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\section*{K3 surface}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{ModuliOfHigherLineBundles}{Moduli of higher line bundles and deformation theory}\dotfill \pageref*{ModuliOfHigherLineBundles} \linebreak
\noindent\hyperlink{as_a_fiber_space_in_string_compactifications}{As a fiber space in string compactifications}\dotfill \pageref*{as_a_fiber_space_in_string_compactifications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{InStringTheory}{In string theory}\dotfill \pageref*{InStringTheory} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{K3 surface} is a [[Calabi-Yau variety]] of [[dimension]] $2$ (a Calabi-Yau [[algebraic surface]]/[[complex surface]]). This means that the [[canonical bundle]] $\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X$ is trivial and $H^1(X, \mathcal{O}_X)=0$.

The term ``K3'' is

\begin{quote}%
in honor of Kummer, K\"a{}hler, Kodaira, and the beautiful K2 mountain in Kashmir


\end{quote}
(\hyperlink{Weil79}{Weil 79, p. 546})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item A cyclic cover of $\mathbb{P}^2$ branched over a curve of degree $6$.


\item A nonsingular degree $4$ hypersurface in $\mathbb{P}^3$, such as the [[Fermat hypersurface|Fermat quartic]] $\{[w,x,y,z] \in \mathbb{P}^3 | w^4 + x^4 + y^4 + z^4 = 0\}$ (in fact every K3 surface over $\mathbb{C}$ is [[diffeomorphism|diffeomorphic]] to this example).


\item The [[flat orbifold]] quotient of the [[4-torus]] by the sign [[involution]] on all four canonical [[coordinates]] is the flat compact 4-dimensional orbifold known as the \emph{Kummer surface} $T^4 \sslash \mathbb{Z}_2$, a singular [[K3-surface]] (e.g. \href{Riemannian+orbifold#BettiolDerdzinskiPiccione18}{Bettiol-Derdzinski-Piccione 18, 5.5})



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{itemize}%
\item All K3 surfaces are [[simply connected]].


\item Over the [[complex numbers]] they are all [[Kähler manifold|Kähler]], and even [[hyperkähler manifold|hyperkähler]].



\end{itemize}
\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

\begin{prop}
\label{IntegralCohomology}\hypertarget{IntegralCohomology}{}
\textbf{([[integral cohomology]] of [[K3-surface]])}

The [[integral cohomology]] of a K3-surface $X$ is

\begin{displaymath}
H^n(X,\mathbb{Z})
  \;\simeq\;
  \left\{
  \itexarray{
    \mathbb{Z} &\vert& n = 0
    \\
    0 &\vert& n = 1
    \\
    \mathbb{Z}^{22} &\vert& n = 2
    \\
    0 &\vert& n = 3
    \\
    \mathbb{Z} &\vert& n = 4
  }
  \right.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{BarthPetersVandenVen84}{Barth-Peters-Van den Ven 84, VIII Prop. 3.2})

\begin{prop}
\label{BettiNumbers}\hypertarget{BettiNumbers}{}
\textbf{([[Betti numbers]] of a [[K3-surface]])}

The [[Hodge diamond]] is completely determined (even in positive [[characteristic]]) and hence the [[Hodge-de Rham spectral sequence]] degenerates at $E_1$. This also implies that the [[Betti numbers]] are completely determined as $1, 0, 22, 0, 1$:

\begin{displaymath}
\itexarray{
    && h^{0,0}
    \\
    & h^{1,0} && h^{0,1}
    \\
    h^{2,0} & & h^{1,1} & & h^{0,2}
    \\
    & h^{2,1} & & h^{1,2}
    \\
    && h^{2,2}
  }
  \;\;\;=\;\;\;
  \itexarray{
    && 1
    \\
    & 0 && 0
    \\
    1 & & 20 & & 1
    \\
    & 0 & & 0
    \\
    && 
    1
  }
\end{displaymath}
\end{prop}
(e.g. \hyperlink{BarthPetersVandenVen84}{Barth-Peters-Van den Ven 84, VIII Prop. 3.3})

\hypertarget{ModuliOfHigherLineBundles}{}\subsubsection*{{Moduli of higher line bundles and deformation theory}}\label{ModuliOfHigherLineBundles}

In [[positive number|positive]] [[characteristic]] $p$:

The [[Néron-Severi group]] of a K3 is a [[free abelian group]]

The [[formal Brauer group]] is

\begin{itemize}%
\item either the formal [[additive group]], in which case it has [[height of a formal group|height]] $h = \infty$, by definition;


\item or its [[height of a formal group|height]] is $1 \leq h \leq 10$, and every value may occur



\end{itemize}
(\hyperlink{Artin74}{Artin 74}), see also (\hyperlink{ArtinMazur77}{Artin-Mazur 77, p. 5 (of 46)})

[[!include moduli of higher lines -- table]]

\hypertarget{as_a_fiber_space_in_string_compactifications}{}\subsubsection*{{As a fiber space in string compactifications}}\label{as_a_fiber_space_in_string_compactifications}

See \emph{[[duality between heterotic and type II string theory]]}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[formal Brauer group]]


\item [[K3-cohomology]]


\item [[F-theory]],


\item [[elliptic fibration]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original sources include

\begin{itemize}%
\item [[Michael Artin]], \emph{Supersingular K3 Surfaces}, Annal. Sc. d, l'\'E{}c Norm. Sup. 4e s\'e{}ries, T. 7, fasc. 4, 1974, pp. 543-568


\item [[Andre Weil]], Final report on contract AF 18 (603)-57. In Scientific works. Collected papers. Vol. II (1951-1964). 1979.



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item W. Barth, C. Peters, A. Van den Ven, chapter VII of \emph{Compact complex surfaces}, Springer 1984

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Daniel Huybrechts]], \emph{Lectures on K3-surfaces} (\href{http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf}{pdf})


\item [[Claire Voisin]], \emph{}


\item [[David Morrison]], \emph{\href{http://www.cgtp.duke.edu/ITP99/morrison/cortona.pdf}{The geometry of K3 surfaces}} Lecture notes (1988)


\item Viacheslav Nikulin, \emph{Elliptic fibrations on K3 surfaces} (\href{http://arxiv.org/abs/1010.3904}{arXiv:1010.3904})



\end{itemize}
Discussion of the [[deformation theory]] of K3-surfaces (of their [[Picard schemes]]) is (see also at \emph{[[Artin-Mazur formal group]]}) in

\begin{itemize}%
\item [[Michael Artin]], [[Barry Mazur]], \emph{Formal Groups Arising from Algebraic Varieties}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 10 no. 1 (1977), p. 87-131 \href{http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0}{numdam}, \href{http://www.ams.org/mathscinet-getitem?mr=56:15663}{MR56:15663}

\end{itemize}
\hypertarget{InStringTheory}{}\subsubsection*{{In string theory}}\label{InStringTheory}

In [[string theory]], the [[KK-compactification]] of [[type IIA string theory]]/[[M-theory]]/[[F-theory]] on K3-[[fibers]] is supposed to exhibit te [[duality between M/F-theory and heterotic string theory]], originally due to

\begin{itemize}%
\item [[Chris Hull]], [[Paul Townsend]], section 6 of \emph{Unity of Superstring Dualities}, Nucl.Phys.B438:109-137,1995 (\href{http://arxiv.org/abs/hep-th/9410167}{arXiv:hep-th/9410167})


\item [[Edward Witten]], section 4 of \emph{[[String Theory Dynamics In Various Dimensions]]}, Nucl.Phys.B443:85-126,1995 (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124})



\end{itemize}
Review includes

\begin{itemize}%
\item [[Paul Aspinwall]], \emph{K3 Surfaces and String Duality}, in [[Shing-Tung Yau]] (ed.): \emph{Differential geometry inspired by string theory} 1-95 (\href{https://arxiv.org/abs/hep-th/9611137}{arXiv:9611137}, \href{http://inspirehep.net/record/426102}{spire:426102})

\end{itemize}
Further discussion includes

\begin{itemize}%
\item [[Paul Aspinwall]], [[David Morrison]], \emph{String Theory on K3 Surfaces}, in [[Brian Greene]], [[Shing-Tung Yau]] (eds.), \emph{Mirror Symmetry II}, International Press, Cambridge, 1997, pp. 703-716 (\href{https://arxiv.org/abs/hep-th/9404151}{arXiv:hep-th/9404151})


\item [[Paul Aspinwall]], \emph{Enhanced Gauge Symmetries and K3 Surfaces}, Phys.Lett. B357 (1995) 329-334 (\href{http://arxiv.org/abs/hep-th/9507012}{arXiv:hep-th/9507012})



\end{itemize}
Specifically in relation to [[orbifold]] [[string theory]]:

\begin{itemize}%
\item [[Katrin Wendland]], \emph{Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry}, in \emph{[[Orbifolds in Mathematics and Physics]]} (\href{https://arxiv.org/abs/hep-th/0112006}{arXiv:hep-th/0112006})

\end{itemize}
Specifically in relation to the putative [[K-theory]]-classification of [[D-brane charge]]:

\begin{itemize}%
\item Inaki Garcia-Etxebarria, [[Angel Uranga]], \emph{From F/M-theory to K-theory and back}, JHEP 0602:008,2006 (\href{https://arxiv.org/abs/hep-th/0510073}{arXiv:hep-th/0510073})

\end{itemize}
Specifically in [[M-theory on G2-manifolds]]:

\begin{itemize}%
\item [[Michael Atiyah]], [[Edward Witten]] section 6.4 of \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})

\end{itemize}
Specifically in relation to [[Moonshine]]:

\begin{itemize}%
\item [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, \emph{K3 String Theory, Lattices and Moonshine} (\href{https://arxiv.org/abs/1612.04404}{arXiv:1612.04404})

\end{itemize}
Specifically in relation to [[little string theory]]:

\begin{itemize}%
\item [[Shamit Kachru]], Arnav Tripathy, Max Zimet, \emph{K3 metrics from little string theory} (\href{https://arxiv.org/abs/1810.10540}{arXiv:1810.10540})

\end{itemize}
[[!redirects K3 surfaces]]

[[!redirects K3]]

[[!redirects K3-surface]] [[!redirects K3-surfaces]]

[[!redirects Kummer surface]] [[!redirects Kummer surfaces]]



\end{document}