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Results tagged with complex-geometry
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user 5259
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.
4
votes
0
answers
148
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Looking for some abelian surface fibration
Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor.
I am not lookin …
8
votes
1
answer
556
views
Is the complex moduli of Quintic Calabi-Yau toric?
Complex moduli space (or Teichmuller space) of a Quintic Calabi-Yau 3-fold is a
101-dimensional complex orbifold. Does it have a toric structure?
1
vote
1
answer
1k
views
How do we write a locally free resolution for...
Let's $C \subset X$ be a smooth curve inside a three dimensional variety with split normal bundle $N_C^X= \nu_1 \oplus \nu_2$. What is a locally free resolution of $\iota_{*}\mathcal{O}_{C}$ ?
1
vote
1
answer
2k
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Section for a given fibration
Let $X\rightarrow S$ be a (projective, flat... or any other assumption which makes you happy) fibration of a smooth threefold over a smooth surface with connected one-dimensional fibers.
As an exampl …
21
votes
3
answers
5k
views
flatness in complex analytic geometry
It is always a pain to move back and forth between definitions in algebraic geometry and complex analytic geometry. Dictionary is much easier when are working with (family of) smooth varieties but the …
0
votes
Cone of movable curves
I think following post in the mathoverflow gives an answer:
Effective versus movable cones of curves
There, people mention that there is an example where the Ample cone is rational polyhedral but mo …
5
votes
2
answers
580
views
Mirror of Flop?
If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?
4
votes
3
answers
901
views
wedge product of second chern class and kahler form on Calabi-Yau 3-folds.
Let $X$ be a smooth Calabi-Yau 3-fold with Kahler form $w$,
It is true that $\int c_2(TX) \wedge w \geq 0$ (for any Kahler form $w$ on $X$).
Proof via algebraic geometry is rather difficult. Some wh …
3
votes
1
answer
958
views
Reference for elliptic 3-folds
I was looking for a reference which studies elliptic 3-folds (Their canonical bundle, second Chern class, singular fibers,...), similar to one for surfaces (Which is available in many books including …
2
votes
0
answers
327
views
surfaces with effective first Chern class
Let $S$ be a smooth complex surface, If $c_1(S) \in N_1(S)$ is nef and non-torsion, then we know that this would imply some restrictions on the cone of effective curves (and surface itself)--see the d …
9
votes
3
answers
3k
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Cone of movable curves
Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure …
2
votes
2
answers
2k
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Toroidal embedding
Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central f …
3
votes
5
answers
3k
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indecomposable vector bundles having proper sub-bundles.
Over rational curve we know that any vector bundle is decomposable to direct sum of line bundles.
In higher dimensions there are examples of indecomposable bundles.
some indecomposable vector bundl …
16
votes
3
answers
5k
views
Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative structur …
12
votes
3
answers
4k
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Lefschetz hyper-plane theorem for singular projective varieties?
Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:
For smooth hyperplane section $Y= X\cap H$, the restriction map
$H^i(X) \rightarrow H^i(Y)$ is an isomo …