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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How to prove that in a Kähler manifold without boundary $\Omega \wedge \cdots \wedge \Omega$ is closed but not exact?

Let $M$ be a compact Kähler manifold without boundary. How can I show that the volume form $$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$ where we have the wedge of $m$ $\Omega$s is ...
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0answers
7 views

anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
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24 views

Genus formula for a curve in a $2$-dimensional complex torus?

For a curve $C$ in a $2$-dimensional complex torus $T$, is there any formula to compute the genus of $C$? Say, in terms of self-intersection number? For a curve in $\mathbb{P}^2$ or a K3 surface, ...
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1answer
27 views

Automorphisms of manifold vs automorphisms of Hodge structure

Let $X$ be a compact Kahler manifold. Let $H^k = H^k_{prim} (X)$ be primitive cohomology of $X$ considered as an object in category of of polarized Hodge structures. We have a map \begin{equation*} ...
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1answer
33 views

Analytic paths through converging sequences in the complex space.

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
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0answers
30 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
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1answer
22 views

If $V$ irreducible an. variety $V=V_1\cup V_2$, then $V_1\cap V_2\subset V_{sing}$

Let $V$ be an irreducible analytic variety, and $V_1, V_2$ analytic subvarieties such that $$V=V_1\cup V_2.$$ In Griffiths-Harris book, it is mentioned that $V_1 \cap V_2$ is a subset of the ...
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0answers
22 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
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0answers
18 views

Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
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1answer
18 views

Center and angle of complex function

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?
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21 views

Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
2
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1answer
33 views

An injective morphism between varieties that is not an immersion

I believe this is relatively elementary, but I'm struggling to think of an example of a morphism $f: X \rightarrow Y$ between varieties which isn't an immersion in the sense of algebraic geometry. ...
2
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0answers
43 views

What is the space $T_Y^*X$?

Help with notation! I'm a physicist, but I've come across the following notation, $T_Y^* X$ where $Y$ is a complex analytic submanifold of $X$. A phrase I've heard used is "conormal bundle" but is ...
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0answers
36 views

Projection of surfaces in $\mathbb{C}^2, \mathbb{C}P^2, \mathbb{C}H^2$ to $\mathbb{R}^3$

As part of my thesis in Riemannian geometry, I study surfaces in $\mathbb{C}^2$, $\mathbb{C}P^2$ and $\mathbb{C}H^2$. Since visulation is always nice, I was wondering if there existed any "nice" ...
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0answers
41 views

Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
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0answers
60 views

Geometry of cubic 3-fold

I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$. Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained ...
2
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1answer
51 views

$R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.

I have some questions on local systems. If $f:X\to Y$ is a morphism of projective complex algebraic varieties, $Y$ being a curve, I want to prove that if the fibers of $f$ are smooth hypersurfaces in ...
3
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0answers
34 views

When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
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0answers
22 views

Notions of convergence for maps of complex manifolds

Suppose that $X_t$ and $Y_t$ are two families of complex manifolds, parametrized by maps $\sigma : \mathcal X \to T$ and $\tau : \mathcal Y \to T$. Suppose too I have holomorphic maps $f_t : X_t \to ...
2
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1answer
15 views

Curvature for tautological bundle of projectivation

I'm trying to compute locally the curvature of $\mathscr{O}_{\mathbb{P}(E)}(-1)$, where $E\to X$ is a holomorphic bundle. The covering is given by $U_i\times \mathbb{P}(E)_j$, where $\{U_i\}$ is a ...
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1answer
17 views

Local expression of hermitian metric

I have really hard times reading Zheng's Complex Differential Geometry and I find the following sentence especially baffling (sec. 7.4, page 170): "Let $M^n$ be a complex manifold. A Hermitian metric ...
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1answer
53 views

Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...
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3answers
48 views

Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
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1answer
50 views

Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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1answer
20 views

The exact sequence $0 \rightarrow L(-1) \rightarrow L(0)^2 \rightarrow L(1) \rightarrow 0$

I am trying to show that there is an exact sequence: The exact sequence $0 \rightarrow L(-1) \rightarrow L(0)^2 \rightarrow L(1) \rightarrow 0$, where $L(-1)$ is the tautological line bundle of ...
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0answers
35 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
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0answers
36 views

Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
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0answers
34 views

What is a real structure on a manifold?

I have been looking at manifolds (twistor spaces) that have a "real structure". I am not quite sure what this means. I've looked on Wikipedia and they have an article that explains real structures on ...
2
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1answer
24 views

A clarification of addition on elliptic curves over the complex numbers

I am trying to prove that the order of the two points $P_{\pm}=(0,\pm\sqrt{-g_3})$ is three on the elliptic curve $y^2=4x^3-g_3$, for $g_3 \not= 0$, defined over $\mathbb{P}^2_{\mathbb{C}}$. Here's an ...
3
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1answer
30 views

The Classification of almost complex structures (almost) tamed by a quadratic form

Preamble I am currently dealing with the almost complex structures associated with a non-degenerate quadratic form. In particular, I am interested the size of the almost complex structures as a ...
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0answers
38 views

$x^3+y^3+z^3 = 0$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega \not= 1\}$

I am rather stuck trying to prove that $x^3+y^3+z^3 = 0$ in $\mathbb{P}^2_{\mathbb{C}}$ is isomorphic to $\mathbb{C}/\Lambda$, where $\Lambda = \{n+m\omega \mid n,m \in \mathbb{Z}, \omega^3=1, \omega ...
2
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1answer
39 views

$j$-invariants of isogenous elliptic curves

Suppose that $E,E'$ are isogenous smooth complex elliptic curves - is there some relation between their $j$-invariants?
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1answer
41 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
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0answers
32 views

About rational curves on elliptic K3 surfaces

I am trying to read this paper http://arxiv.org/pdf/math/9902092.pdf. I have some trouble at the end in Corollary 3.27, and also Proposition 3.24. 0)Morally speaking my main trouble is that i don't ...
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2answers
28 views

Holomorphic curve with unit norm

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \parallel \gamma(z) \parallel =1$. if the answer is yes, can the ...
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1answer
88 views

‘Integral’ of a Weierstrass $ \wp $-function.

I'm revising for my finals and I've seen a question which asks: Is there a meromorphic function $f: \mathbb{C}/\Lambda \to \mathbb{P}^1$ such that $f' = \wp$? There is a hint which says consider the ...
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1answer
46 views

What exactly is the Kahler class of a torus?

Is there an intuitional way to understand what the Kahler class of $T^2$ actually is? It would be extremely useful to me if you could provide me some intuition behind it!
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0answers
80 views

What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to X$ a ...
2
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1answer
30 views

Description of $(T\Bbb{CP}^1)^\perp$

Is there a nice "concrete" description (i.e., coordinates) of the normal bundle of $\Bbb{CP}^1$ when is considered as a submanifold of $\Bbb{CP}^n$? Or, at least, $\Bbb{CP}^2$?
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2answers
20 views

Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
5
votes
2answers
171 views

References to understand $K3$ surface as a double cover of $\mathbb{P}^2$ ramified along a sextic

My goal is to understand that $2:1$ cover of $\mathbb{P}^2(\mathbb{C})$ ramified along a sextic is a $K3$ surface. My main problem is in understanding the theory of ramified covering of ...
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0answers
23 views

Dévissage for complex manifolds

In algebraic geometry one has the following result: Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} ...
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1answer
54 views

What is the meaning of this statement about complex structure?

I get confused when in papers it is said that: "Something is holomorphic (Complex, symplectic, etc ...) in some Complex structure" What is the meaning of this in general? For example in a paper ...
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1answer
92 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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1answer
61 views

Change of Eigenvectors by Hadamard product.

Please forgive me if my question is not clear and appropriate but please do not give negative marking as I am trying very hard to answer this question; thank you. My question is related to the ...
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0answers
34 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
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2answers
95 views

Does a complex manifold always admit an acyclic cover for the sheaf of holomorphic functions?

Let $\mathcal{F}$ be a sheaf on a topological space $X$. An open cover $\mathcal{U} = \{U_i\}_{i\in I}$ of $X$ is called acyclic for $\mathcal{F}$ if for all $i_0, \dots, i_p \in I$, ...
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0answers
28 views

When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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72 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
3
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1answer
133 views

Classification of line bundles by Griffiths and Harris

I am reading pages $132$ and $133$ of Principles of Algebraic Geometry by Griffiths and Harris. They consider a holomorphic line bundle $L \to M$ over a manifold $M$ and an open cover $\left\{ ...