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Results tagged with complex-geometry
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user 108274
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.
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Zariski connectedness theorem: from analytic & topological viewpoint
Let $p:Y \to X$ be a proper, flat (see later why) surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ …
- 63.9k
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answer
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Almost Complex Structure extending to Complex Structure, aka "Integrable"
Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 …
- 5,986
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Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states th …
- 5,986
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answers
183
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Zariski Connectedness Theorem in Complex Geometry
Let $f: X \to Y$ be a proper surjective morphism of complex irreducible varieties such that general fibre of $f$ is connected and $Y$ integrally closed\normal. Say, we even assume wlog $Y=\text{Spec}( …
- 5,986
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164
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Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's original form …
CommunityBot
- 1
2
votes
1
answer
212
views
Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi …
- 35.2k
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votes
1
answer
322
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Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\ope …
- 5,986
2
votes
1
answer
350
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Comparison of classical and Zariski topologies with constructible sets
In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the
proof of (3.25) Specialization principle on page 53 contains an argument
I not understand.
General assumptions: all our v …
- 68
2
votes
1
answer
133
views
Generically finite projection $\pi_L: X \to \mathbb{P}^2$ from plane $L$ and critical points
(In following we are working in "classical" complex setting: i.e. all involved schemes are considered to be varieties over $k=\mathbb{C}$)
Let $X \subset \mathbb{P}^n$ be irreducible surface and $L $ …
- 149k
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1
answer
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Example motivating mixed Hodge structures
The suggested intuition behind mixed Hodge structures - developed
in particular to generalize Hodge decomposition of cohomology
groups from complex smooth complete varieties to more general algebraic …
- 149k
3
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answers
220
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Historical proof of Leschetz Hyperplane Theorem
I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
hypersur …
- 5,986
0
votes
0
answers
201
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Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)
In Riemann's publication about Abelian functions
'Theorie der Abelschen Functionen' (Here the original paper in german)
at the end of Chapter 4, part 2 is clamed that for every Riemann
surface $T$ and …
- 6,377
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290
views
Chow's Lemma: Mumford's and Grothendieck's (?) definitions
David Mumford gives in his book Algebraic Geometry I, Complex Projective Varieties
on page 61 a definition of Chow's Lemma which has at least for me not a
usual form:
If says that a closed $^*$-analyt …
- 5,986
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answers
522
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Polarizations in algebraic and symplectic geometry
In context of Abelian varieties there are a couple of equivalent ways to
introduce the polarization of a algebraic variety. One way is to
choose a line bundle $\mathcal{L}$ which satisfies certain ide …
- 5,986
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1
answer
278
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Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations arou …
- 7,320