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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 ...
Mtheorist's user avatar
  • 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 \...
John Baez's user avatar
  • 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 ...
AmorFati's user avatar
  • 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 ...
user avatar
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 ...
gsvr's user avatar
  • 235
6 votes
0 answers
176 views

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 ...
isekaijin's user avatar
  • 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 ...
Rogelio Yoyontzin's user avatar
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 ...
Guangbo Xu's user avatar
  • 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 ...
Christian's user avatar
  • 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 ...
Rogelio Yoyontzin's user avatar
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 \...
wonderich's user avatar
  • 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 ...
HNuer's user avatar
  • 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 ...
Basics's user avatar
  • 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 $...
Neeroen123's user avatar
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 ...
Mtheorist's user avatar
  • 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 ...
Mtheorist's user avatar
  • 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, \,\...
Philip Engel's user avatar
  • 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 ...
Youloush's user avatar
  • 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 ...
Pooya's user avatar
  • 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 ...
user avatar
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 ...
Angelo's user avatar
  • 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 $...
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