Here we recall the classification theory of complex K3 sufaces (see [PSS] and its extension to non-algebraic case in [LP]). Let $X$ be a complex K3 surface, i.e. smooth connected complex manifold with $\dim_{{\bf C}}X=2$ which admits a nowhere vanishing holomorphic $2$-form $\Omega$, and such that $H^{1}(X,{{\bf Z}})=0$.

It is known that the group $H^{2}(X,{{\bf Z}})$ endowed with the Poincare pairing $(\cdot,\cdot)$ is isomorphic to the lattice

$$\Lambda_{K3}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\oplus\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\oplus\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\oplus\left(-E_{8}\right)\oplus\left(-E_{8}\right)$$

of signature $(3,19)$.

Complex $1$-dimensional vector space $H^{2,0}(X)={{\bf C}}\cdot[\Omega]\subset H^{2}(X,{{\bf Z}})\otimes{{\bf C}}$ satisfies the condition $(v,v)=0,(v,\overline{v})>0$ for any non-zero vector $v$. Finally, it is known that $X$ admits a Kähler metric, and Kähler cone ${\cal K}_{X}\subset H^{2}(X,{{\bf R}})$ of all Kähler metrics on $X$ is an open subset of $C_{X}:=\{[\omega]\in H^{2}(X,{{\bf Z}})\otimes{{\bf R}}|([\omega],[\Omega])=0,([\omega],[\omega])>0\}$. In fact ${\cal K}_{X}$ is a connected component of the set $C_{X}\setminus\cup_{v\in H^{2}(X,{{\bf Z}}),(v,v)=-2,(v,[\Omega])=0}H_{v}$, where $H_{v}$ is the hyperplane orthogonal to $v$.

Axiomatizing these data we arrive to the following definition.

The K3 period data form a groupoid. On the other hand, K3 surfaces also form a groupoid (morphisms are isomorphisms of K3 surfaces). Then classical global Torelli theorem can be formulated in the following way.

In particular the automorphism group of a K3 surface is isomorphic to the automorphism group of its period data.

More generally one can speak about holomorphic families of K3 surfaces over complex analytic spaces. For a K3 surface over an analytic space $M$ the period data consist of a local system of integral lattices $(\Lambda,(\cdot,\cdot))$ pointwise isomorphic to $\Lambda_{K3}$, a holomorphic line subbundle $H^{2,0}$ of $\Lambda\otimes_{{\bf Z}}{\cal O}_{M}$ which is isotropic with respect to the symmetric pairing $(\cdot,\cdot)$, and satisfies pointwise the condition $(v,\overline{v})>0,v\in H^{2,0}_{x}\setminus\{0\},x\in M^{red}$, and an open subset of the total space of the bundle over $M^{red}$ with the fibers $\Lambda_{x}\otimes{{\bf R}}\cap(H^{2,0})^{\perp}$ ($(H^{2,0})^{\perp}$ is the orthogonal complement) satisfying pointwise the condition 4) from the definition of K3 period data. Then Torelli theorem holds for families as well.

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Addresses:

M.K.: IHES, 35 route de Chartres, F-91440, France

maxim@ihes.fr

Y.S.: Department of Mathematics, KSU, Manhattan, KS 66506, USA

soibel@math.ksu.edu