\(
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\)
Basic Theory
Definition 35.6 (The spinor norm and the special orthogonal group) Let \((L,\beta_L)\) be a lattice. Every \(g\in\Orth(L_\RR)\) is a product of reflections \[
g=s_{w_1}\cdots s_{w_m}
\] in anisotropic vectors \(w_i\in L_\RR\), where \[
s_w(x)=x-2\frac{\beta_L(x,w)}{\beta_L(w,w)}w.
\] The real spinor norm is \[
\spinornorm_\RR(g)
\definedas
\prod_{i=1}^m\frac{-\beta_L(w_i,w_i)}2
\in
\unitsof{\RR}/(\unitsof{\RR})^2
\cong\theset{\pm1},
\] independent of the chosen reflection decomposition. The special orthogonal group is \[
\SO(L)\definedas\ker\qty{\det\colon\Orth(L)\too\theset{\pm1}}.
\] For a lattice of signature \((2,n)\), \(\Orth^+(L)\) denotes the subgroup preserving either connected component of \(\Omega_L\); with the convention above this is the trivial-real-spinor-norm subgroup.
Definition 35.7 (K3 families, markings, periods, and monodromy) Let \[
\lkt\definedas U^3\oplus E_8^2
\] with its integral symmetric bilinear form; this is the K3 lattice of signature \((3,19)\).
For a connected complex manifold \(S\), let \(F_{\Kthree}(S)\) be the groupoid of smooth proper holomorphic maps \[
f\colon\mathcal X\too S
\] whose fibers \(X_s\) are K3 surfaces, with morphisms the cartesian isomorphisms over \(S\). No polarization, marking, or level structure is part of an object of \(F_{\Kthree}\).
For \(f\in F_{\Kthree}(S)\), set \[
\mathbb V_f\definedas R^2f_*\underline{\ZZ}_{\mathcal X}.
\] Cup product gives a morphism of integral local systems \[
\beta_f\colon
\mathbb V_f\tensor_{\underline{\ZZ}_S}\mathbb V_f
\too
\underline{\ZZ}_S,
\] whose fiber at \(s\in S\) is the K3 lattice \[
\qty{H^2(X_s,\ZZ),\smile}.
\] The weight-two Hodge filtration has \[
F^2\qty{\mathbb V_f\tensor_\ZZ\mathcal O_S}
=
f_*\Omega^2_{\mathcal X/S},
\] a line bundle whose fiber is \(H^{2,0}(X_s)=\CC\omega_s\).
The marked K3 functor \(F_{\Kthree}^{\mathrm{marked}}(S)\) has objects \((f,\eta)\) with \(f\in F_{\Kthree}(S)\) and \[
\eta\colon
\qty{\mathbb V_f,\beta_f}
\iso
\qty{\underline{\lkt}_S,\beta_{\lkt})
\] an isometry of local systems. Equivalently, \(\eta\) is a compatible family of fiber markings \[
\eta_s\colon
\qty{H^2(X_s,\ZZ),\smile}
\iso
\lkt.
\] A global marking trivializes the integral local system; monodromy below is therefore defined first for an arbitrary object of \(F_{\Kthree}\), using only a marking of one chosen fiber.
For any lattice \(L\), write \[
\Omega_L
\definedas
\theset{
[\omega]\in\PP(L_\CC)
\st
(\omega,\omega)=0,\ (\omega,\bar\omega)>0
}.
\] If \((f,\eta)\in F_{\Kthree}^{\mathrm{marked}}(S)\), its period map is \[
\operatorname{per}_{f,\eta}\colon
S\too\Omega_{\lkt},
\qquad
s\longmapsto
[\eta_{s,\CC}(\omega_s)],
\] where \(0\neq\omega_s\in H^{2,0}(X_s)\). The notation \([\omega]\) always denotes the projective class of a nonzero holomorphic symplectic form after the relevant marking.
Now let \(f\in F_{\Kthree}(S)\), choose \(s_0\in S\), and choose a marking of the single fiber \[
\eta_{s_0}\colon
H^2(X_{s_0},\ZZ)\iso\lkt.
\] The local system \(\mathbb V_f\) has monodromy representation \[
\rho_{f,s_0}\colon
\pi_1(S,s_0)
\too
\Orth\qty{H^2(X_{s_0},\ZZ),\smile}.
\] Transporting through \(\eta_{s_0}\) gives \[
\rho_{f,s_0}^{\eta_{s_0}}
\definedas
\eta_{s_0}\rho_{f,s_0}\eta_{s_0}^{-1}
\colon
\pi_1(S,s_0)
\too
\Orth(\lkt).
\] Define the monodromy group of the marked base fiber of the family by \[
\operatorname{Mon}(f,s_0,\eta_{s_0})
\definedas
\im\rho_{f,s_0}^{\eta_{s_0}}
\iscontainedin
\Orth(\lkt).
\] Thus monodromy is attached to the local system \(\mathbb V_f\); the marking only identifies its fiber with the fixed lattice \(\lkt\).
Finally, define \[
\operatorname{Mon}(\lkt)
\] to be the subgroup of \(\Orth(\lkt)\) generated by the images \(\operatorname{Mon}(f,s_0,\eta_{s_0})\) as \(f\), \(S\), \(s_0\), and the base-fiber marking vary.
Definition 35.8 (The orientation character of the K3 lattice) Set \[
\widetilde C_{\lkt}
\definedas
\theset{x\in\lkt_\RR\st(x,x)>0}.
\] For a positive-definite three-plane \(W_3\containedin\lkt_\RR\), the inclusion \[
W_3\sm\theset{0}\injects\widetilde C_{\lkt}
\] is a homotopy equivalence: in the orthogonal decomposition \[
\lkt_\RR=W_3\oplus W_3^\perp,
\] scaling the \(W_3^\perp\)-component to zero preserves positivity. Hence \[
H_2\qty{W_3\sm\theset{0},\ZZ}
\iso
H_2\qty{\widetilde C_{\lkt},\ZZ}
\iso
\ZZ.
\] An orientation of \(W_3\) determines one of the two generators; these are the orientation classes of \(\widetilde C_{\lkt}\). The action on orientation classes defines \[
\sigma_{\lkt}\colon
\Orth(\lkt_\RR)
\too
\theset{\pm1}.
\] This is the positive-three-plane form of the real spinor-norm character; set \[
\Orth^+(\lkt)
\definedas
\ker\sigma_{\lkt}.
\] This is the convention used for the K3 lattice in Huybrechts, Lectures on K3 Surfaces, §5.4, and is the orientation-class formulation used in Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, §4.
Proposition 35.9 (K3 monodromy preserves the positive-three-plane orientation) Let \(f\in F_{\Kthree}(S)\) with \(S\) connected, let \(s_0\in S\), and choose a fiber marking \[
\eta_{s_0}\colon H^2(X_{s_0},\ZZ)\iso\lkt.
\] Then \[
\operatorname{Mon}(f,s_0,\eta_{s_0})
\iscontainedin
\Orth^+(\lkt).
\] Consequently \[
\operatorname{Mon}(\lkt)
\iscontainedin
\Orth^+(\lkt).
\]
Proof. For each \(s\in S\), choose \(0\neq\omega_s\in H^{2,0}(X_s)\) and a Kähler class \[
H_s\in H^{1,1}(X_s,\RR).
\] The real three-plane \[
W_s
\definedas
\Span_\RR\qty{\Re\omega_s,\Im\omega_s,H_s}
\containedin
H^2(X_s,\RR)
\] is positive definite for the cup-product form. The ordered basis \[
\qty{\Re\omega_s,\Im\omega_s,H_s}
\] defines an orientation class of the positive cone of \(H^2(X_s,\RR)\).
This class is independent of the choices: replacing \(\omega_s\) by \(a\omega_s\) with \(a\in\CC^*\) acts orientation-preservingly on the real plane \(H^{2,0}(X_s)\oplus H^{0,2}(X_s)\), and the Kähler cone is connected. In a local trivialization of \(\mathbb V_f\), the Hodge line and a Kähler class vary continuously, so these orientation classes form a locally constant section of the two-element orientation local system associated to the positive cones.
Let \(\gamma\in\pi_1(S,s_0)\). Gauss–Manin parallel transport along \(\gamma\) therefore preserves the distinguished orientation class at \(s_0\). After applying \(\eta_{s_0}\), this says \[
\sigma_{\lkt}\qty{\rho_{f,s_0}^{\eta_{s_0}}(\gamma)}=+1.
\] Hence \[
\im\rho_{f,s_0}^{\eta_{s_0}}
\iscontainedin
\ker\sigma_{\lkt}
=
\Orth^+(\lkt).
\] Taking the subgroup generated by all such family monodromy images gives the second containment.
Definition 35.10 (Degree-$2d$ K3 families and the primitive period lattice) Fix a primitive vector \[
h\in\lkt,
\qquad
h^2=2d,
\] and set \[
\lkttd
\definedas
\generators{h}^{\perp\lkt}
\cong
\generators{-2d}\oplus U^2\oplus E_8^2.
\] Thus \(\lkttd\) is the degree-\(2d\) primitive K3 lattice already used for \(\ftd\).
For a connected complex manifold \(S\), an object of \(\ftd(S)\) is a pair \[
\qty{f\colon\mathcal X\too S,\mathcal L}
\] with \(f\in F_{\Kthree}(S)\) and \(\mathcal L\) a relatively ample primitive line bundle satisfying \[
c_1(\mathcal L_s)^2=2d
\] on every fiber. The classes \(c_1(\mathcal L_s)\) form a flat section \[
c_1(\mathcal L)\in\Gamma(S,\mathbb V_f),
\] so the primitive local system \[
\mathbb V_{f,\mathcal L}^{\mathrm{prim}}
\definedas
c_1(\mathcal L)^\perp
\iscontainedin
\mathbb V_f
\] is monodromy invariant.
A marking of the polarized family is a marking \[
\eta\colon\mathbb V_f\iso\underline{\lkt}_S
\] such that \[
\eta\qty{c_1(\mathcal L)}=h.
\] It restricts to \[
\eta^{\mathrm{prim}}\colon
\mathbb V_{f,\mathcal L}^{\mathrm{prim}}
\iso
\underline{\lkttd}_S.
\] The marked polarized period map is therefore \[
\operatorname{per}_{f,\mathcal L,\eta}
\colon
S\too\Omega_{\lkttd},
\qquad
s\longmapsto
[\eta_{s,\CC}(\omega_s)].
\]
For an unmarked polarized family, the monodromy representation of \(\mathbb V_{f,\mathcal L}^{\mathrm{prim}}\) lands, after a base-fiber marking carrying \(c_1(\mathcal L_{s_0})\) to \(h\), in \[
\Orth^+(\lkttd),
\] equivalently in the subgroup preserving a connected component of the type-IV domain \(\Omega_{\lkttd}\). This is the polarized form of the positive-orientation statement above.
Proposition 35.11 (Polarized K3 monodromy preserves the primitive period component) Let \((f\colon\mathcal X\to S,\mathcal L)\in \ftd(S)\) with \(S\) connected, choose \(s_0\in S\), and choose a fiber marking \[
\eta_{s_0}\colon H^2(X_{s_0},\ZZ)\iso\lkt
\] with \[
\eta_{s_0}\qty{c_1(\mathcal L_{s_0})}=h.
\] Then the monodromy representation of the primitive local system \[
\mathbb V_{f,\mathcal L}^{\mathrm{prim}}
\] lands, after the induced identification with \(\lkttd=h^{\perp\lkt}\), in \[
\Orth^+(\lkttd).
\]
Proof. Because \(c_1(\mathcal L)\) is a flat section of \(\mathbb V_f\), every element \[
g\in\operatorname{Mon}(f,s_0,\eta_{s_0})
\iscontainedin\Orth(\lkt)
\] fixes \(h\). By Proposition 35.9, \(g\in\Orth^+(\lkt)\). Let \[
P\containedin\lkttd_\RR
\] be an oriented positive two-plane. Since \(h^2>0\) and \(P\perp h\), the direct sum \[
\RR h\oplus P\containedin\lkt_\RR
\] is a positive three-plane. Orient it by the positive vector \(h\) followed by an oriented basis of \(P\). Because \(g(h)=h\), preservation by \(g\) of the orientation of \(\RR h\oplus P\) is equivalent to preservation by \(g\vert_{\lkttd}\) of the orientation of \(P\). Hence \[
g\vert_{\lkttd}\in\Orth^+(\lkttd),
\] which is equivalently the subgroup preserving a connected component of \(\Omega_{\lkttd}\).
Theorem 35.14 (Classification of primitive isotropic vectors by type) Let \(S\) be an even hyperbolic 2-elementary lattice with invariants \((r, a, \delta)\), let \(v\in S\) be a primitive isotropic vector, and write \(\oS\definedas v^{\perp S}/v\). Then exactly one of the following holds, according to the type of \(v\) in the sense above, and in each case \(S\) splits off the indicated rank-two summand containing \(v\):
odd: \(S \cong U\oplus\oS\), with \(a_{\oS} = a\) and \(\delta_{\oS} = \delta\);
even ordinary: \(S \cong U(2)\oplus\oS\), with \(a_{\oS} = a - 2\) and \(\delta_{\oS} = \delta\);
even characteristic: \(S \cong \latI_{1,1}(2)\oplus\oS\), with \(a_{\oS} = a - 2\); this case forces \(\delta = 1\) and gives \(\delta_{\oS} = 0\).
In all three cases \(\oS\) is even negative definite of rank \(r - 2\). Classifying the primitive isotropic vectors of \(S\) up to isometry is therefore equivalent to classifying the even negative definite lattices with invariants \((r-2, a, \delta)\), \((r-2, a-2, \delta)\) and \((r-2, a-2, 0)\).
Alexeev, Valery, and Philip Engel. 2022. Compactifications of Moduli Spaces of K3 Surfaces with a Nonsymplectic Involution.
Dawes, Matthew. 2022.
Orbits in Lattices.
https://doi.org/10.48550/arXiv.2205.10601.
Nikulin, V. V. 1980.
“Integral Symmetric Bilinear Forms and Some of Their Applications.” Mathematics of the USSR-Izvestiya 14 (1): 103.
https://doi.org/10.1070/IM1980v014n01ABEH001060.
Sikirić, Mathieu Dutour, and Klaus Hulek. 2023.
Moduli of Polarized Enriques Surfaces – Computational Aspects.
https://doi.org/10.48550/arXiv.2302.01679.
Vinberg, E. B. 1975. “Some Arithmetical Discrete Groups in Lobachevskii Spaces.” In Discrete Subgroups of Lie Groups and Applications to Moduli. Oxford University Press.
Vinberg, È. B. 1985.
“Hyperbolic Groups of Reflections.” Rossíiskaya Akademiya Nauk 40 (1(241)): 29–66, 255.
https://doi.org/10.1070/RM1985v040n01ABEH003527.