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\begin{document}

\begin{center}{\LARGE \bf  The singular set of triholomorphic maps into quartic K3 surface}
\end{center}


\begin{center}
Ling He
\footnote{
L. He

Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, P. R. China

e-mail: heling11b@mails.ucas.ac.cn
} and Jiayu Li
\footnote{
J. Li
School of Mathematical Sciences, University of Science and Technology of China,
Hefei, 230026;
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, P. R. China

e-mail: jiayuli@ustc.edu.cn
}
\end{center}




\bigskip

\no
{\bf Abstract.}
We prove that any weakly triholomorphic map from a compact hyperk\"{a}hler surface to
an algebraic K3 surface defined by a
homogeneous polynomial of degree 4 in $\mathbb{C}P^3$
has only isolated singularities.\\

\no
{\bf{Keywords and Phrases.}} triholomorphic map, hyperk\"{a}hler manifold, K3 surface, regularity.\\

\no
{\bf{Mathematics Subject Classification (2010).}} Primary 53C42, 58E20.

\bigskip

\section{Introduction}\label{sec1}

Triholomorphic maps between hyperk\"ahler manifolds arose from the study of higher dimensional
gauge theory (\cite{Chen},\cite{FKS}). Weakly triholomorphic maps and their regularity were
first studied in \cite{Chen-Li2000} and \cite{Li-Tian}.

Recall that a hyperk\"ahler manifold is a Riemannian manifold $(M,g)$ with three parallel
complex structures $\left\{J^1,J^2,J^3\right\}$ compatible with the metric $g$ such that
$(J^1)^2=(J^2)^2=(J^3)^2=J^1J^2J^3=-\text{id}$.
A hyperk\"ahler manifold is of dimension $4m$ and has a natural $S^2$-family
of complex structures, which is called hyperk\"ahler $S^2$.
The simplest hyperk\"ahler manifold is the Euclidean space $\mathbb{R}^{4m}$.
It is well-known that the only compact hyperk\"ahler manifolds of dimension $4$
are $K3$ surfaces and complex tori.

A weakly triholomorphic map between two hyperk\"ahler manifolds $(M,I^1,I^2,I^3)$ and $(N,J^1,J^2,J^3)$
is a $W^{1,2}$-map $u:\Omega\subset M\rightarrow N$ whose restriction to any ball $B\subset\Omega$ is
a limit of smooth maps from $B$ to $N$ in the $W^{1,2}$-topology and which satisfies
\begin{equation}\label{eq00-1}
du=a_{ij}J^{j}\circ du\circ I^i,
\end{equation}
where $A=\left(a_{ij}\right)_{1\leq i,j\leq3}\in SO(3)$ is a
constant matrix which is a stationary harmonic map (c.f. \cite{Chen-Li2000}).

An interesting question is to analyze the structure of the singularity of a weakly triholomorphic map.
This question has been explored in \cite{Chen-Li2000}.

Note $u$ is smooth outside a closed subset $\text{Sing}_u$ defined by
$$
\text{Sing}_u=\left\{x\in\Omega\mid\lim_{r\rightarrow 0}r^{2-4m}\int_{B(x,r)}|\nabla u|^2\geq\epsilon\right\},
$$
where $\epsilon$ is the constant given by the $\epsilon$-regularity of Bethuel (\cite{Bethuel}).

By Bethuel's theorem, the singular set $\text{Sing}_u$ has vanishing $(4m-2)$-dimensional Hausdorff measure.
In fact, it is expected that $\text{Sing}_u$ has Hausdorff codimension at least three or possibly four
(\cite{Bellettini-Tian},\cite{Tian}) in some cases.



Let $\phi:S^2\rightarrow N$ be a smooth map.
A map $u:\mathbb{R}^4\rightarrow N$ is defined by
$$
u(x,x^4)=\phi(\frac{x}{|x|}) ~\text{for any} ~x\in \mathbb{R}^3\backslash \{0\}.
$$
Obviously $u$ is a map of homogeneous degree zero with $1$-dimensional singular
set.
From \eqref{eq00-1} we know that $u:\mathbb{R}^4\rightarrow N$ is a triholomorphic
map with the $x^4$-axis as its singular set if and only if the corresponding map $\phi$ satisfies
\begin{equation}\label{eq2}
d\phi\circ J_{S^2}=-a_{ij}x^{j}J^{i}\circ d\phi,
\end{equation}
where $A=\left(a_{ij}\right)_{1\leq i,j\leq3}\in SO(3)$ is a
constant matrix, and $J_{S^2}$ is the standard complex structure on $S^2$. Such a smooth map $\phi$ is called
in \cite{Chen-Li2000} a holomorphic
$S^2$ with respect to a complex structure in the hyperk\"ahler $S^2$ of $N$.
Chen and Li (\cite{Chen-Li2000}) prove that if $N$ does not admit a holomorphic $S^2$ with respect to a complex
structure in the hyperk\"ahler $S^2$ of $N$, then the singular set of a weakly triholomorphic map from $M$ to $N$ is of
codimension 4.

The only known nontrivial example of $\phi$ satisfying \eqref{eq2} is constructed in the case of a non-compact target
\cite{Chen-Li}.
In the monograph \cite{Atiyah-Hitchin}, Atiyah and Hitchin considered the space $M_2^0$ of
centered $2$-monopoles on $\mathbb{R}^3$ with finite action. It is a complete hyperk\"ahler manifold
of dimension $4$. $SO(3)$ acts on $M_2^0$ isometrically and this action lifts to a double
covering $\widetilde{M}_2^0$, which is also a complete hyperk\"ahler manifold
of dimension $4$. In \cite{Chen-Li}, Chen and Li showed that there does exist a nontrivial map $\phi$
from $S^2$ to $\widetilde{M}_2^0$ such that the extended map $u$ from $\phi$ is a triholomorphic
map from $\mathbb{R}^4$ to $\widetilde{M}_2^0$ with the entire $x^4$-axis as singular set.

In this paper, we will focus on the case of compact target.
 Assume $N$ is an algebraic K3 surface defined by a
homogeneous polynomial of degree 4 in $\mathbb{C}P^3$.
It is a nonsingular quartic in $\mathbb{C}P^3$.
For convenience, we call it quartic K3 surface denoted by $X$.
It is a compact hyperk\"ahler manifold
of dimension $4$, i.e. a hyperk\"ahler surface.
A particularly interesting special case is
the Fermat quartic $F\subset \mathbb{C}P^3$ defined by the equation
$$
x_0^4+x_1^4+x_2^4+x_3^4=0,
$$
where $[(x_0,x_1,x_2,x_3)]\in \mathbb{C}P^3$.

Looking at the map
\begin{eqnarray}\label{eq00-2}
h:&&\mathbb{C}^2(=\mathbb{R}^4)\rightarrow F
\nonumber\\
&&
(z_1,z_2)\mapsto [z_1,e^{\frac{\pi}{4}i}z_1,z_2,e^{\frac{\pi}{4}i}z_2],
\end{eqnarray}
which is holomorphic, naturally triholomorphic and singular at the origin,
one has to expect at most any weakly triholomorphic map into $K3$ surface
has isolated singularities.

Through proving that the quartic K3 surface does not admit a holomorphic $S^2$ with respect to a complex
structure in the hyperk\"ahler $S^2$, we obtain
our main result as follows
\begin{theorem}\label{thm1}
Any weakly triholomorphic map from a compact hyperk\"ahler surface to
a quartic K3 surface defined by a
homogeneous polynomial of degree 4 in $\mathbb{C}P^3$
has only isolated singularities.
\end{theorem}

Our bound of the size of the singular set, in view of the example \eqref{eq00-2},
is clearly optimal. It is reasonable to expect this result to be extendable to
any compact target.
Generally,
our result supports the expectation that $\text{Sing}_u$ has Hausdorff codimension four
in the case of a compact target.


\section{Quartic K3 surface}
Let $X$ be a nonsingular quartic surface given by a homogeneous polynomial of degree $4$ in $\mathbb{C}P^3$.
Let $x_i,~0\leq i\leq 3$ be homogeneous coordinates of the complex projective space $\mathbb{C}P^3$.
Let $f(x_0,x_1,x_2,x_3)$ be a homogeneous polynomial of degree $4$ in $\mathbb{C}P^3$, i.e.
$$
f(tx_0,tx_1,tx_2,tx_3)=t^4f(x_0,x_1,x_2,x_3),
$$
for all $t\in\mathbb{C}$.\\
Then
$$
X=\left\{[(x_0,x_1,x_2,x_3)]\in \mathbb{C}P^3\mid f(x_0,x_1,x_2,x_3)=0\right\}.
$$
It is well-known that $X$ is $K3$ and every $K3$ surface is hyperk\"{a}hlerian
(recall that a manifold is hyperk\"{a}hlerian if it admits a hyperk\"{a}hler metric)
(c.f. \cite{Besse},\cite{Huybrechts},\cite{Siu}).

Set $U_i=\left\{[(x_0,x_1,x_2,x_3)]\in \mathbb{C}P^3\mid x_i\neq 0\right\}$ and
$\widetilde{U}_i=X\cap U_i$ ($i=0,1,2,3$).
Then $\bigcup_{i=0}^3\widetilde{U}_i$ is an open covering of $X$.
Let $z_i=\frac{x_i}{x_0},~i=1,2,3$. Then $z=(z_1,z_2,z_3)$ form natural holomorphic
coordinates on $U_0(\cong\mathbb{C}^3)$.
On $\widetilde{U}_0$, denote
\begin{equation}\label{eq0-1}
f(1,z_1,z_2,z_3)=\frac{1}{x_0^4}f(x_0,x_1,x_2,x_3),~f_{z_i}=\frac{\partial f(1,z_1,z_2,z_3)}{\partial z_i}.
\end{equation}
Then,
\begin{equation}\label{eq0-3}
f(1,z_1,z_2,z_3)=0 ~\text{on} ~\widetilde{U}_0.
\end{equation}
Applying the exterior differential $d$ to \eqref{eq0-3}, we obtain
\begin{equation}\label{eq0-4}
f_{z_1}dz_1+f_{z_2}dz_2+f_{z_3}dz_3=0 ~\text{on} ~\widetilde{U}_0,
\end{equation}
which implies
\begin{equation}\label{eq0-5}
f_{z_1}dz_1\wedge dz_2=f_{z_3}dz_2\wedge dz_3~\text{and}~f_{z_1}dz_1\wedge dz_3=-f_{z_2}dz_2\wedge dz_3 ~\text{on} ~\widetilde{U}_0.
\end{equation}
Define a $2$-form $\Omega_0$ on $\widetilde{U}_0$ as follows:
\begin{equation}\label{eq0-6}
\Omega_0\mid_{V_1}=\frac{dz_2\wedge dz_3}{f_{z_1}},
~\Omega_0\mid_{V_2}=\frac{dz_3\wedge dz_1}{f_{z_2}},
~\Omega_0\mid_{V_3}=\frac{dz_1\wedge dz_2}{f_{z_3}},
\end{equation}
where $V_i=\widetilde{U}_0\cap\left\{z\in\widetilde{U}_0\mid f_{z_i}\neq 0\right\},~(i=1,2,3)$.
Here $\Omega_0\mid_{V_i}$ means the restriction of the $2$-form $\Omega_0$ to the open subset $V_i$.

Since $X$ is non-singular, we have $\widetilde{U}_0=V_1\cup V_2\cup V_3$.
Then $\Omega_0$ is well-defined on $\widetilde{U}_0$.
From \eqref{eq0-5} we know
\begin{equation}\label{eq0-7}
d(\Omega_0\mid_{V_1})
=d(\Omega_0\mid_{V_2})
=d(\Omega_0\mid_{V_3})=0.
\end{equation}
Therefore, $\Omega_0$ is a holomorphic symplectic form on $\widetilde{U}_0$.

On $\widetilde{U}_1$, let $y_i=\frac{x_i}{x_1}$ for $i=0,2,3$.
Then
\begin{equation}\label{eq0-7-1}
f(y_0,1,y_2,y_3)=\frac{1}{x_1^4}f(x_0,x_1,x_2,x_3).
\end{equation}
Define a $2$-form $\Omega_1$ on $\widetilde{U}_1$ as follows:
\begin{equation}\label{eq0-8}
\Omega_1\mid_{A_0}=\frac{dy_3\wedge dy_2}{f_{y_0}},
~\Omega_1\mid_{A_2}=\frac{dy_0\wedge dy_3}{f_{y_2}},
~\Omega_1\mid_{A_3}=\frac{dy_2\wedge dy_0}{f_{y_3}},
\end{equation}
where $A_i=\widetilde{U}_1\cap\left\{z\in\widetilde{U}_1\mid f_{y_i}\neq 0\right\},~(i=0,2,3)$.

On $\widetilde{U}_2$, let $w_i=\frac{x_i}{x_2}$ for $i=0,1,3$.
Then
\begin{equation}\label{eq0-8-1}
f(w_0,w_1,1,w_3)=\frac{1}{x_2^4}f(x_0,x_1,x_2,x_3).
\end{equation}
Define a $2$-form $\Omega_2$ on $\widetilde{U}_2$ as follows:
\begin{equation}\label{eq0-9}
\Omega_2\mid_{B_0}=\frac{dw_1\wedge dw_3}{f_{w_0}},
~\Omega_2\mid_{B_1}=\frac{dw_3\wedge dw_0}{f_{w_1}},
~\Omega_2\mid_{B_3}=\frac{dw_0\wedge dw_1}{f_{w_3}},
\end{equation}
where $B_i=\widetilde{U}_2\cap\left\{z\in\widetilde{U}_2\mid f_{w_i}\neq 0\right\},~(i=0,1,3)$.

On $\widetilde{U}_3$, let $v_i=\frac{x_i}{x_3}$ for $i=0,1,2$.
Then
\begin{equation}\label{eq0-9-1}
f(v_0,v_1,v_2,1)=\frac{1}{x_3^4}f(x_0,x_1,x_2,x_3).
\end{equation}
Define a $2$-form $\Omega_3$ on $\widetilde{U}_3$ as follows:
\begin{equation}\label{eq0-10}
\Omega_3\mid_{C_0}=\frac{dv_2\wedge dv_1}{f_{v_0}},
~\Omega_3\mid_{C_1}=\frac{dv_0\wedge dv_2}{f_{v_1}},
~\Omega_3\mid_{C_2}=\frac{dv_1\wedge dv_0}{f_{v_2}},
\end{equation}
where $C_i=\widetilde{U}_3\cap\left\{z\in\widetilde{U}_3\mid f_{w_i}\neq 0\right\},~(i=0,1,2)$.

Define a form $\Omega$ on $X$ such that
\begin{equation}\label{eq0-11}
\Omega\mid_{\widetilde{U}_i}=\Omega_i.
\end{equation}
Then we claim that $\Omega$ is a well-defined holomorphic symplectic form on $X$.
In fact, on $\widetilde{U}_0\cap \widetilde{U}_1$, we know
\begin{equation}\label{eq0-12}
\begin{cases}
y_0=\frac{x_0}{x_1}=\frac{1}{z_1}\\
y_2=\frac{x_2}{x_1}=\frac{z_2}{z_1}\\
y_3=\frac{x_3}{x_1}=\frac{z_3}{z_1}
\end{cases}
\begin{cases}
z_1=\frac{1}{y_0}\\
z_2=\frac{y_2}{y_0}\\
z_3=\frac{y_3}{y_0}
\end{cases}
\begin{cases}
dy_2=\frac{z_1dz_2-z_2dz_1}{z_1^2}\\
dy_3=\frac{z_1dz_3-z_3dz_1}{z_1^2},
\end{cases}
\end{equation}
and
\begin{equation}\label{eq0-12-1}
f(y)=y_0^4f(z),
\end{equation}
which implies
\begin{equation}\label{eq0-12-2}
-z_1^3f_{y_0}=z_1f_{z_1}+z_2f_{z_2}+z_3f_{z_3}.
\end{equation}
Assume $V_1\cap A_0\left(\subset\widetilde{U}_0\cap \widetilde{U}_1\right)\neq \emptyset$.
Thus,
\begin{eqnarray}\label{eq0-13}
\Omega_1\mid_{V_1\cap A_0}
&=&
\frac{dy_3\wedge dy_2}{f_{y_0}}
\nonumber\\
&=&
\frac{z_1dz_3\wedge dz_2-z_2dz_3\wedge dz_1-z_3dz_1\wedge dz_2}{z_1^3f_{y_0}}
\nonumber\\
&=&
-\frac{z_1f_{z_1}+z_2f_{z_2}+z_3f_{z_3}}{z_1^3f_{y_0}}
\cdot\frac{dz_2\wedge dz_3}{f_{z_1}}
\nonumber\\
&=&
\frac{dz_2\wedge dz_3}{f_{z_1}}
\nonumber\\
&=&
\Omega_0\mid_{V_1\cap A_0}.
\end{eqnarray}
As the same way, we can also check the other cases.




For each point $z=(z_1,z_2,z_3)\in V_1\subset\widetilde{U}_0$,
the tangent space of $X$ is as follows,
\begin{eqnarray}\label{eq1}
T_z X
&=&\left\{a_1\frac{\partial}{\partial z_1}+a_2\frac{\partial}{\partial z_2}
+a_3\frac{\partial}{\partial z_3}\in T_z \mathbb{C}P^3
\big| a_1f_{z_1}+a_2f_{z_2}+a_3f_{z_3}=0,a_1,a_2,a_3\in\mathbb{C}\right\}
\nonumber\\
&=&\left\{(a_1,a_2,a_3)\in T_z \mathbb{C}P^3
\big| a_1=-\frac{f_{z_2}}{f_{z_1}}a_2-\frac{f_{z_3}}{f_{z_1}}a_3,a_1,a_2,a_3\in\mathbb{C}\right\}
\nonumber\\
&=&\left\{a_2\left(-\frac{f_{z_2}}{f_{z_1}},1,0\right)+a_3\left(-\frac{f_{z_3}}{f_{z_1}},0,1\right)
\big| a_2,a_3\in\mathbb{C}\right\}
\nonumber\\
&=&\text{span}_{\mathbb{C}}\left\{V_1,V_2\right\},
\end{eqnarray}
where $V_1=\left(-\frac{f_{z_2}}{f_{z_1}},1,0\right),~V_2=\left(-\frac{f_{z_3}}{f_{z_1}},0,1\right)$.
\\Thus the corresponding cotangent space of $X$ is
$(T_z X)^*=\text{span}_{\mathbb{C}}\left\{\varphi^1,~\varphi^2\right\}$,
where $\varphi^1=
-\overline{\left(\frac{f_{z_2}}{f_{z_1}}\right)}\alpha dz_1
+(1-\left|\frac{f_{z_2}}{f_{z_1}}\right|^2\alpha)dz_2
-\overline{\left(\frac{f_{z_2}}{f_{z_1}}\right)}\frac{f_{z_3}}{f_{z_1}}\alpha dz_3$
and
$\varphi^2=
-\overline{\left(\frac{f_{z_3}}{f_{z_1}}\right)}\alpha dz_1
-\frac{f_{z_2}}{f_{z_1}}\overline{\left(\frac{f_{z_3}}{f_{z_1}}\right)}\alpha dz_2
+(1-\left|\frac{f_{z_3}}{f_{z_1}}\right|^2\alpha)dz_3$
with $\alpha=1/\left(1+\left|\frac{f_{z_2}}{f_{z_1}}\right|^2+\left|\frac{f_{z_3}}{f_{z_1}}\right|^2\right)$,
satisfying $\varphi^i(V_j)=\delta_{ij}$.

Then it follows from \cite{Park} that
\begin{theorem}\label{thm2}
Let $X$ be a quartic K3 surface defined by a
homogeneous polynomial $f$ of degree 4 in $\mathbb{C}P^3$.
Then any hyperk\"ahler metrics $h$ on the complex
manifold $X$ can be locally written down as follows.

For each $z=(z_1,z_2,z_3)\in V_1\subset\widetilde{U}_0\subset X$,
let $S=\frac{|f_{z_1}|^2+|f_{z_2}|^2+|f_{z_3}|^2}{|f_{z_1}|^2}$ and $f_{z_1}=|f_{z_1}|e^{i\theta_1}$. Then we have
$$
h=2Re\sum_{i,j=1}^{2}h_{i\overline{j}}\varphi^{i}\overline{\varphi}^{j},
$$
with
$$
\begin{cases}
h_{1\bar{1}}=\frac{\rho S}{2|f_{z_1}|}\lambda\\
h_{1\bar{2}}=-\frac{\rho S}{2|f_{z_1}|}\tau e^{i\theta_1}\\
h_{2\bar{1}}=\overline{h_{1\bar{2}}}\\
h_{2\bar{2}}=\frac{\rho S}{2|f_{z_1}|}\mu
\end{cases}
$$
for some positive constant $\rho$ and some positive real-valued functions $\lambda,~\mu$
and complex-valued function $\tau$ on $V_1$ satisfying
the following conditions:
$$
\lambda\mu=|\tau|^2+1.
$$

Moreover, the corresponding hyperk\"{a}hler structure $\{J^1,J^2,J^3\}$
which is compatible with the metric $h$ is as follows.
\begin{equation}\label{eq1-1}
\begin{cases}
J^1V_1=\sqrt{-1}V_1,
~J^1V_2=\sqrt{-1}V_2,
\\
J^2V_1=\tau\overline{V}_1+\lambda e^{-i\theta_1}\overline{V}_2,
~J^2{V}_2=-\mu e^{-i\theta_1}\overline{V}_1-\overline{\tau}e^{-2i\theta_1}\overline{V}_2,
\\
J^3=J^1J^2.
\end{cases}
\end{equation}
\end{theorem}


\section{Proof of main theorem}
At first, we give three important lemmas which will be applied in the
proof of main theorem.
\begin{lemma}[\cite{Chen-Li2000}]\label{lem1}
Let $M$ be a compact hyperk\"{a}hler surface and let $N$ be a compact
hyperk\"{a}hler manifold. If $N$ does not admit holomorphic $S^2$
with respect to a complex structure in the hyperk\"{a}hler $S^2$,
and $u:M\rightarrow N$ is a weakly (stationary) triholomorphic map, then it
is smooth outside of a finite set of points.
\end{lemma}

\begin{lemma}[B\'{e}zout's theorem \cite{Kirwan}]\label{lem2}
If $C$ and $D$ are two projective curves of degrees $n$ and $m$ in
$\mathbb{C}P^2$ which have no common component then they have precisely
$nm$ points of intersection counting multiplicities.
\end{lemma}

\begin{lemma}\label{lem3}
Let $\phi:\Sigma\rightarrow N$ be a smooth isometric immersion from an
oriented surface $\Sigma$ to a 4-dim hyperk\"{a}hler manifold $N$.
Assume $h$ is the hyperk\"{a}hler metric on $N$ and $\left\{J^1,J^2,J^3\right\}$
are three parallel
complex structures compatible with the metric $h$.
Let $\alpha_p~(p=1,2,3)$ be three corresponding K\"ahler angles
of $\Sigma$ in $N$.
Then we have
\begin{equation}\label{eq20}
\cos^2\alpha_1+\cos^2\alpha_2+\cos^2\alpha_3=1.
\end{equation}
\end{lemma}

\begin{proof}
Fix $x\in \Sigma$. We choose the local frame of
$\Sigma$ around $x$ $\{e_1, e_2, e_3, e_4\}$ such that $\{e_1, e_2\}$ is
the frame of the tangent bundle $T\Sigma$ and $\{e_3, e_4\}$ is the
frame of the normal bundle $(T\Sigma)^{\bot}$.
The K\"ahler angle $\alpha_p$ of $\Sigma$ in $N$ is
defined by
$$\cos\alpha_p=\omega_{J^p}(e_1, e_2)=\langle {J^p}e_1, e_2\rangle.$$
We choose suitable $\{e_3, e_4\}$ such that the complex structure ${J^1}$ has the following form
\begin{eqnarray}\label{eq18}
{J^1}=\left (\begin{array}{clcr} 0 &\cos\alpha_1 &\sin\alpha_1 &0 \\
-\cos\alpha_1 &0 &0 &-\sin\alpha_1\\
-\sin\alpha_1 &0 &0 &\cos\alpha_1\\
0 &\sin\alpha_1 &-\cos\alpha_1 &0 \end{array}\right).
\end{eqnarray}
Let $J^2,~J^3$ be the following form respectively
\begin{eqnarray}\label{eq19}
{J^2}=\left (\begin{array}{clcr} 0 &\cos\alpha_2 &b_{13} &b_{14} \\
-\cos\alpha_2 &0 &b_{23} &b_{24}\\
-b_{13} &-b_{23} &0 &b_{34}\\
-b_{14} &-b_{24} &-b_{34} &0 \end{array}\right),
~{J^3}=\left (\begin{array}{clcr} 0 &\cos\alpha_3 &c_{13} &c_{14} \\
-\cos\alpha_3 &0 &c_{23} &c_{24}\\
-c_{13} &-c_{23} &0 &c_{34}\\
-c_{14} &-c_{24} &-c_{34} &0 \end{array}\right).
\end{eqnarray}
Assume $\sin\alpha_1\neq 0$.
Since $J^3=J^1J^2$, then from \eqref{eq18} and \eqref{eq19} we have
$$
b_{13}=-b_{24}=-\frac{\cos\alpha_2\cos\alpha_1}{\sin\alpha_1},
~b_{14}=b_{23}=\frac{\cos\alpha_3}{\sin\alpha_1},
~b_{34}=\cos\alpha_2,
$$
and
$$
c_{13}=-c_{24}=-\frac{\cos\alpha_3\cos\alpha_1}{\sin\alpha_1},
~c_{14}=c_{23}=-\frac{\cos\alpha_2}{\sin\alpha_1},
~c_{34}=\cos\alpha_3.
$$
Thus we obtain \eqref{eq20} by $(J^2)^2=-\text{id}$.

Moreover,
\begin{eqnarray}\label{eq21}
{J^2}=\left (\begin{array}{clcr} 0 &\cos\alpha_2 &-\frac{\cos\alpha_2\cos\alpha_1}{\sin\alpha_1} &\frac{\cos\alpha_3}{\sin\alpha_1} \\
-\cos\alpha_2 &0 &\frac{\cos\alpha_3}{\sin\alpha_1} &\frac{\cos\alpha_2\cos\alpha_1}{\sin\alpha_1}\\
\frac{\cos\alpha_2\cos\alpha_1}{\sin\alpha_1} &-\frac{\cos\alpha_3}{\sin\alpha_1} &0 &\cos\alpha_2\\
-\frac{\cos\alpha_3}{\sin\alpha_1} &-\frac{\cos\alpha_2\cos\alpha_1}{\sin\alpha_1} &-\cos\alpha_2 &0 \end{array}\right),
\end{eqnarray}
\begin{eqnarray}\label{eq22}
{J^3}=\left (\begin{array}{clcr}
0 &\cos\alpha_3 &-\frac{\cos\alpha_3\cos\alpha_1}{\sin\alpha_1} &-\frac{\cos\alpha_2}{\sin\alpha_1} \\
-\cos\alpha_3 &0 &-\frac{\cos\alpha_2}{\sin\alpha_1} &\frac{\cos\alpha_3\cos\alpha_1}{\sin\alpha_1}\\
\frac{\cos\alpha_3\cos\alpha_1}{\sin\alpha_1} &\frac{\cos\alpha_2}{\sin\alpha_1} &0 &\cos\alpha_3\\
\frac{\cos\alpha_2}{\sin\alpha_1} &-\frac{\cos\alpha_3\cos\alpha_1}{\sin\alpha_1} &-\cos\alpha_3 &0
\end{array}\right).
\end{eqnarray}

\end{proof}

In the following, we give the proof of our main theorem.
\begin{theorem}\label{thm3}
Any weakly triholomorphic map from a compact hyperk\"{a}hler surface to
a quartic K3 surface defined by a
homogeneous polynomial of degree 4 in $\mathbb{C}P^3$
has only isolated singularities.
\end{theorem}

\begin{proof}
Let $X$ be a quartic K3 surface defined by a
homogeneous polynomial $f$ of degree 4 in $\mathbb{C}P^3$, which is
characterized in section 2.
Let $\phi:S^2\rightarrow X$ be a smooth map satisfying \eqref{eq2}.
In the following, we will prove that such smooth map $\phi$ is a constant map,
which implies that $X$ does not admit holomorphic $S^2$
with respect to any complex structure in the hyperk\"{a}hler $S^2$.
Then from Lemma \ref{lem1}, we know that
any weakly (stationary) triholomorphic map from a compact hyperk\"{a}hler surface to
$X$ is smooth outside a finite set of points.
Thus we will finish the proof of our main theorem.

Let $z=\frac{x^1+\sqrt{-1}x^2}{1-x^3}$ be the local holomorphic
coordinate on $S^2\backslash \{(0,0,1)\}$.
Then the standard metric of constant curvature $1$ on $S^2$ is as follows.
$$
g=\frac{4}{(1+z\overline{z})^2}dzd\overline{z}.
$$
Denote
$$\pa=\frac{\pa}{\pa z}, \quad \ov{\pa}=\frac{\pa} {\pa\ov{z}}.$$
Let $\phi=[(1,\phi^1,\phi^2,\phi^3)]$, then
\begin{eqnarray}\label{eq5}
d\phi\left(\frac{\partial}{\partial z}\right)
&=&
\partial\phi^1\frac{\partial}{\partial z_1}
+\partial\phi^2\frac{\partial}{\partial z_2}
+\partial\phi^3\frac{\partial}{\partial z_3}
+\partial\overline{\phi}^1\frac{\partial}{\partial \overline{z}_1}
+\partial\overline{\phi}^2\frac{\partial}{\partial \overline{z}_2}
+\partial\overline{\phi}^3\frac{\partial}{\partial \overline{z}_3}\nonumber\\
&=&
\partial\phi^2V_1
+\partial\phi^3V_2
+\partial\overline{\phi}^2\overline{V}_1
+\partial\overline{\phi}^3\overline{V}_2
\end{eqnarray}
From \eqref{eq1-1} and \eqref{eq5}, we know that \eqref{eq2} is equivalent to
\begin{equation}\label{eq4}
\begin{cases}
\sqrt{-1}\left(1+a_{1j}x^j\right)\partial \phi^2=-\left(a_{2j}x^j+\sqrt{-1}a_{3j}x^j\right)
\left(\overline{\tau}\partial \overline{\phi}^2-\mu e^{i\theta_1}\partial \overline{\phi}^3\right)
\\
\sqrt{-1}\left(1+a_{1j}x^j\right)\partial \phi^3=-\left(a_{2j}x^j+\sqrt{-1}a_{3j}x^j\right)
\left(\lambda e^{i\theta_1}\partial \overline{\phi}^2-\tau e^{2i\theta_1}\partial \overline{\phi}^3\right).
\end{cases}
\end{equation}

A direct calculation shows
\begin{eqnarray}\label{eq12}
\phi^*h
&=&
\frac{\rho S}{\left|f_{z_1}\right|(1+a_{1j}x^j)}
\nonumber\\
&&
\cdot
\left(\lambda\left|\partial\overline{\phi}^2\right|^2
+\mu\left|\partial\overline{\phi}^3\right|^2
-\tau e^{i\theta_1}
\overline{\partial}\phi^2\partial\overline{\phi}^3
-\overline{\tau}e^{-i\theta_1}
\overline{\partial}\phi^3\partial\overline{\phi}^2
\right)dzd\overline{z},
\end{eqnarray}
which implies the corresponding volume form of the pull-back metric
$\phi^*h$ is as follows.
\begin{eqnarray}\label{eq13}
d V_{\phi^*h}
&=&
\frac{\sqrt{-1}}{2}
\frac{\rho S}{\left|f_{z_1}\right|(1+a_{1j}x^j)}
\nonumber\\
&&
\cdot
\left(\lambda\left|\partial\overline{\phi}^2\right|^2
+\mu\left|\partial\overline{\phi}^3\right|^2
-\tau e^{i\theta_1}
\overline{\partial}\phi^2\partial\overline{\phi}^3
-\overline{\tau}e^{-i\theta_1}
\overline{\partial}\phi^3\partial\overline{\phi}^2
\right)dz\wedge d\overline{z}.
\end{eqnarray}

Let $\omega_{J^p}$ be the corresponding k\"ahler form which is compatible with
the corresponding complex structure $J^p$ on $(X,h)$, where $p=1,2,3$. Then
we have $\omega_{J^1}=\frac{\sqrt{-1}}{2}h_{i\overline{j}}\varphi^i\wedge\overline{\varphi}^j$.
Let $\Omega$ be the unique (up to a nonzero constant) holomorphic symplectic form of
the quartic K3 surface $X$.
Then we get $\Omega=\omega_{J^2}+\sqrt{-1}\omega_{J^3}$. From the above section we know
$\Omega\mid_{V_1}=\frac{dz_2\wedge dz_3}{f_{z_1}}$.
Then we have
\begin{eqnarray}\label{eq14}
\phi^*\omega_{J^1}
&=&
\frac{\sqrt{-1}}{2}\frac{\rho S}{2\left|f_{z_1}\right|}
\left(
\lambda d\phi^2\wedge d\overline{\phi}^2
+\mu d\phi^3\wedge d\overline{\phi}^3
-\tau e^{i\theta_1}d\phi^2\wedge d\overline{\phi}^3
-\overline{\tau} e^{-i\theta_1}d\phi^3\wedge d\overline{\phi}^2
\right)
\nonumber\\
&=&
\frac{\sqrt{-1}}{2}
\frac{\rho S \left(-a_{1j}x^j\right)}{\left|f_{z_1}\right|(1+a_{1j}x^j)}
\nonumber\\
&&
\cdot
\left(\lambda\left|\partial\overline{\phi}^2\right|^2
+\mu\left|\partial\overline{\phi}^3\right|^2
-\tau e^{i\theta_1}
\overline{\partial}\phi^2\partial\overline{\phi}^3
-\overline{\tau}e^{-i\theta_1}
\overline{\partial}\phi^3\partial\overline{\phi}^2
\right)dz\wedge d\overline{z}
\nonumber\\
&=&
\left(-a_{1j}x^j\right)d V_{\phi^*h},
\end{eqnarray}
and

\begin{eqnarray}\label{eq15}
\phi^*\Omega
&=&
\frac{d\phi^2\wedge d\phi^3}{f_{z_1}}
\nonumber\\
&=&
\frac{1}{f_{z_1}}\left(\partial{\phi}^2\overline{\partial}\phi^3
-\partial{\phi}^3\overline{\partial}\phi^2\right)dz\wedge d\overline{z}
\nonumber\\
&=&
(-\sqrt{-1})\frac{a_{2j}x^j+\sqrt{-1}a_{3j}x^j}{\left|f_{z_1}\right|(1+a_{1j}x^j)}
\nonumber\\
&&
\cdot
\left(\lambda\left|\partial\overline{\phi}^2\right|^2
+\mu\left|\partial\overline{\phi}^3\right|^2
-\tau e^{i\theta_1}
\overline{\partial}\phi^2\partial\overline{\phi}^3
-\overline{\tau}e^{-i\theta_1}
\overline{\partial}\phi^3\partial\overline{\phi}^2
\right)dz\wedge d\overline{z}
\nonumber\\
&=&
-2\frac{a_{2j}x^j+\sqrt{-1}a_{3j}x^j}{\rho S}d V_{\phi^*h}.
\end{eqnarray}

The k\"ahler angle $\alpha_p$~$(p=1,2,3)$ of $S^2$ in $X$ was defined by Chern-Wolfson\cite{Chern-Wolfson}
\begin{equation}\label{eq16}
\omega_{J^p}=\cos\alpha_p d V_{\phi^*h}.
\end{equation}
It follows from \eqref{eq14} and \eqref{eq15} that
\begin{equation}\label{eq17}
\begin{cases}
\cos\alpha_1=-a_{1j}x^j,
\\
\cos\alpha_2=-2\frac{a_{2j}x^j}{\rho S},
\\
\cos\alpha_2=-2\frac{a_{2j}x^j}{\rho S}.
\end{cases}
\end{equation}
From \eqref{eq17} and \eqref{eq20}, we get
\begin{equation}\label{eq23}
S=\frac{2}{\rho}=\text{constant}.
\end{equation}

Assume $\rho=2$. Then we find
\begin{equation}\label{eq24}
S=\frac{|f_{z_1}|^2+|f_{z_2}|^2+|f_{z_3}|^2}{|f_{z_1}|^2}=1,
~\text{on} ~\phi(S^2)\cap V_1\subset \widetilde{U}_0\subset X,
\end{equation}
which implies
\begin{equation}\label{eq25}
f_{z_2}=f_{z_3}=0,
~\text{on} ~\phi(S^2)\cap V_1.
\end{equation}
Since $f_{z_1}\neq 0$, from \eqref{eq0-4} we get
\begin{equation}\label{eq26}
z_1=\text{constant},
~\text{on} ~\phi(S^2)\cap V_1.
\end{equation}
Without loss of generality, assume $z_1=\sigma~(\text{constant})$.
Then $\phi(S^2)\cap V_1$ is contained in a projective curve of degree 4
in $\mathbb{C}P^2$.

Let
\begin{equation}\label{eq26-1}
g(1,z_2,z_3)=f(1,\sigma,z_2,z_3).
\end{equation}
Then we have
\begin{equation}\label{eq26-2}
g(1,z_2,z_3)=0,~g_{z_2}=f_{z_2}=0,~g_{z_3}=f_{z_3}=0,~\text{on}~\phi(S^2)\cap V_1.
\end{equation}
Define
\begin{equation}\label{eq26-3}
C=\left\{[1,z_2,z_3]\in \mathbb{C}P^2: ~g(1,z_2,z_3)=0\right\},
\end{equation}
\begin{equation}\label{eq27}
D=\left\{[1,z_2,z_3]\in \mathbb{C}P^2: ~g_{z_2}=0\right\},
\end{equation}
and
\begin{equation}\label{eq28}
E=\left\{[1,z_2,z_3]\in \mathbb{C}P^2: ~g_{z_3}=0\right\}.
\end{equation}
Then $C,~D,~E$ are three projective curves of degree $4,3,3$
in $\mathbb{C}P^2$ respectively.
Moreover we get
\begin{equation}\label{eq28-1}
\phi(S^2)\cap V_1\subset C\cap D\cap E.
\end{equation}

In the following, we prove that $C\cap D\cap E$ has only finite points
by considering three different cases about their common components.\\
\textbf{Case 1:} Assume $C,~D,~E$ have common component.
Then the corresponding polynomials $g(1,z_2,z_3),~g_{z_2},~g_{z_3}$
have common nonconstant polynomial factor $P$.
Since any two projective curves in $\mathbb{C}P^2$ intersect in at least one point,
we know that the projective curve given by $P$ and the one given by $f_{z_1}(1,\sigma,z_2,z_3)$
intersect in at least one point denoted by $[1,b,c]\in \mathbb{C}P^2$.
Then the point $[1,\sigma,b,c]\in \mathbb{C}P^3$ is a singularity of the polynomial
$f(1,z_1,z_2,z_3)$, which contradicts to the fact that quartic K3 surface given by $f(1,z_1,z_2,z_3)$
is nonsingular. So there doesn't exist this case.
\\
\textbf{Case 2:} Assume two and only two curves of $C,~D,~E$ have common component.
Without loss of generality, assume $C$ and $D$ have common component.
Then $C$ and $E$ have no common component.
From Lemma \ref{lem2}, we know that $C$ and $E$ have only finite points of intersection.
So $C\cap D\cap E$ has only finite points in this case.
\\
\textbf{Case 3:} Assume any two curves of $C,~D,~E$ have no common component.
Obviously $C\cap D\cap E$ has only finite points or $C\cap D\cap E=\emptyset$
in this case.


Now, we have proven that $C\cap D\cap E$ has only finite points.
It follows from \eqref{eq28-1} that
$\phi(S^2)\cap V_1$ has only finite points.
Similarly, we know that $\phi(S^2)\cap V_2$
and $\phi(S^2)\cap V_3$ have only finite points respectively.
Then we get that $\phi(S^2)\subset X$ has only finite points.
But $\phi$ is smooth. Hence
$\phi$ must be a constant map.

\end{proof}




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\end{document}

1604.02648.extracted_bib
{Atiyah-Hitchin M. Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton University Press, 1988.
1604.02648.extracted_bib
{Bellettini-Tian C. Bellettini and G. Tian, Compactness results for triholomorphic maps, arXiv:1507.06558v2, 2015.
1604.02648.extracted_bib
{Besse A. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987.
1604.02648.extracted_bib
{Bethuel F. Bethuel, On the singular set of stationary harmonic maps, Manu. Math., 78(1993), 417-443.
1604.02648.extracted_bib
{Chen J.Y. Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic curves, Commun. Math. Phys., 201(1999), 201-247.
1604.02648.extracted_bib
{Chen-Li2000 J.Y. Chen and J.Y. Li, Quaternionic maps between hyperk\"ahler manifolds, J. Diff. Geom., 55(2000), 355-384.
1604.02648.extracted_bib
{Chen-Li J.Y. Chen and J.Y. Li, Quaternionic maps and minimal surfaces, Ann. Scuola. Sup. Pisa Cl. Sci., 4(5)(2005), 375-388.
1604.02648.extracted_bib
{Chern-Wolfson S.S. Chern and J. Wolfson, Minimal surfaces by moving frames, Am. J. Math., 105(1983), 59-83.
1604.02648.extracted_bib
{FKS J.M. Figueroa-O'Farrill, C.Khl and B.Spence, Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves, Nuclear Phys., B521(3)(1998), 419-443.
1604.02648.extracted_bib
{Huybrechts D. Huybrechts,Compact hyperk\"ahler manifolds: basic results, Invent. Math., 135(1999), 63-113.
1604.02648.extracted_bib
{Kirwan F. Kirwan, Complex algebraic curves, London Mathematical Society, 1992.
1604.02648.extracted_bib
{Li-Tian J.Y. Li and G. Tian, A blow-up formula for stationary harmonic maps, Internat.Math.Res.Notices, 14(1998), 735-755.
1604.02648.extracted_bib
{Park K.S. Park, A Local construction of hyperk\"ahler metrics on the Fermat surface, Journal of Dynamical Systems $\&$ Geometric Theories, 9(2)(2011), 185-200.
1604.02648.extracted_bib
{Siu Y.T. Siu, Every $K3$ surface is k\"ahler, Invent. Math., 73(1983), 139-150.
1604.02648.extracted_bib
{Tian G. Tian,Some Regularity Problems of Stationary Harmonic Maps, Contemp. Math., 350(2004), 245-252.