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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 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}( …
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156
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Construct Torsion element in $H^2(X,\mathbb{Z})$ with Ambrose–Singer theorem
Let $X$ be a Kahler manifold. Using the exponential sequence one obtains a homomorphism $c_1:H^1(X,\mathcal{O}_X^*)\rightarrow H^2(X,\mathbb{Z})$. This is associating to a holomorphic line bundle $L$ …
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Holomorphic map proper after shrinking (Kollar's Lecture on Resolution of Singularities)
I'm reading Janos Kolloar's Lecture on Resolution of Singularities and have some problems to understand a detail in the proof of Thm. 1.5 on page 10:
Thm 1.5 (Riemann) Let $F(x,y)$ be an irreducibl …
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0
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106
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Isomorphic Jacobians for different choices of basis of $1$-forms
In Otto Forster's Lectures on Riemann Surfaces on page 170 Jacobi Variety is defined in 21.6:
Suppose $X$ is a compact
Riemann surface of genus $g$ and $ \omega_1,..., \omega_g $ is a basis
of $\Omega …
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3
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answers
346
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When are two complex Tori biholomorphic
Let $g \ge 1$ be a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
…
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0
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0
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290
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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 …
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2
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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 …
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1
vote
0
answers
693
views
Questions on Néron–Severi group
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134.
Let $X$ be a compact Kähler manifold. Consider …
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2
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1
answer
133
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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 $ …
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2
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1
answer
594
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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 …
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0
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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 …
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0
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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 …
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0
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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 …
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3
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0
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 …
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2
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0
answers
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Explicit construction of Fubini Study Metric
I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$
from Notes on canonical Kähler metrics
on page 8 is remarked (Example 2.12 4.):
Fix a Hermitian innerproduct on $\mathbb{C}^ …
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