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Results tagged with complex-geometry
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user 38823
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.
9
votes
1
answer
789
views
Quotients of $K3$ surfaces by finite groups
Let $G$ be a finite subgroup of the group of automorphsims of a $K3$ surface $S$.
Consider the quotient $S/G$.
I am interested in the collection of such qutients:
$$\{ S/G \mid S\text{ is a K3 surf …
- 1,841
6
votes
1
answer
265
views
Examples of non-Kähler compact complex manifolds of dimension four with some properties
I am looking for examples of non-Kähler compact complex manifolds of dimension four with trivial canonical class and $H^i(M, {\mathcal O}_M) = H^0(M, \Omega^i_M) =0$ for $0< i < \dim M$.
In dimensio …
- 1,841
6
votes
2
answers
308
views
Possible number of components of anticanonical sections of projective manifolds
Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$).
Let $k$ be the number of components of $D$.
Some cheap thoughts give:
If $M$ is a Fano manif …
- 1,841
5
votes
0
answers
295
views
Examples or references for this claim about elliptic Calabi-Yau threefolds
In this article (page 2) , the authors say:
"it is expected, based on known examples, that Calabi–Yau threefolds of large Picard
rank are always elliptically fibered, perhaps after flopping a finite n …
- 1,841
5
votes
1
answer
301
views
Mirror symmetry for K3 fibered Calabi-Yau threefolds
By a K3 fibered Calabi-Yau threefold, I mean a smooth projective threefold $X$ with trivial canonical class and
$h^{1,0}(X) =h^{2,0}(X) = 0$ that has a fibration $X \rightarrow \mathbb P^1$ whose gen …
- 1,841
5
votes
1
answer
459
views
Threefolds with the same Betti numbers and the same Chern numbers
By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the …
- 1,841
4
votes
1
answer
169
views
Big divisors and projectivity
Let $M$ be a compact complex manifold of dimension three.
Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that
$$ h^0(M, \mathcal O_M(nD)) > C n^3 $$
for sufficiently la …
- 1,841
3
votes
1
answer
429
views
Examples of CY fibrations over $\mathbb P^1$
We work over $\mathbb C$ and let us call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O_M ) = 0$ for $0 < i < \dim(M)$.
In this d …
- 1,841
3
votes
1
answer
178
views
Projective manifold whose anticanonical section is composed of two components
Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K_M$).
In an answer to a previous question,
It is told that $D$ may have at most two components.
An …
- 1,841
3
votes
1
answer
231
views
Irrationality of some threefolds
Consider a smooth projective threefold $\overline W$, constructed in section 4 of this paper.
This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb …
- 1,841
3
votes
0
answers
184
views
Smallest Hodge numbers of Calabi-Yau threefolds ever found
By a Calabi-Yau threefold, I mean a simply-connected smooth compact K"ahler threefold with trivial canonical class.
It has two independent Hodge numbers $h^{1,1}$ and $h^{1,2}$.
What is the smallest …
- 1,841
3
votes
0
answers
343
views
Fundamental group of blow-ups
Let $M$ be a simply-connected compact complex manifold of dimension three and $C$ is a smooth complex curve in $M$.
Let $M'$ be the blow-up of $M$ along $C$.
My question is:
Is $M'$ also simply-conne …
- 1,841
2
votes
0
answers
147
views
Minimal Betti numbers of simply-connected threefolds with trivial canonical class
By a threefold, I mean a compact complex manifold of dimension three.
For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy:
$$b_2 \ge 0, b_3 \ge 2.$$
I am wondering …
- 1,841
2
votes
0
answers
198
views
Betti numbers of threefolds with trivial canonical class
I am interested in a simply-connected compact complex manifold $M$ of dimension three with trivial canonical class.
Note that if it is K"ahler, then it is a Calabi-Yau threefold.
Its independent Bett …
- 1,841