Questions tagged [complex-geometry]
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
3,122
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Non-isomorphic geometric objects obtained by cutting the Möbius band [closed]
How many non-isomorphic geometric objects can be obtained by cutting a Möbius band while keeping parallel on the substrate?
2
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
3
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1
answer
179
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A complex version of the Cahiers topos
Has anyone tried defining a complex version of the Cahiers topos?
If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
2
votes
1
answer
137
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Gluing local holomorphic connections
On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
1
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0
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39
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Looking at a frequency reassignment rule as a Möbius transform
Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed.
I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
4
votes
2
answers
404
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Nearby cycles for stacks
Let $X$ be a variety over $\mathbb C$. Let $f\colon X \to \mathbb{A}^1$ be a regular function. I understand that there is an analytic nearby cycles functor, defined in terms of the exponential map. I ...
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How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?
I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface, ...
5
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185
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
2
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0
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43
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Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
4
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1
answer
214
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Spin$^c$ structures induced by an almost complex structure
Let $M$ be a closed spin$^c$ $4$-manifold with determinant line bundle $L$.
If $c_1^2(L)=2\chi(M)+3\tau(M)$, where $\chi$ and $\tau$ denote the Euler characteristic and signature of $M$ respectively, ...
1
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0
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44
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Elliptic surfaces with monodromy in Borel subgroup
Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...
8
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251
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Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...
1
vote
1
answer
104
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Nonequidimensional birational Mori contractions
I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image.
To agree with the setup I like, the ...
0
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0
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135
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“Holomorphic” bump function
I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...
4
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0
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135
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$\pm 1$-equivariant perverse sheaves on the affine line
Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
1
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0
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185
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Constructing curves with large tangent space in complex variety
Suppose $M$ is a (singular) complex analytic/algebraic variety. Then for every $p\in M$ there exists a (possibly reducible) curve $C \subset U\subseteq M$ containing $p$ such that $T_pC=T_pM$, where $...
3
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0
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123
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Splitting of normal bundle exact sequence and Holomorphic neighbourhood retract
Let $X$ be a compact complex manifold and $Y\subset X$ a complex submanifold of $X$.
Consider the two following conditions:
The exact sequence $0\to TY\to TX|_{Y}\to N_Y\to 0$, where $TX$, $TY$ ...
7
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0
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226
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Is a smooth projective variety over $\mathbb{C}$ dominated by a Ball?
Suppose that $X$ is a smooth projective variety of dimension $d$ over the complex numbers.
Is it true that there is a ball $\Delta_d=\{ z\in \mathbb{C}^d / \lvert z\rvert<1\}$ and a surjective ...
1
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1
answer
419
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
2
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162
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Splitting of de Rham cohomology for singular spaces
I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
2
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0
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120
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Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
1
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1
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102
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Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
2
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192
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Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
4
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0
answers
207
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Classification of affine varieties over the affine line whose central fiber is $\mathbb{C}$ and general fiber is $\mathbb{C}^*$
Consider an affine variety $Y$ equipped with a morphism $\pi: Y \rightarrow \mathbb{C}$. The conditions we have are that $\pi^{-1}(0)=\mathbb{C}$, and for any $x$ not equal to zero, $\pi^{-1}(x)=\...
3
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0
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73
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
0
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0
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111
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Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
13
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2
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Is the Gödel universe Wick rotatable?
Take Wick rotatability being as the way defined in the following article by Helleland and Hervik:
Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
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Complex geodesic coordinate, local ramified map, and the conic metric
Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance
Let $X$ be a compact Kaehler ...
3
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0
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99
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Request for non-Einstein positive constant scalar curvature Kähler surfaces
I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature.
There are of course the Fano (del Pezzo) Kähler-...
5
votes
0
answers
273
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Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
1
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0
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Kähler potential for ALEs from resolving $\mathbb{C}^2/\mathbb{Z}_2$:
I am reading the famous paper of Kronheimer “The construction of ALE spaces as hyperkähler quotients”
I want to calculate explicitly the metric on the ALE spaces, obtained by a family of resolution of ...
2
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0
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344
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Is a Wick rotatable spacetime necessarily strongly causal?
There are a few viable ways to formulate Wick rotatability that preserve distinct features.
One is mentioned in the post:
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
There's also ...
0
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0
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202
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Definition of motivic/coming from geometry
Suppose $X$ is a projective smooth connected curve, $S\subset X$ is a finite set of points and $U=X\setminus S$.
I encountered the following definiton:
We say a $\mathbb Q$-local system $\mathcal F$ ...
3
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0
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154
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Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
0
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0
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88
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Torsion free Chern connections and Kähler manifolds
Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
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0
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81
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Functional inequality with complex variables
I am interested in considering a function $C(\tau)$ of the complex variable $\tau = t + i\eta$ such that
$C(\tau)$ is analytic for $\Re(\tau)=t>t_0\ge0$
$\exists$ a constant $C_0$ and a function $...
0
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0
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78
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Separating orthogonal vectors in $\mathbb{C}^2$
Is it possible to partition $\mathbb{C}^2$ into two sets $S$ and $S'$ such that, given any two nonzero orthogonal vectors $\mathbf{v}$ and $\mathbf{w}$ of $\mathbb{C}^2$, one of them lies in $S$ and ...
1
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0
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65
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Exponentiate additive transition functions for $\mathbb{A}^1$-bundles
Consider an smooth complex elliptic curve $E$ glued from two affine curves ($p\in\mathbb{C}\setminus0$)
$$C_{(x,y)}: y^2=x^3+px\\C_{(s,t)}: t^2=ps^3+s$$
via the coordinate change $s=1/x,t=y/x^2$.
It ...
1
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0
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104
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Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties
Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
0
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0
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268
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Space which is diffeomorphic to CP^2 # -CP^2
The manifold $\mathbb{CP}^2 \# -\mathbb{CP}^2$, the non-trivial $\mathbb{S}^2$ bundle over $\mathbb{S}^2$, is known
to be diffeomorphic to the space that we will now describe. Represent $\mathbb{S}^3 ...
2
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0
answers
49
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Lefschetz operator on bundle-valued forms
For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
1
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0
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87
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Effective Torelli theorem for K3 surfaces
The proof of the Torelli theorem I've seen goes something like:
Put $M$ the moduli space of marked $K3$ surfaces, and $D$ the period domain s.t there is a natural map $$P: M \to D$$.
Up to lies, here ...
2
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0
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79
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Does Kobayashi isometry map preserve complex geodesics?
Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
3
votes
1
answer
212
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Hyperelliptic integrals
I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
2
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1
answer
197
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Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
3
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0
answers
182
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An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
6
votes
1
answer
280
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Is this $\mathbb C$-fibration over compact Riemann surface trivial?
I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:
$p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...
-1
votes
1
answer
116
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A complex complex integral operation [closed]
This question mainly asks about the integral of complex numbers. This question originates from the optical properties of the axicon angle.$\lambda, f,R,ρ_{0},k$
are a constant.
$$m(r,\varphi)=\frac{1}{...
1
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0
answers
117
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Is every curve on a projective three-fold a homology-theoretic complete intersection of sorts?
Let $C$ be a curve on a smooth projective three-fold $M$ equipped with the restriction of the Fubini-Study metric $\omega$. I'd like to know if there exists a surface $S$ such that for every closed $(...
2
votes
1
answer
219
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The indecomposable bundle on an elliptic curve
M. Atiyah (Theorem 5, p. 432 of "Vector bundles on an elliptic curve") defines an indecomposable bundle of degree $0$ that has a global section for each rank $r$ (I'm thinking on an elliptic ...