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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 …
Basics's user avatar
  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 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 …
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  • 1,841