All Questions
Tagged with k3-surfaces complex-geometry
25
questions
9
votes
2
answers
752
views
Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?
Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?
What I know so far is as follows:
In this paper (https://arxiv.org/pdf/hep-th/9512195.pdf) by Verbitsky, it is claimed that ...
- 1,105
8
votes
3
answers
1k
views
Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces
For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface $X$, an elliptic fibration $\pi \colon X \...
- 21k
7
votes
1
answer
450
views
Do non-projective K3 surfaces have rational curves?
Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
- 1,269
6
votes
1
answer
330
views
automorphism group of K3 surfaces
It is known that smooth complex hypersurfaces with degree bigger than 2 and dimension bigger than 1 have finite automorphism groups, except for K3 surfaces.
But the group of polarised automorphisms ...
anon
6
votes
1
answer
253
views
Loci in the moduli space of K3 surfaces associated to lattices
The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a ...
- 235
6
votes
0
answers
347
views
Quantifying the failure of geometric formality in K3 surfaces
It is known that K3 surfaces are never geometrically formal [1]. That is, the wedge product of two harmonic forms on an arbitrary K3 surface is in general not harmonic, or equivalently, the space $\...
- 271
6
votes
0
answers
176
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Find an explicit quasi-smooth embedding $X_{38} \subset \mathbb P(5, 6, 8, 19)$
This question is not quite about research-level mathematics, so I apologize for bringing it here. I asked it in Math.SE first, but I got no answers, and only a suggestion to ask it here.
Consider the ...
- 183
6
votes
0
answers
570
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 ...
5
votes
3
answers
2k
views
K3 surface of genus 8
Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$.
Why the degree ...
- 1,197
5
votes
1
answer
544
views
Is the automorphism group of a Calabi-Yau variety an arithmetic group
Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a ...
- 193
5
votes
1
answer
494
views
Existence of logarithmic structures and d-semistability
I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a ...
3
votes
1
answer
353
views
(1/2) K3 surface or half-K3 surface: Ways to think about it?
I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:
Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...
- 10.3k
3
votes
1
answer
333
views
Euler characteristic of nodal K3 surfaces (as in singular)
This is probably easy, but I was just wondering if there is a nice and easy formula for the topological Euler characteristic of a K3 surface $X$ with say $k$ nodes. If there is no general formula, is ...
- 2,078
3
votes
1
answer
230
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 ...
- 1,821
2
votes
1
answer
265
views
Common gerbes over two K3 surfaces
Let $X$ and $Y$ be K3 surfaces over the complex numbers.
Under what assumptions, do there exist
a finite group $G_X$
a finite group $G_Y$
a $G_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology)
a $...
- 107
2
votes
0
answers
115
views
Is there a hyperkaehler manifold whose mirror is the total space of a tangent/cotangent bundle?
I am looking for an example of a hyperkaehler manifold $Y$ whose mirror is the total space of a tangent bundle $TX$ or a cotangent bundle $T^*X$, where $X$ can be any Riemannian manifold.
Is such a ...
- 1,105
2
votes
0
answers
236
views
Is the mirror of a noncompact hyperkaehler manifold also hyperkaehler?
This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that ...
- 1,105
2
votes
0
answers
146
views
Open Period Integrals of Elliptically Fibered K3 surfaces
Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
- 1,493
1
vote
1
answer
756
views
Picard group of a K3 surface generated by a curve
In Lazarsfeld's article "Brill Noether Petri without degenerations" he mentions the fact that for any integer $g \geq 2$, one may find a K3 surface $X$ and a curve $C$ of genus $g$ on $X$ such that ...
- 355
1
vote
2
answers
459
views
Isotrivial K3 family and Picard number
Is it true that any family of K3 surfaces over $\mathbb{C}$ whose Picard number is constant is isotrivial? Here isotrivial means locally analytically trivial.
Speculation: Let $\mathcal{M}$ be the ...
- 11
1
vote
1
answer
228
views
One-dimensional family of complex algebraic K3 surfaces
Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in ...
user494851
1
vote
1
answer
371
views
Understand the Mukai vector
Let $S$ be a K3 surface and $h:=c_1(i^*\mathcal{O}_{\mathbb{P^3}}(1))$, then we can compute that $c_1(S)=0,c_2(S)=6h^2$. Hence
\begin{align}
\sqrt{\text{td}(S)}=1+\frac{c_2(S)}{24}=1+\frac{1}{4}h^2
\...
user479269
1
vote
1
answer
118
views
A question on real surfaces on K3 surfaces.
Let $X$ be a K3 surface and $\omega$ be a nowhere vanishing 2-form on $X$. Suppose $Y\subset X$ be a smooth real surface. How can one see that $\omega|_Y=0$ implies $Y$ is a complex submanifold (a ...
- 11
1
vote
1
answer
355
views
Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory
I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces.
I got quite stuck in Corollary 3.27 ...
- 13
1
vote
0
answers
128
views
complex K3 surfaces with automorphisms of given orders
Concerning complex K3 surfaces, there are various methods to show the non-existence of an automorphism of certain orders. The usually way is to investigate the action of the automorphism on the space $...
user498585