Questions about K3 surfaces, which are smooth complex surfaces $X$ with trivial canonical bundle and vanishing $H^1(O_X)$. They are examples of Calabi-Yau varieties of dimension $2$.

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8
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0answers
337 views

Torelli-like theorem for K3 surfaces on terms of its étale cohomology

Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology? For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...
5
votes
0answers
138 views

on the automorphisms of the transcendental Hodge structure of a K3 surface

Let $S$ be a complex projective K3 surface and consider the sub-Hodge structure $$ T(S) \subset H^2(S, \mathbb{Q}) $$ consisting of transcendental cycles. Let $\varphi$ be an automorphism of Hodge ...
5
votes
0answers
154 views

Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface $$ S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1) $$ is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...
4
votes
0answers
272 views

Possible automorphism groups of a K3 surface

Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)? ...and does the answer change is I remove "polarized"? (polarized = equipped with an ample line bundle)
3
votes
0answers
116 views

Integral cohomology of the hilbert scheme of points on a k3

i'm reading the famous article "Varietes kahleriennes dont la premiere classe de chern est nulle" by Beauville, in particular proposition 6, which characterizes the second cohomology group for the ...
3
votes
0answers
203 views

Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
3
votes
0answers
136 views

Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$). K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
3
votes
0answers
319 views

The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $

Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem. This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...
2
votes
0answers
100 views

Coherent systems on K3 surfaces

Does anyone know whether the theory of coherent systems on $K3$ surfaces has been studied and, if yes, can you give me a reference? In particular, is there an analogue of Gieseker stability and of ...
2
votes
0answers
257 views

Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
1
vote
0answers
113 views

Moduli space of K3 surfaces and Bogomolov-Tian-Todorov theorem

The famous Bogomolov-Tian-Todorov theorem says that the moduli space of Calabi-Yau manifold is smooth, that is locally a complex manifold. Doesn't this contradicts to the fact that the moduli space ...
1
vote
0answers
99 views

Maximally unipotent monodromy point of a K3 surface

I have a question on maximally unipotent monodromy point (or large complex structure limit) of the family of polarized K3 surfaces $(X,L)$. It is known that the moduli space of such pair is given by ...
1
vote
0answers
147 views

Moduli Space of an Algebraic K3 surface with singularities.

Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a ...
0
votes
0answers
104 views

K3 surface minus finite set

Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that $$ H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z}) $$ for $0\le i \le 2$. Then the Euler ...
0
votes
0answers
143 views

$T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...