Questions tagged [k3-surfaces]

For all questions about *K3 surfaces*, that is complex or algebraic smooth surfaces which are regular with trivial canonical bundle.

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Global sections of a line bundle associated to a smooth irreducible curve on a K3 surface.

I would like to know it the following proof is correct: Lemma: Let $C$ be a smooth, irreducible curve of genus $g$ over a complex projective K3 surface $X$. Let $L:=\mathcal{O}_X(C)$. Then $C^2=2g-2$...
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40 views

Completions of Stalks of a Surface

Let S be a surface (so a 2-dimensional proper k-scheme). My question arises from following another thread of mine: https://mathoverflow.net/questions/331251/intuition-behind-rdp-rational-double-points ...
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40 views

Kähler Potential on Blowup of $\mathbb{C}/\{\pm 1\}$

The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example: Consider $\mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted ...
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1answer
88 views

blowing-up preserves the first Betti number?

Let $G$ be a finite group and $S$ be a K3 surface. $G$ acts effectively and symplectically(fix the nowhere vanishing 2-form of $S$) on $S$. Since the action is symplectic,quotient surface $S/G$ has ...
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33 views

Any concrete example(s) of $K3$ surfaces with Picand number 18 and does not admit Shioda-Inose structure?

I am looking for some explicit examples of (elliptic) $K3$-families defined over a number field (better to be over $\mathbb{Q}$) with Picard number $18$ but does not admit Shioda-Inose structure, i.e. ...
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109 views

Is the Calabi-Yau property of a smooth manifold a differential invariant?

In this question I asked if it could happen that two complex manifolds are homeomorphic, and one of them is a Calabi-Yau manifold but the other isn't. It turns out that there are complex surfaces that ...
8
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1answer
165 views

Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme ...
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2answers
223 views

Let $X$ be a K3 surface, show that $H_1(X,\mathbb Z)=0$

Let $X$ be a(n algebraic) K3 surface, i.e., $X$ is a smooth algebraic surface with trivial canonical bundle and $H^1(X,\mathcal{O})=0$. This assumption directly implies that $H^1(X,\mathbb C)=0$, so $...
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80 views

References for properties of a $K3$ surface

I have read that the surface $K3$ in $\mathbb{CP}^3$ defined by the equation $x^4+y^4+z^4+w^4=0$ is a spin manifold whose $\hat{A}$-genus is non-vanishing. However, not knowing algebraic geometry, I ...
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1answer
140 views

Two questions on three quadrics in $P^5$ whose intersection is a genus $5$ K3 surface.

It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ). Question I: Does anyone have an example (...
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770 views

Why the Picard group of a K3 surface is torsion free

Let $X$ be a K3 surface. I want to prove that $Pic(X)\simeq H^1(X,\mathcal{O}^*_X)$ is torsion free. From D.Huybrechts' lectures on K3 surfaces I read that if $L$ is torsion then the Riemann-Roch ...