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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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Saturated ideals in computational algebra
Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals.
The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal
$$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$
where $(I:J^n)= \ …
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3
votes
0
answers
346
views
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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3
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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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answers
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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2
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0
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 …
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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 $ …
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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 …
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2
votes
1
answer
594
views
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 …
- 5,966
2
votes
1
answer
212
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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 …
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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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2
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answers
164
views
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 …
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2
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answers
183
views
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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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) $U \subset X$ …
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answers
693
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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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