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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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How is real and imaginary part of a complex polynomial defined

I have been reading a survey on polynomial optimization, where on page 15, Lemma 2.5, the author used notations such as $Re(p_{z})$ and $Im(p_{z})$, where $p_{z} \in \mathbb{C}[\mathbf{x}]$, for $\...
wsz_fantasy's user avatar
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Poincare residue and trivializing section of canonical bundle of plane cubic.

I am trying to get my hands dirty and do the following computation, but I don't feel like I'm doing it right. Help would be very much appreciated! I will tell you the setup of the calculation, and ...
maxo's user avatar
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Choice of local holomorphic coordinates on Hermitian manifolds

I'm reading Fangyang Zheng's Complex Differential Geometry. I have a question about the details while reading a proof of a lemma. This question can be described as follows: Let $(M, h)$ and $(N, g)$ ...
HeroZhang001's user avatar
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What is the meaning of "trace of a form with respect to the Kähler form"?

I'm reading Fangyang Zheng's Complex Differential Geometry. I have a problem with terminology while reading the following lemma. Lemma 7.22 (Lu's Inequality). Let $ (M, h) $ and $ (N, g) $ be ...
HeroZhang001's user avatar
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Example of semi-stable non sufficiently smooth vector bundle

Let $X$ be a smooth projective complex variety and $H = c_1(\mathcal{O}_X(1))$ an ample class (if you prefer, you can take more generally $X$ to be a compact Kähler manifold with $H = [\omega]$ its ...
Cactus's user avatar
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Normalization of Algebraic Curves

I'm reading through Simon Donaldson's Riemann Surfaces, and am confused about his construction for normalizing algebraic curves. He gives the example of normalizing $w^2 - z^2 (1 - z)$, giving the ...
Nes37's user avatar
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Uniformization Theorem and Non Existence of Family of Elliptic Curves over Riemann Sphere

A question concerning a statement from these notes introducing/motivating period map. On first page, left column, last sentence states: Since $\Bbb{H}$ (=complex upper half plane) is biholomorphic to ...
user267839's user avatar
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Uniqueness of the decomposition of rank 2 holomorphic line bundle over $\mathbb{CP}^1$

Let $\mathcal{O}(1)$ be the hyperplane bundle over $\mathbb{CP}^1$ and $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}$. I knew that any rank 2 holomorphic line bundle over $\mathbb{CP}^1$ is isomorphic to ...
Rosalina's user avatar
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Show that $A(C)$ is a vertical line or half circle orthogonal to $\mathbb R.$

Here is the question I am trying to tackle: Let $C \subset \mathcal{H}^2$ be a half circle orthogonal to $\mathbb R$ or a vertical line and let $A: \mathcal{H}^2 \to \mathcal{H}^2$ be a Möbius ...
Intuition's user avatar
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Sheaf $\mathscr{F}_S$ for which $\mathscr{F}_S(U)$ consists of holomorphic sections that vanish on $S \cap U$ isomorphic to $\mathscr{O}(L^*_Y)$

Let $M$ be a complex manifold and $Y \subset M$ a closed hypersurface and $L_Y$ the holomorphic line bundle associated with $Y$. How can I show that the sheaf $\mathscr{F}_Y$ for which $\mathscr{F}_Y(...
Elena's user avatar
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How can both the Čech complex and the alpha complex have the same homotopy type as the union of balls if they are constructed differently?

I understand that, according to the nerve theorem, both Čech and alpha complexes have the same homotopy type as the union of balls. However, consider the following four points: A = $(1,0)$ B=$(-1,0)$,...
QuinnTheEskimo's user avatar
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Map of global sections surjective under a local condition

Let $M$ be a compact complex manifold and $p \in M$. Let $L \to M$ be a line bundle on $M$ and $\mathcal{F}_{\{p\}}$ be the sheaf of holomorphic sections of $L$ that vanish at $p$. Locally around a ...
Elena's user avatar
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Kahler-Einstein metric on complex projective space

I think this question may be well-known to the experts; or someone may have already asked the following question in this website. Since I couldn't figure it out myself and I couldn't find a related ...
Tong's user avatar
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Decomposition of a homology class $C \in H_2(X, \mathbb{Q})$ of a smooth projective complex algebraic variety

Let $ X $ be a smooth projective complex algebraic variety of dimension $ n $. Prove that for any homology class $ C \in H_2(X, \mathbb{Q})$, the following two conditions are equivalent: $$\int_C \...
Alex Gomez's user avatar
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Are all complex algebraic varieties in 2+ variables unbounded as a result of Hartogs' extension theorem?

Hartogs' extension theorem states that for any $n\geq 2$, $U\subseteq\mathbb C^n$, $K\subset U$ compact (in $\mathbb C^n$) and a holomorphic function $f:U\setminus K\to\mathbb C$, it can always be ...
Boris Dimitrov's user avatar
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Isomorphism of real vector spaces $\Omega^1(\mathfrak u(n)) \cong \Omega^{0,1}(\mathfrak{gl}(n,\mathbb C))$

Consider a compact Riemann surface $X$. In [1, p. 570] the isomorphism of real vector spaces $$ \Omega^1_{\mathbb R}(X,\mathfrak u(n)) \cong \Omega^{0,1}(X,\mathfrak{gl}(n,\mathbb C)) $$ is mentioned. ...
mifrandir's user avatar
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1 answer
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Confusion about (closed) submanifolds of Stein manifolds

I'm starting to study Stein manifolds and I think I'm confused by something that is probably very elementary (which possibly highlights how little I truly understand about basic manifold theory in ...
Maths Matador's user avatar
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Chern connection and Levi-Civita connection

It is stated on Voisin's book in complex algebraic geometry that on a Kahler manifold, the Chern connection and the Levi--Civita connection coincides on $T_X$. I wonder what this exactly means, as ...
fyx1123581347's user avatar
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Algebraic Geometry: Some questions about a example of irreducible hypersurface and singular points

Some definitions: Let $ M^{n} $ be a complex manifold of dimension $ n$. For an open subset $ U \subset M$, denote by $ \mathcal{O}(U) $ the ring of all holomorphic functions on $ U$. A hypersurface ...
HeroZhang001's user avatar
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3 votes
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Kahler differentials and etale fundamental group

In Algebraic geometry we can define two notions that generalize tools from topology/differential geometry: the etale fundamental group and the module/sheaf of Kähler differentials. In the complex ...
user720386's user avatar
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Order 2 branch cut and different sheet structures on Riemann surfaces

I am trying to understand some simple branch structures of Riemann surfaces with order 2 ramification / branch points. Let's talk about surfaces in $\Sigma \subset \mathbb{C}^2$ cut out by a ...
Samuel Crew's user avatar
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1 answer
65 views

Is there an analogy of associated points for sheaves on complex analytic spaces?

Let $\mathcal F$ be a coherent sheaf on a complex analytic space $X$. I was wondering if there is an analogy to what is called associated points in scheme theory. Searching the internet didn't yield ...
red_trumpet's user avatar
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2 votes
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Jet bundle question

Let $E \to M$ be a holomorphic vector bundle. Is there a metric on the first jet bundle $J^1 E$ that can be defined in terms of metrics on $E$ and $M$?
Hammerhead's user avatar
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Error in "Principles of Algebraic Geometry" by Griffiths and Harris

At page $148$ of "Introduction to Algebraic Geometry", Griffiths and Harris define a positive line bundle as a line bundle $L\to M$ with a metric such that $(i/2\pi)\Theta$ is a positive $(1,...
Temoi's user avatar
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Understanding of the set of all Kähler forms on a compact complex manifold $X$ is an open convex cone in $\{\omega\in\mathcal A^{1,1}(X)|d\omega=0\}$

I am reading Complex Geometry by Daniel Huybrechts. I have a problem when reading the proof of the following corollary: Proposition. The set of closed positive definite (i.e. $\omega$ is locally of ...
HeroZhang001's user avatar
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5 votes
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Curvature form and Curvature tensor in complex vector bundle

Let $X$ be a complex manifold and $\pi:E\rightarrow X$ be a complex vector bundle. A connection on $E$ is a $\mathbb C$-linear operator: $$\nabla:\mathcal C^\infty(X,E)\rightarrow\mathcal C^\infty(X,\...
N00BMaster's user avatar
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Computing the second Chern character of $\text{End}(E)$

Let $E \to M$ be a complex vector bundle with a hermitian metric $h$ and $\nabla$ a connection on $E$. Its quite straightforward to conclude that $c_1(\text{End}(E)) = 0$. I'm trying to verify the ...
Jonathan's user avatar
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Singular locus of analytic variety

Suppose $X$ is a complex manifold and $V \subseteq X$ is an analytic subvariety of $X$, i.e., $V$ is locally the zero locus of a finite collection of holomorphic functions. A point $x \in V$ is a ...
Frank's user avatar
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3 votes
1 answer
118 views

Do coherent sheaves on compact Kähler manifolds admit locally free resolution?

Let $X$ be a compact Kähler manifold, and $F$ a coherent sheaf. Does $F$ admit a locally free resolution $$E^* \to F \to 0?$$ Of course it will be enough to construct a surjection $$E^0 \to F \to 0$$ ...
red_trumpet's user avatar
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Does $\hat{\mathscr{O}}_{X,x}\simeq \hat{\mathscr{O}}_{Y,y}$ imply a local ismorphism in the analytic category?

Let $f\colon X \rightarrow Y$ be a morphism between two $\mathbb{C}$-varieties. If the induced map $\hat{\mathscr{O}}_{Y,y}\rightarrow \hat{\mathscr{O}}_{X,x}$ is an isomorphism for two closed points $...
notime's user avatar
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1 answer
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Zero set of analytic function $f$, $g$ on $\mathbb{C}^n$ concides implies that $f$ divides $g$.

If I have two analytic functions $f$ and $g$ on $\mathbb{C}^n$ such that $\{g=0\}\subset\{f=0\}$, what condition does it need to imply that $\frac{f}{g}$ is analytic? I think here we may need some ...
Holden Lyu's user avatar
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Definitions of complex analytic spaces

I have come across two different definitions of complex analytic spaces. One is from the nlab: A complex analytic test space is a common vanishing locus of a set of holomorphic functions $\mathbb{C}^...
Maksim's user avatar
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Obtaining a holomorphic bundle structure on $E$ from $D'' \circ D'' = 0$

I'm trying to prove the following theorem: Let $E$ be a $C^\infty$ complex vector bundle over a complex manifold $M$. If $D$ is a connection in $E$ such that $D'' \circ D'' = 0$, then there is a ...
Tepes's user avatar
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0 answers
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When is a real analytic subvariety complex?

Let $X$ be a complex manifold and $Y\subset X$ a real analytic subvariety, not necessarily smooth. Suppose that there is a dense open subset $U\subset Y$ such that $U$ is a complex analytic subvariety ...
fgh's user avatar
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Fundamental group of a smooth projective surface over $\mathbb{R}$.

Let $X_{\mathbb{R}}$ be a smooth, projective surface over $\mathbb{R}$. Then its complexification, $X$, is a smooth complex projective surface. $X_\mathbb{R}$ is a manifold, so we can talk about its ...
unsure's user avatar
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1 answer
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Can an ample line bundle equipped with a metric which is negative somewhere?

Let $X$ be a compact complex manifold and $L$ a holomorphic line bundle on $X$. As is wellknown, $L$ is ample if and only if $L$ admits a positive Hermitian metric (i.e., its curvature form is ...
Lelong  Wang's user avatar
1 vote
0 answers
16 views

Shilov boundary of unit sphere

Let $B=\{z=(z_1, z_2) : |z_1|^2+|z_2|^2<1\}$ denote the unit ball in $\mathbb{C}^2$ and let $\partial B$ be its topological boundary, i.e. the unit sphere in $\mathbb{C}^2$. One of the texts by J. ...
Curious's user avatar
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2 votes
0 answers
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Pseudoconvex domains in one complex variables

I am trying to prove that every domain in $\mathbb{C}$ with $C^2$ boundary is (Levi) pseudoconvex. For that, suppose $\Omega$ is defined as $\rho(z)<0$, where $\rho$ is $C^2$ defining function in a ...
Soumya Ganguly's user avatar
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35 views

The Laplacian of Kähler potential on a complex-$1$-dimensional Kähler manifold $M$ must be $2$

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have encountered a problem, but if my argument below is correct, then the ...
HeroZhang001's user avatar
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1 vote
1 answer
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How to find out metric for different line bundles over complex projective space

(Tautological line bundle on $\mathbb{C P}^n$) The point on $\gamma_n$ is $(\ell, z) \in \mathbb{C P}^n \times \mathbb{C}^{n+1}$. It is natural to define a Hermitian metric $h$ on $\gamma_n$ by $h(\...
falamiw's user avatar
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1 vote
1 answer
99 views

the existence of solution satisfying a ddbar equation

Denote by $\mathbb D$ (resp. $\mathbb D^*$) the open disc (resp. the punctured disc) in $\mathbb C$. Let $u$ be a smooth function over the total disc $\mathbb D$ (which is not harmonic usually). I am ...
Invariance's user avatar
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1 answer
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Does any complex analytic space has the differential structure?

As well-known a complex manifold always has the differential manifold structure. In fact, in the book "Complex Geometry An Introduction" of Huybrechts, he defines a complex manifold as ...
Lelong  Wang's user avatar
1 vote
0 answers
15 views

Semi-continuity of Lelong number

Demailly gives the following some-continuity result for Lelong numbers: Proposition. Let $T_k$ be a sequence of closed positive currents of bidimension $(p, p)$ converging weakly to a limit $T$. ...
eulershi's user avatar
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Questions about the definition of generalized Lelong number by Demailly

According to Demailly, the definition of Lelong number is local, so assume $X$ is a Stein manifold, $\varphi$ is a continuous psh function (which means $e^{\varphi}$ is continuous. Let $T$ be a closed ...
eulershi's user avatar
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What is C minus infinitely many disjoint "Jordan Balls" conformally equivalent to?

So I recently came across a Theorem due to Koebe proved in 1918 that states that an M connected domain $\mathbb{D}$ in $\mathbb{C}$, that is a a domain such that $\partial \mathbb{D}=\bigcup_{i=1}^{k}\...
Bigalos's user avatar
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1 answer
53 views

A formula resembling the integral mean value on Kähler manifolds

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem: Theorem. ...
HeroZhang001's user avatar
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0 votes
0 answers
58 views

Is any blowup a composition of finitely many blow-ups with smooth centers?

This question may be a well-known fact. However, I cannot find it in any reference. We always assume that the center of a blowup is a closed subvariety in this context. Let $f: X \rightarrow Y$ be a ...
Lelong  Wang's user avatar
2 votes
1 answer
84 views

Showing that a particular function is holomorphic

Here is the problem: I have a diffeomorphism $F: \mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^n \times \mathbb{C}^m$ over the projection $\mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^m$, so in ...
Emory Sun's user avatar
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1 vote
1 answer
68 views

Biholomorphic maps on projective variety

Let $X$ be a projective variety, and suppose it comes with some embedding $X\subset \mathbb{P}^n$. If $f:X\to X$ is biholomorphic, can we say anything like:f is the restriction to $X$ of some ...
unsure's user avatar
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1 vote
1 answer
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Hamiltonian Group Actions on Calabi-Yau Cones

Let $(M, g, J, \omega, \Omega)$ be a Calabi-Yau cone (where $\Omega \in \Gamma(K_M)$ is the parallel holomorphic volume form), and assume we have a Hamiltonian group action $G \circlearrowright M$ ...
Albert Wood's user avatar

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