# 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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### two different definition of relative differential are the same?

Given a smooth morphism $f:(X,\mathcal{O}_X)\to (S,\mathcal{O}_S)$ between two smooth manifolds .I came across some different definition of relative differential in different context define it as ...
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### understanding algebraic proof in complex geometry

I am studying "Condensed Mathematics and Complex Geometry" by Dustin Clausen, Peter Scholze. I came across this theorem and this proof: I don't understand a lot of steps in the proof. I ...
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Let $X$ be a compact Riemann surface, and $f: X \to \mathbb{P}^1(\mathbb{C})$ be a holomorphic branched cover. This induces a finite extension of their fields of meromorphic functions $f^*: \mathbb{C}(... • 3,666 0 votes 1 answer 123 views ### Proof of Spherical metric on Riemann Sphere While studying stereographic projection of extended complex on unit sphere$S$in$\mathbb{R^3}$we get two metrics one is chordal metric and second one is spherical metric. The spherical metric$d_s$... • 71 1 vote 1 answer 61 views ### Why is the quotient of a manifold by a group action singular at a point with non-trivial stabilizer? My question is essentially the same as the question here, and I'm not sure if I fully understand that answer. I am reading Don Zagier's chapter in The 1-2-3 of Modular Forms, and the author says The ... • 1,218 2 votes 1 answer 146 views ### Maximising the sum of squares of distance between 3 points on 3 concentric circles$\textbf{Question : }$Given three concentric circles$C_1,C_2,C_3$of radius$r_1,r_2,r_3\ (r_1<r_2<r_3)$and a point on each of the circle's circumference then what is the maximum value of the ... • 367 0 votes 0 answers 28 views ### Vector bundle decomposition via holonomy group I'm following Kobayashi's "Differential geometry of complex vector bundles". In section IV.2 (p. 107) we have the following data: An holomorphic vector bundle$(E,h)\longrightarrow (M,g)$... • 91 1 vote 0 answers 26 views ### Number of dyadic squares to cover curve of finite length I have been working on this for a while without much luck. Suppose we have a Jordan arc on the complex plane$\mathbb{C}$, i.e. the image of the unit interval$[0,1]$under a homeomorphism, with ... • 61 1 vote 0 answers 34 views ### How to prove the decomposition of k-forms over a complexified vector space? I am studying the book "Complex Geometry, an introduction", page 27. We consider the complexification of a vector space V, namely$V_\mathbb{C}. If we take the decomposition $$V = W_1 + W_2.... 1 vote 0 answers 40 views ### Primitive of a Kähler Form on a punctured complex plane Consider a Kähler metric g on M = \mathbb C \setminus \{z_0, \dots, z_n\} with the standard complex structure. We know that with respect to the usual real coordinates x,y the metric is given by ... 2 votes 0 answers 39 views ### For \alpha\in A^{p,q}(X)\cap\ker d, is \partial\mathcal H_{\bar\partial}\alpha=0? Let X be a compact complex manifold with a Hermitian metric h, then we can define \bar\partial^* as the adjoint of \bar\partial, define \Delta_{\bar\partial}:=\bar\partial\bar\partial^*+\bar\... • 479 1 vote 1 answer 84 views ### Disconnected fiber of regular birational map between complex projective varieties Let f: X \to Y be a regular birational morphism of projective varieties over k= \mathbb{C}, q \in Y a point. Claim: If the fiber f^{-1}(q) is disconnected, then q is a singular point of Y (... 1 vote 1 answer 74 views ### Support of the direct image sheaf equals the image? \def\sO{\mathcal{O}} \def\supp{\operatorname{Supp}} \def\sI{\mathcal{I}} \def\sC{\mathcal{C}} \def\colim{\operatorname{colim}}I am studying complex spaces using Grauert, Remmert, Coherent Analytic ... 0 votes 0 answers 31 views ### How to compute a Riemannian metric from a hermitian metric I am working with the complex projective space \mathbb{CP}^n as a real manifold of dimension 2n. I want to evaluate vectors in the tangent space at a point z \in \mathbb{C} with the Fubini-Study ... • 1,138 2 votes 1 answer 71 views ### What is the relation between the residue of a meromorphic connection and the degree of the underlying vector bundle? We consider a rank n holomorphic vector bundle E over a Riemann sphere \mathbb{P}^1, and let \nabla be a meromorphic connection on E with poles at a_1,...,a_m\in\mathbb{P}^1, where each ... 2 votes 1 answer 156 views ### Does integration over algebraic varieties make sense? Suppose X is a smooth projective variety over \mathbf{C} of complex dimension n. Let [\omega] \in H^{2n}_{\text{dR}}(X) be a de Rham cohomology class on X. If \omega is a representative of ... • 2,308 3 votes 1 answer 98 views ### Stokes type-formula Let X be a complex manifold. Is the following identity true?$$\int_X\overline {\partial} \alpha=\int_{\partial X} \alpha$$where \alpha is a differential form on X, and \partial X is the ... 0 votes 1 answer 43 views ### Difference between strongly and strictly pseudoconvex domains in \mathbb{C}^n Could anyone help me find the difference between strictly and strongly pseudoconvex domains in \mathbb{C}^n? I managed to find in the literature only the definition of strictly pseudoconvex domains (... 0 votes 1 answer 57 views ### Are complex functions whose limit at infinity is zero bounded? Let f(z) : X \to \mathbb{C} be an (analytic) complex function, X is \mathbb{C} except finite sets. If \lim_{z\to \infty}f(z)=0, then f is bounded? This question originates from following ... • 1,504 1 vote 0 answers 42 views ### Does the dual of a line bundle with no sections have a section? [duplicate] Let L \to X be a holomorphic line bundle over a compact complex manifold. Suppose L is non-trivial and has no non-trivial sections. Let me ask the following (hopefully not entirely trivial) ... 1 vote 1 answer 70 views ### Standard Hermitian metric for \mathbb C In the middle of Page 42 of Ballmann's book, the author defines the Hermitian metric by h=g+i\omega, where g is the compatible Riemannian metric which is a Riemannian metric g satisfying g(X,Y)... • 479 0 votes 0 answers 40 views ### A feasible sequel to Stewart and Tall: Complex Analysis I really do appreciate the book by Ian Stewart and David Tall on Complex Analyis. (sure, it has some minor flaws which are, nonetheless, not that relevant here). I especially like that they mention ... 6 votes 1 answer 97 views ### Holomorphic maps into ruled surfaces Let \Sigma be a compact Riemann surfaces. Let L \to \Sigma be a holomorphic line bundle. This gives rise to a ruled surface \mathbb{P}(\underline{\mathbb{C}} \oplus L) \to \Sigma, where \mathbb{... • 3,320 0 votes 1 answer 42 views ### Motivation behind g(X,Y)=g(JX,JY). In Ballmann's book p.23, there is Let M be a complex manifold with corresponding complex structure J. We say that a Riemann metric g is compatible with J if$$g(X,Y)=g(JX,JY)$$for all vector ... • 479 1 vote 0 answers 54 views ### different norms of \mathbb R^3 transformed by 'stereographic projection' The standard norm also called Euclidean norm of v=(x,y,z)\in\mathbb R^3 is$$\|v\|_E=\sqrt{x^2+y^2+z^2}$$Then a stereographic projection is given by$$\begin{align} x=\xi_0^2-\xi_1^2,\\ y=i(\xi_0^... 0 votes 0 answers 44 views ### shifted sheaves Consider a sheaf\mathcal{O}_{\mathbb{P}^n}$a sheaf of functions over the complex projective space. What is the precise definition to$\mathcal{O}_{\mathbb{P}^n}(\mathcal{l})$, for$\mathcal{l}\$ an ...
If I cite my textbook the complex exponential function is defined as: $$e^{\theta i} = cos \theta + i \sin \theta$$ Is this just an arbitrary definition or is it possible to prove this somehow?