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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12
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1answer
181 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 ...
5
votes
1answer
215 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 ...
1
vote
1answer
73 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 ...
4
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0answers
105 views

Is there a relation between Super Riemannian manifolds and Kähler 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 Kähler manifolds, or at ...
2
votes
0answers
39 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 ...
0
votes
1answer
41 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 ...
3
votes
2answers
64 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 ...
2
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0answers
43 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
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0answers
46 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 ...
1
vote
1answer
54 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. ...
5
votes
0answers
122 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 ...
4
votes
2answers
59 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 ...
6
votes
1answer
258 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
1
vote
1answer
59 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 ...
1
vote
0answers
87 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
104 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
158 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 ...
5
votes
4answers
376 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 ...
1
vote
1answer
47 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 ...
2
votes
0answers
53 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 ...
4
votes
1answer
147 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? ...
3
votes
2answers
122 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 ...
0
votes
0answers
87 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
158 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
171 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 ...
3
votes
1answer
93 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 ...
1
vote
1answer
76 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 $z$ ...
2
votes
0answers
67 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
128 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
2answers
131 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 ...
2
votes
1answer
78 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
57 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 ...
5
votes
1answer
118 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 ...
1
vote
1answer
52 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?
2
votes
3answers
90 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 ...
5
votes
1answer
126 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 ...
5
votes
2answers
138 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 ...
7
votes
2answers
262 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 ...
12
votes
2answers
754 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 ...
7
votes
2answers
129 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 ...
8
votes
1answer
197 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 ...
0
votes
2answers
73 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?
4
votes
2answers
136 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?
0
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0answers
40 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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0answers
60 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
4
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1answer
160 views

When are two tori biholomorphic? [duplicate]

If $\Lambda \subset \mathbb{C}$ is a lattice, let $T_{\Lambda}$ be the torus $\mathbb{C}/\Lambda$. My question is: If $\Lambda_1, \Lambda_2 \subset \mathbb{C}$ are two lattices, when are ...
1
vote
1answer
39 views

Proving $\arg(a)\equiv \alpha\Longleftrightarrow z_1z_2 \in \mathbb{R}$

Suppose the complex equation $iz^2+(2-i)az-(1+i)a^2=0$ as $a\in \mathbb{C}^{*}$. $z_1$ and $z_2$ are the solution of this equation and we have also $z_1*z_2 = a^2(i-1)$. How can I prove that ...
2
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1answer
93 views

question about conformal map

Thank you for let me ask question I am really enjoy with this website. It is great website I have question about geometry for expert geometry what is the definition of conformal map and the condition? ...
-1
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1answer
51 views

Set of points $M(z)$

Suppose $z \in \mathbb{C}$ and $a=-1+i$ and $\forall z \in \mathbb{C}\setminus{a}$ $\quad f_a(z)=\dfrac{az}{z-a} $ we suppose in a plan $(P)$: $(D)=\{{M(z) \in (P), ...
0
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0answers
34 views

Pullback of a family of curves via a covering map.

Let $X$ be a smooth compact projective manifold and $\pi:Y\rightarrow X$ a Galois covering map. Is it always possible to pull back a family of curves on $X$ to $Y$?