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Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

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Do complexification and exterior power commute?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Are $(\bigwedge^k V)^{\mathbb{C}}$ and $\bigwedge^k (V^{\mathbb{C}})$ naturally isomorphic? They both have the same complex ...
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Algebraic forms on an elliptic curve

On an elliptic curve defined by the equation, $$E:y^2=x^3+a x +b$$ The algebraic form $dx/y$ is defined on the elliptic curve and it is a non-vanishing section of the (trivial) canonical bundle. From ...
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Classification of Complex Surfaces. Illustrations of the failure of genus to provide a good classification.

Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which ...
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1answer
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This covering map is homeomorphism

Suppose that $f:\mathbf RP^2\rightarrow X$ is a covering map and $X$ is a CW-complex. Show that $f$ is homoemorphism. We know covering map is continuous and onto,so we should show that $f$ is one-...
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What is the codimension of the set of non-node singularites?

Let $F$ be a given homogeneous polynomial in $\mathbb C[x_0,\ldots,x_n]$. It defines some hypersurface $X=Z(F)$ in $\mathbb {P}^n$. Let $$U:=\{x\in X:\text{$x$ is a non-node singular point in $X$}\}.$...
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Direct proof that a parallel almost complex structure is integrable

Let $(M,g,J)$ be an almost Hermitian manifold nad $\nabla$ be the Levi-Civita connection on $M$. If $\nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which ...
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1answer
35 views

Is a projective algebraic manifold irreducible algebraic set in $P^n$?

A compact complex manifold $X$ which admits an embedding into $P^n(\mathbb{C})$ (for some $n$) is called a projective algebraic manifold. And by a theorem of Chow, every complex submanifold $V$ of $P^...
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42 views

Genus of a smooth projective curve

I was trying to prove that the genus of a smooth projective complex curve $F=0$ of degree $d$ is $(d-1)(d-2)/2$. My attempt was to take the standard projection $$\pi:\mathbb{P}^2 \to \mathbb{P}^1$$ $$...
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Apparent flaw in Harris and Griffiths' proof of the Proper Mapping Theorem

I have found an apparent contradiction in the proof of the Proper Mapping Theorem given by Griffiths and Harris in their book "Principles of Algebraic Geometry". It probably is not a contradiction but ...
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Analytic sets in a Manifold

For a holomorphic mapping $ f:M\to N$ of manifolds, where $k\in\mathbb{N}$ is fixed. The set $\{x\in M:rk_xf\le k\}$ is a analytic by the Rank Theorem. I dont see why this result holds trivially, ...
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Meaning of $h_*$ complex map.

I have come across the following: Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and let $h: \Omega \rightarrow \mathbb{C}^n$ (or $\mathbb{R}^n$) be a smooth mapping with $||h_*||= sup_x || h_*(x)|...
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Unique complex structure on the modular curve $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$

Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$) According to ...
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Explicit equation for the dual hypersurface

Let $X$ be a hypersurface in $\mathbb P^n$. Assume $X$ is defined by some homogenous polynomial $F$. Then, is there a way to write the equation for $X^\vee$ explicitly? (This is related to my ...
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1answer
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Proof regarding Stein's and Sharachi's proof of the unit circle mapping onto the upper half plane - complex analysis

I am currently working with Stein's and Shakarchi's Complex Analysis and trying to comprehend the proofs of certain theorems. But unfortunately i can't get behind the equation used in the proof of ...
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Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $...
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homeomorphic self maps of open unit disk?

It is well known that the conformal, bijective self maps of open unit are given by the set of Möbius transformations of the form $$e^{i\theta}\left(\dfrac{z-a}{1-\overline{a}z}\right).$$ Now I am ...
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Constructing diffeomorphisms of moduli spaces of $J$-holomorphic curves

Let $M^{2n}$ be a smooth manifold admitting two almost complex structures $J_0$ and $J_1$. Suppose that $J_0$ and $J_1$ are both regular in the sense that the moduli space $$ \mathcal{M}_i:=\mathcal{...
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Can we ignore the ``holomorphic trivialisation'' in the definition of a holomorphic vector bundle?

I have learnt two definitions about holomorphic vector bundles over a complex manifold $M$. $E\to M$ is a smooth complex vector bundle with a trivialisation such that the transition functions are ...
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argument of complex number for bode plot

Could someone help me please. I'm looking for the value of $\omega$ $\text{Argument}(\frac{1,6}{(1+0,004\text{j}\cdot\omega)(1+0,04\text{j}\cdot\omega)})=-135° $ $\text{Argument}({1,6})-\text{...
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CR holomorphic functions

Let $\Omega \subset \mathbb{C}$ be a domain, $\mathcal{O}(\Omega)$ denote holomorphic functions on $\Omega$ and $\mathcal{C}^{\infty}(\overline{\Omega})$ functions smooth up to the boundary. I'm ...
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Construct bundles/sheaves associated to a lower dimensional subvariety

For simplicity, suppose $X$ is a smooth projective variety of dimension $n \geq 3$, and $Y$ is a smooth subvariety of $X$. If the dimension of $Y$ is $n-1$, i.e. $Y$ is a divisor, then we can ...
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Singular locus of dual hypersurfaces

Everything is over field $\mathbb C$. Let $X$ be a hypersurface of degree $d$ in $\mathbb P^n$. We know that if $X$ is smooth, then its dual $X^\vee$ is still a hypersurface in $(\mathbb P^n)^\vee$, ...
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Books for Complex Differential Geometry: index approach

What are some books (if any) that tackle complex differential geometry with tensors in index notation? My studies have led me to an area where I think I must proceed with that field; however, I have ...
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Automorphism of an open subset with completement of codimension $2$

Let $\mathbb P^n=\mathbb {CP^n}$, I guess the following is true: Let $D\subset \mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $\mathbb P^n-D$ is linear, i.e. ${...
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Why is the Albanese map well-defined?

Let $X$ be a compact Kähler manifold of complex dimension $n$. $Alb(X):=\frac{H^0(X, \Omega_X)^*}{\rho(H_1(X,\mathbb{Z}))}$, where $\rho:H_1(X, \mathbb{Z}) \to H^0(X, \Omega_X)^*$ is given by $[r]\...
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26 views

Counterexample of Remmert's theorem without properness

The Remmert's proper mapping theorem states Remmert's proper mapping theorem: Let $f:X\to Y$ be a proper holomorphic map of complex spaces and $A\subset X$ a closed analytic subset. Then $f(A)$ is ...
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60 views

Infinite cyclic cover corresponding to non-zero cohomology class $\alpha \in H^1(x,\mathbb Z)$

I want to understand the following sentence: Let X a compact (complex) manifold which has a non-zero cohomology class $\alpha \in H^1(X,\mathbb Z)$. Let $\pi: \bar X\to X$ be the corresponding ...
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A homotopy where all intermediate maps have holomorphic antiderivatives

I will denote by $\mathbb{C}^*$ the punctured complex plane, $\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane, $w: \mathbb{C}^* \to \mathbb{C}$, such that the ...
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69 views

Deforming antiholomorphic involutions

Let $(M,J)$ be a compact smooth almost complex manifold. We can "deform" $J$ as follows: if$A$ is a smooth section of the endomorphism bundle $\mathrm{End}(TM)\to M$ satisfying $ AJ=-JA, $ it follows ...
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1answer
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What does it mean for a complex differential form on a complex manifold to be real?

I am trying to read Kobayashi's "Differential geometry of complex vector bundles". There are many places where a complex differential form is referred to as being real. e.g Chapter I, Proposition 7....
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Constructing a homotopy of nonzero holomorphic functions using local homotopies

I'll denote by $\mathbb{C}^*$ the punctured complex plane $\mathbb{C} \setminus \{0\}$. Say that I've got some open cover $\{V_j\}_{j \in J}$ of the closed unit interval $[0,1]$, and continuous ...
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1answer
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Questions regarding Connections, and in particular, Hermitian Connections

I have been reading Chapter 0 of Griffiths' and Harris' Principles of Algebraic Geometry, in particular, the section on Vector Bundles, Connections, and Curvature. I have three questions: A ...
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1answer
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Showing that an elliptic function has no poles

Let $\Lambda = \{m \omega_1+n\omega_2; m,n \in \mathbb{Z}\}$ with $\omega_i \in \mathbb{C}$ with $\omega_2/\omega_1 \notin \mathbb{R}$ be a lattice. Define the Weierstrass $\mathscr{P}$ function on ...
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Todd class of a two-dimensional bundle

Let $M$ be a complex 2-dimensional manifold. I am suggested to prove that the Todd class of $TM$ is $1$, but i can't quite believe that it indeed holds. Given a bundle $\xi: E \rightarrow X$ of rank $...
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Is there a pattern for closed and co-closed $n$-forms on $\mathbb{R}^{2n}$?

Consider $\mathbb{R}^{2n}$ with its standard Euclidean Riemannian metric. Let $\omega \in \Omega^n(\mathbb{R}^{2n})$ be an $n$-form. I am trying to understand if there is a succinct way to express ...
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1answer
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Computing Fubini-Study metric from the formal definition

Definition: the Fubini-Study metric $g_{FB}$ on $\mathbb{CP}^n$ is the only metric which makes the projection $\pi:(\mathbb{S}^{2n+1},g)\to(\mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is ...
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Condition for $(1,1)$-classes on complex tori

Let $X = V/\Lambda$ be a complex torus, where $V$ is a complex vector space and $\Lambda \subset V$ is a full-rank lattice. We can identify the cohomology group $H^{2}(X, \mathbb{Z})$ with ...
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1answer
31 views

The homomorphism induced by holomorphic map preserves Hodge decomposition

Let $f:X\to Y$ be a holomorphic map between two compact Kähler manifolds. Then the induced map $f^*:H^1(Y,\mathbb{C})\to H^1(X,\mathbb{C})$ preserves the Hodge decomposition. Is there a reference for ...
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Non split real form of projective space

On the complex projective space $\mathbb{P}^1_\mathbb{C}$ we have an involution $z\mapsto -\frac{1}{\bar{z}}$. Using this as descent datum we should end up with a real form, which is not split (this ...
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Holomorphic forms: a very basic notion

Let $M$ be a holomorphic (complex) manifold with: $$(1)\quad dim_{C}(M)=m. $$ What I understand regarding a holomorphic form is that it is: $$(2)\quad \alpha^{(r,0)}=\frac{1}{r!}f_{\mu_1,\dots,\mu_r}...
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Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
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Use of the Rellich Lemma in the proof of the Hodge Theorem

I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100. The method of proof is to establish a weak solution of ...
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1answer
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Kähler manifolds are formal

I want to understand why Kähler manifolds are formal. This was first proved by Deligne, Griffiths, Morgan, Sullivan Let $\mathcal M$ be a minimal differential algebra and $H^*(\mathcal M)$ the ...
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1answer
66 views

Globally Generated Vector Bundle on a Riemann surface

This is a very vague question, in fact not really a question at all more of a search. I am studying some vector bundle theory on Riemann surfaces and would just like some non-trivial example of ...
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Every holomorphic vector bundle on a stein manifold is Nakano positive?

I encounter this statement on page 53 of Takeo Ohsawa's L2 Approaches in Several Complex Variables but I don't know how to prove it. Could anyone explain it to me or give me a reference for it?
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The topology of nonsingular projective variety and its corresponding manifold.

A nonsingular projective variety is a manifold, then if the topology of the variety can be identified with the topology on the corresponding manifold? in other words, if any open set $U$ of the ...
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A geometric proof of Picard's little theorem

I'm preparing a presentation where I'd like to present a proof of Picard's little theorem using hyperbolic geometry. Picard's little theorem states that the range of an entire function can omit at ...
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Reference for Hodge decomposition for flag variety

Any references for proof of the following facts: The cohomology of the (complex) flag variety is always in $(p, p)$-type of Hodge Decomposition. The natural map $G/T → G_\mathbb{C}/B$ is a ...
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When can compact group actions be complexified?

Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the ...
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1answer
152 views

Positivity of Currents

I already asked about this a couple of weeks ago but had introduced some rather annoying notation. I decided to reformulate the question in a more compact format. (edit: old post taken down as it is ...