コンパクト複素曲線 Aが real K3 surface (X, \varphi) with non-symplectic holomorphic involution $\tau$ of type (S, θ) における反シンプレクティック正則対合$\tau$の不動点集合（固定点曲線）である場合を考える．
このとき，Aがdividingであるかどうかは，
条件(S, θ)付対合付格子(H_2(X; Z), \varphi_*)の不変量
δ_{\varphi S}
によって決まることが判明している：
Theorem (Nikulin and Saito, Proc.LMS, 2007)
Let (X, \varphi) be a real K3 surface with non-symplectic holomorphic involution $\tau$ of type (S,\theta)
and A be the fixed point set of $\tau$ in X.
Assume that A is non-empty. Then,
(A, \varphi) is dividing if and only if δ_{\varphi S} = 0.
Proof
The main technique is a so-called Donaldson's trick
(see {D} or the book {DIK2000}).
We know the K3 surface $X$ is algebraic.
(See {Yoshi2004}, for example.)
We can take a hyperplane section class $h$ in $S$.
Since $\varphi_*(h) = -h$,
$h$ is contained in $S_- = L^{\tau} \cap L_{\varphi}$.
Moreover, $h$ is contained in the K\"ahler cone
$\mathcal{C}^+_X$ of $X$ (See {BHPV}).
Thus we see
$S_- \cap \mathcal{C}^+_X \neq \emptyset$.
Let $I$ be the complex structure of the K3 surface $X$ and
take a K\"ahler class $c$ of $(X,I)$
in $((L^{\tau} \cap L_{\varphi}) \otimes \br) \cap \mathcal{C}^+_X$.
Since $c_1(X)=0$, by the Calabi-Yau theorem
(see, for example, {Besse}, Theorem 11.15)
$0$ is the Ricci form of one and only one K\"ahler metric in the class $c$.
Thus we can take the unique K\"ahler form $P$ with zero Ricci form in the class $c$, where $c=[P]$.
Let $g$ be the K\"ahler metric whose K\"ahler form is $P$,
i.e., $P = g(I( ), )$.
By the uniqueness of the Ricci flat K\"ahler form,
we have $\tau^*P = P$ and $\varphi^*P = - P$.
Since $\tau$ is holomorphic on $(X,I)$, we have $\tau^*g = g$.
And, since $\varphi$ is anti-holomorphic on $(X,I)$,
we have $\varphi^*g = g$.
Now we can take a nowhere vanishing holomorphic $2$-form
$\omega_I$ on $(X,I)$
such that $\tau^* \omega_I = - \omega_I$ and $\varphi^*(\omega_I) = \overline{\omega_I}$.
We define the real $2$-forms $Q$ and $R$ on $X$ to be
$\omega_I = Q + \sqrt{-1}R$.
Then we have
$\tau^* Q = -Q, \tau^* R = -R, \varphi^* Q = Q and
\varphi^* R = -R.$
Moreover, we may asuume
$2(P\wedge P) = \omega_I \wedge \overline{\omega_I}$,
equivalently,
$Q \wedge Q = P \wedge P$.
We define the new complex structures $J$ and $K$ on $X$ by
$Q=g(J( ), ) and R=g(K( ), ).$
(Then $g$ is a hyperK\"ahler metric on $X$.
See {Huybrechts02}, Remark 2.4.)
By the above we can verify that
$\tau$ is anti-holomorphic and $\varphi$ is holomorphic for $(X,J)$.
We see $\omega_J := R + \sqrt{-1}P$ is
a nowhere vanishing holomorphic $2$-form on $(X,J)$.
We have $\varphi^*(\omega_J)= - \omega_J$, i.e.,
$\varphi$ is non-symplectic on the K3 surface $(X,J)$.
We get a new real K3 surface with a non-symplectic involution
$((X,J),\varphi,\tau)$.
Then the fixed point set $X(\br)$ of $\varphi$
is a complex 1-dimensional submanifold of $(X,J)$
(see {Nikulin81}, p.1424).
The fixed point set of $\tau$ in $X(\br)$ is $A(\br)$.
Thus we can apply {NikSaito2005}, Theorem 16
to the K3 surface $(X,J)$, the anti-holomorphic involution $\tau$ and the complex curve $X(\br)=:C$.
Thus we conclude that the following two conditions are equivalent.
(1) [X(\br)] ・ x \equiv 0 mod 2 for all x in L_\tau.
(2) [A(\br)] = 0 in H_1(A; Z/2Z).
The condition (1) is equivalent to $δ_{\varphi S} = 0$.
This completes the theorem. □
Remark
この証明でわかるのは，Xの複素構造を取り替えることにより，証明したい定理が，{NikSaito2005}のTheorem 16と等価となることである．
(Therem 16 については，後日述べます)
このTheorem 16を証明するときにキーとなったのは，
論文{Kharlamov75a}，{Kharlamov76}，{Mangolte97}であったことも想起したい．
References
{Besse}
A.L. Besse,
Einstein Manifolds,
Ergeb. Math. Grenzgeb. (3), {\bf 10}, Springer-Verlag, 1987.
{BHPV}
W. Barth, Hulek, C. Peters, A. Van de Ven,
Compact Complex Surfaces,
Ergeb. Math. Grenzgeb. (3), {\bf 4}, Springer-Verlag.
{DIK2000}
A. Degtyarev, I. Itenberg, V. Kharlamov,
Real Enriques Surfaces}
Lect. Notes in Math., {\bf 1746} (2000), 259 pp.
{D}
S.K. Donaldson,
Yang--Mills invariants of smooth four-manifolds,
Geometry of low-dimensional manifolds
(S.K. Donaldson, C.B. Thomas, eds.), Vol. 1.
Cambridge University Press, Cambridge (1990), 5--40.
{Huybrechts02}
Daniel Huybrechts,
Moduli spaces of hyperk\"ahler manifolds and mirror symmetry,
math.AG/0210219 v3.
{Kharlamov75a}
V.M. Kharlamov,
Additional congruences for the Euler characteristic of real
algebraic manifolds of even dimensions,
Funkt. Anal. Prilozhen.,{\bf 9} (1975), 51--60.
English transl. in Funct. Anal. Appl., {\bf 9} (1975), 134--141.
{Kharlamov76}
V.M. Kharlamov,
The topological types of nonsingular surfaces in
$\br\bp^3$ of degree four,
Funkt. Anal. Prilozhen.,{\bf 10} (1976), 55--68.
English transl. in Funct. Anal. Appl., {\bf 10} (1976), 295--305.
{Mangolte97}
F. Mangolte,
Cycles alg\'ebriques sur les surfaces {K}3 r\'eelles,
Math. Z., {\bf 225} (1997), 559--576.
{Nikulin81}
V.V. Nikulin,
Factor groups of the automorphism groups of
hyperbolic forms by the subgroups generated by 2-reflections,
Algebraic-geometric applications,
Current Problems in Math., Vsesoyuz. Inst. Nauch. i Tekhn. Informatsii, Moscow,
{\bf 18} (1981), 3--114.
English transl. in J. Soviet Math., {\bf 22} (1983), 1401--1476.
{NikSaito2005}
V.V. Nikulin and S. Saito,
Real K3 surfaces with non-symplectic involutions and applications,
Proc. LMS 2005..
{Yoshi2004}
Ken-Ichi Yoshikawa,
K3 surfaces with involution, equivariant analytic torsion,
and automorphic forms on the moduli space,
Invent. Math. {\bf 156-1} (2004), 53--117.