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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Why do algebraic varieties contain curves passing through two given points
Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme.
Someone told me that such varieties have the following ...
2
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0answers
17 views
Maps between total spaces of holomorphic vector bundles
I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles.
Let me outline a situation that is a bit more concrete, to help focus ...
4
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1answer
33 views
Complexified tangent vector, complex manifold
Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
4
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1answer
58 views
Non-vanishing 2-form on quartic surface.
Let $S\subset \mathbb P^3$ be a quartic surface defined by a homogeneous degree 4 polynomial $F\in k[x_0,x_1,x_2,x_3]$. $S$ is a K3 surface, so it has a unique non-vanishing $(2,0)$-form $\omega$ up ...
4
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3answers
54 views
Good source to learn about surface singularities?
I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there.
I checked out Miles Reid his lectures on ...
11
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1answer
93 views
Introduction to the trace formula for people outside number theory
I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula.
I am aware that there are "standard references" on the ...
3
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0answers
75 views
Why is Griffiths Transversality part of the definition of a variation of Hodge structures?
If $X \to S$ is a family of compact Kahler manifolds, then parallel transport with respect to the Gauss-Manin connection on the relative cohomology bundle does not respect the Hodge filtration, e.g. a ...
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vote
1answer
25 views
Complex 3-D Euclidean space - inner product
1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors:
2nd question:
Is there a ...
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40 views
Is there a relation between Super Riemannian manifolds and Kahler manifolds?
(This question has a physics motivation).
Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kahler manifolds, or at ...
2
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0answers
21 views
Algorithm for checking that a singularity is an ordinary double point
Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
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votes
1answer
34 views
Polynomials in the complex ring of 2 variables
Given $I = \left<x^2+y^2-1, x^2-y+1, xy-1\right>$ show that this generates $\mathbb{C}[x,y]$.
I have tried pages and pages of writing a linear combination of these such that the combination is ...
2
votes
2answers
57 views
Help with unknown notation
In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere):
"writing $z=x+iy$, we can verify that: $$x:y:-1=x_1:x_2:x_3-1, $$ and this means that the points ...
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0answers
29 views
Are complex submanifolds necessarily closed?
In the excellent book From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert, if I follow the definition and properties of analytic subsets and the definition of a complex ...
2
votes
0answers
31 views
Riemann mapping between arbitrary triangles
Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)?
Comment---I look for the conformal equivalence of interiors promised ...
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vote
1answer
23 views
Positive curvature on holomorphic vector bundles
There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
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0answers
39 views
Which is the correct universal line bundle: the tautological bundle or its dual?
With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$
In ...
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votes
2answers
49 views
The zero set of $z_0^2+z_1^2-1$ in $\mathbb{C}^2$.
Recently, I read the notes "Vector bundles on Riemann surfaces" by Sabin Cautis (http://www-bcf.usc.edu/~cautis/classes/notes-bundles.pdf). On the sixth page of these notes, there is a statement ...
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0answers
40 views
Degree of morphism of quotient of upper half-plane
Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
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0answers
54 views
A funny condition for ampleness on a curve
Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
8
votes
1answer
82 views
Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free
Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
8
votes
1answer
132 views
Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)
It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
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votes
4answers
127 views
Reference request: Chern classes in algebraic geometry
I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean.
I am looking for a reference that ...
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vote
1answer
30 views
What does $T_z\mathbb{R}^2\otimes\mathbb{C}$ in p. 2 of Huybrechts' book mean?
I apologize for my lack of imagination and the likely silliness of this question, but what does $T_z\mathbb{R}^2\otimes\mathbb{C}$ mean here (last paragraph)?
And how does that extension work?
Thank ...
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0answers
37 views
Torus biholomorphic to smooth cubic curve?
I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ )
I think I ...
3
votes
1answer
90 views
Geometric meaning of Line-bundle product
I was wondering,
What could be a geometric/intuitive meaning of taking the (tensor) product of two line bundles on a smooth variety. Could you depict this to me with some example?
...
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votes
2answers
76 views
Why are the fibers of the Albanese map of a nonrational ruled surface copies of $\mathbb{P}^1$?
I'm currently reading "Rational surfaces with many nodes" by Dolgachev et al., avaliable here:
http://www.math.lsa.umich.edu/~idolga/lisbon.pdf
A "surface" is always smooth and projective and let us ...
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0answers
53 views
K3-surface is not the blow-up of any other smooth complex surface?
Good evening,
I'm stuck in the following exercise in Huybrechts, Complex Geometry, chapter 2, page 103.
Let $X$ be a K3 surface, i.e. X is a compact complex surface with $K_X \cong \mathcal{O}_X$ ...
6
votes
1answer
82 views
Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf
Let $B$ a smooth projective connected variety over $\mathbf C$.
Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero.
Does the converse hold? That is, suppose that $B$ ...
7
votes
2answers
108 views
Are endomorphisms of degree one always automorphisms
Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one.
Do I understand correctly that $\sigma$ is an automorphism?
I believe this ...
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votes
1answer
76 views
Moduli Spaces of Higher Dimensional Complex Tori
I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.
Similarly, I have ...
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1answer
31 views
Automorphisms of the algebraic Torus
Any (holomorphic) group homomorphism $f:\mathbb{C}^\ast\rightarrow\mathbb{C}^\ast$ is of the form $f(z)=z^k$ ?
Is this true? I tried this: differentiating $f(zw)=f(z)f(w)$ with respect to $z$ ...
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votes
0answers
51 views
A question on Chern character computation
Let $C$ be a smooth curve in a complex threefold $X$. How can I see that
$$
\mathrm{ch}(\mathcal{O}_C)=(0,0,[C],\chi(\mathcal{O}_C))\in H^0\oplus H^2 \oplus H^4\oplus H^6,
$$
where $H^0\cong ...
2
votes
2answers
64 views
Reference for definition and more of Galois covering
I encountered the term "galois covering" in Beauville's book on algebraic surfaces, as well as in the article "rational surfaces with many nodes" by Dolgachev et al.
However, i have not yet found a ...
3
votes
1answer
70 views
topic for presenting in hyperbolic geometry
For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing ...
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votes
1answer
51 views
A question on the map associated to a divisor on an algebraic surface.
Let $S$ be a K3 surface and $E\subset S$ be a genus 1 smooth curve. By Riemann-Roch, $h^0(S,\mathcal{O}(E))=2$ and hence there is a map $\phi_{E}:S\rightarrow \mathbb{P}^1$. How do we know that this ...
2
votes
1answer
50 views
Fundamental group of the following disc
What is the fundamental group of the following space in $\mathbf C^n$?
This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
4
votes
1answer
73 views
Alternative Almost Complex Structures
Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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vote
1answer
45 views
A question on the fiber class of a fibration.
Let $X$ be compact complex manifold and $\phi:X\rightarrow C$ be a holomorphic map to a smooth curve $C$. Denote the general fiber of $\phi$ by $Y$. How can one see that $Y|_Y$ is a torsion divisor?
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votes
3answers
67 views
Affine variety over a field
Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
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votes
0answers
55 views
Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem
From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
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votes
2answers
99 views
Showing an analytic map has closed irreducible image
Let $X,Y$ be complex algebraic varieties with $X$ (algebraically hence also analytically irreducible), $\pi : Y \to X$ an algebraic map with each fiber a finite set, and $g:X \to Y$ an analytic map ...
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1answer
108 views
Derived Category of Coherent Sheaves on Elliptic Curves
I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was ...
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2answers
174 views
What exactly is a Kähler Manifold?
Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing.
I am a physics undergrad, trying to carry out some research work on ...
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votes
2answers
80 views
Adjunction for varieties with higher codimension
For $X \subseteq \mathbb{P}^n$ a smooth hypersurface, the canonical divisor $K_X$ can be computed as
$$
K_X = (K_{\mathbb{P}^n} + X)|_X.
$$
Is there a similar formula where $X$ is of higher ...
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1answer
82 views
Torsion Chern class?
Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using ...
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2answers
50 views
Graph of a curve
Today in my test, there was a question which had contour C: $|z+\dfrac{1}{z}| = 2$. What does the curve represent? Is it a discrete set of points or really a curve?
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2answers
81 views
Automorphism group any bounded domain of $\mathbb{C}$
So far the automorphism group I have calculated for known domain is a Lie Group,so Automorphism group any bounded domain of $\mathbb{C}$ is a lie group?
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0answers
30 views
References for the conformal equivalence of the space of complex 1-tori and C?
What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
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44 views
Normal Bundle of Twistor lines
I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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19 views
Complex Coordinate Foot Perpendicular
We suppose $A$ and $M$ two points their complex coordinates respectively are $a=1$ and $z(\theta)$ as $z(\theta)=\dfrac{1}{2}(1+e^{i\theta})^2$ and $\theta \in (-\pi;\pi)$.
What are the complex ...
