, which does not define irreducible analytic germs at all of its points

Is the set $\mathbb C^2 \supset\{ y^2=x^2(1+x) \} $ a good example of irreducible analytic set, which does not define irreducible analytic germs at all of its points? I tried to construc it with some ...
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29 views

is this definition of meromorphic function missing a requirement?

Looking at Definition 1.1.33 in the book Complex geometry by Huybrechts: Def. Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere ...
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49 views

complement of zero set of holomorphic function is connected

I'm stuck with the following part of exercise 1.1.8 in Hubrechts book Complex geometry: Prove that, if $U \subset \mathbb C^n$ is open connected, then $U \setminus Z(f)$, the complement of zero set ...
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74 views

Calabi-Yau manifolds and immersion in real space [closed]

I'm reading some papers how to test extra dimensions in LHC experiments and they suggests CY manifolds as starting point. Is it possible that accelerator itself is made in higher-dimensional geometry ...
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16 views

Minimum height of convex area, with minimum area dependant on number of fixed length sides

I've come across a problem while coding that can be solved simply, to an adequate standard, but I'm sure has a very interesting perfect solution. The problem is as follows: Given N lines of ...
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75 views

Physical or geometric meaning of complex derivative?

As in, the real derivative of a function at a point is a slope of a function at that point. What is the physical or geometric meaning of complex derivative of a function at a point? Any help is ...
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53 views

truncate power series to approximate holomorphic function by polynomial

Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
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1answer
41 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
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39 views

Do the induced metrics on the dual/tensor product bundle behave well with each other?

Let $E,F$ be complex (holomorphic) vector bundles over a smooth complex manifold $M$. Assume $E$ and $F$ are equipped with Hermitian metrics $h$ and $k$. This induces a metric on $E\otimes F$ namley ...
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121 views

Zero moment of arc length measure

Suppose $\gamma$ is a simple smooth closed curve and is not a circle. Does there exist a monomial $z^n$ so that $\int_{\gamma}z^n ds(z)=0$ for some positive integer $n$? (In here, $ds$ is the arc ...
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1answer
39 views

Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
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50 views

Hard Lefschetz Thereom and the Cohomology of Flag Varieties

Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag manifold. Now $F$ is a compact Kahler ...
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239 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
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67 views

$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
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2answers
50 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
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1answer
47 views

What is meant by “rigidity of a geometric structure”?

I often heard description like "complex structures are much more rigid than smooth structures", but I have never managed to understand this notion of rigidity. When I asked what "rigidity" exactly ...
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42 views

Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
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32 views

References for the moduli space of conformal structures on a disk minus boundary points

In an article I'm reading, it is said that the moduli space of conformal structures on a disk minus $k+1$ boundary points is a ($k-2$)-dimensional manifold. I want to understand why and have done some ...
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38 views

About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
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2answers
82 views

Group of automorphisms of a compact hyperbolic Riemann surface is finite

Let $M$ be a compact hyperbolic Riemann surface. Is there a simple way to show that the automorphism group $Aut(M)$ of conformal self-mappings of $M$ is a finite group? Recall that a hyperbolic ...
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1answer
89 views

Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
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1answer
55 views

Definition of Hodge structure: is torsion allowed?

I am trying to understand the definition of an integral Hodge structure. Apparently, for $X$ a compact Kahler manifold, $H^n(X,\mathbb R)$, the lattice $H^n(X,\mathbb Z)$ and the Hodge filtration give ...
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27 views

A have a quick question about moishezon manifold

Is moishezon manifold general type?
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1answer
57 views

Examples of varieties with torsion in their integral Hodge structure

I am not so used to thinking about integral Hodge structures, so this question might be completely trivial. What are easy and interesting examples of smooth projective connected varieties $X$ with ...
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1answer
68 views

Some questions about complex curves in $\mathbb CP^2$

I would like to ask for some clarifications in the following questions about complex curves. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a ...
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63 views

Arc length inequality on ellipse

Suppose $E$ is the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1.$ For $P,Q\in E$ let $\sigma(P,Q)$ denote the length of the minor arc connecting $P$ to $Q$. Find the maximum value for $A>0$ such that ...
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Does the conformal class of a complex projective curve contain the Fubini-Study metric?

Let $X \subset \mathbb CP^2$ be a complex curve with metric $g$ induced by the Fubini-Study metric on $\mathbb CP^2$. Since in the case of two-dimensional real manifolds a complex structure is ...
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1answer
45 views

Intuition for the limit of complex functions

We have intuition and somehow geometrical point of view about the limit in the Real functions.I mean we can think of the limit on the graph of the Real function and imagine how close we can get on the ...
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34 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
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18 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
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68 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
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49 views

A reference on Kodaira-Spencer deformation theory

I am looking for an introduction to Kodaira-Spencer deformation theory. I have a background in Teichmüller theory, but I know almost nothing of (and am not so interested in) algebraic geometry. Do ...
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2answers
46 views

Is it possible to realize a general compact Riemann surface in $\mathbb CP^2$?

Let $X$ be a compact Riemann surface with smooth boundary $\partial X$. Is it always possible to realize $X$ as a complex submanifold of $\mathbb CP^2$? In other words, is it true that there exists a ...