CAN WE LIFT A FAMILY OF K3 OVER A PROPER CURVE?
JIE XIA
1. I NTRODUCTION
In this draft, we study the deformation of a proper smooth curve in positive characteristics within the moduli of K3. We expect that the curve can be lifted to the Witt ring with
generic fiber a Shimura curve.
Given an algebraic closed field k, a double-point K3 surface over k is a projective surface
with at worst rational double point singularities whose minimal resolution is a (smooth)
K3 surface over k. Let π : X −→ C −→ F̄p be a primitively polarized family of doublepoint K3 surfaces and W = W (F̄p ). There exists a ramified Galois covering S −→ C, a
family of (smooth) K3 surfaces Y −→ S over S which is a simultaneous resolution of X:
Y
h
/
/
X
S
C
We denote the top arrow as h.
0
For each such h, let Vh be Hcris
(S, R2 fcris,∗ (OY ))F =p . Then Vh ⊗OS/W is a subcrystal in free
sheaves of R2 fcris,∗ (OY ). Let Eh := (Vh ⊗ OS/W )⊥ ⊂ R2 fcris,∗ (OY ) and Q be the quadratic
form on Eh induced by the cup product. Then Q induces a symmetric bilinear form (|) on
Eh .
Let θij : Rj π∗ (ΩiY /S ) −→ Rj−1 π∗ (Ωi+1
Y /S ) be the Kodaira-Spencer maps associated to Y −→
S. Since Eh has weight 2, the iterated Kodaira-Spencer map is the composition
θ11 ◦ θ20 : π∗ (ωY /S ) −→ R2 π∗ (OY ) ⊗ Ω⊗2
S .
Recall a K3 surface over a finite field is ordinary if its height is 1.
Theorem 1.1. Given a primitively polarized family X −→ C of double-point K3 surfaces over
F̄p , p > 3 if there exists a resolution h : Y −→ X over S −→ C and m ∈ Z>0 such that
(1) the iterated Kodaira-Spencer map of Y /S is an isomorphism,
(2) Y /S is generically ordinary,
(3) dim Vh = 22 − 3m and there exists a degree m extension F of Q, unramified over p such
that
0
Hcris
(S, End(Eh , Q))F ⊗ Qp ∼
= F ⊗ Qp .
Then there exist
(a) a finite etale covering S̃ −→ S,
(b) a unique lifting S̃ 0 of S̃ to W such that S̃ 0 admits a lifting X 0 of XS̃ 0 as a polarized family of
double-point K3 surfaces.
Further, X 0 is unique up to an isomorphism.
1
1.2. Outline of the proof. Since F is unramified over p, F splits to product of fields after
f
a finite extension of Qp . There exists f > 0 such that Eh ∼
= ⊕m
i=1 Wi as F -isocrystals. Since
F preserves Q, each Wi admits a symmetric bilinear form (see Lemma 4.1). In particular,
Wi are self-dual F f -isocrystals.
Since Eh is generically ordinary, the generic Newton slopes of ⊕Wi is {0, f, · · · , f, 2f }.
Note each Wi is self-dual, and hence the generic slopes of Wi are one {0, f, 2f }, others
{f, f, f }. Assume W1 has slopes {0, f, 2f }. Then it is easy to prove (Proposition 4.2) F
induces a Frobenius on W1 .
Furthermore, since all the slopes are nonnegative, it is possible to choose F −crystal
models of W1 and ⊕i>1 Wi . The rank one filtration lies in W1 .
The isomorphic iterated Kodaira-Spencer map implies V1 corresponds to a versally deformed Barsotti-Tate group G over C̃. Then the result in [8] shows that C̃ admits a unique
deformation C̃ 0 to W over which the deformation of G is unobstructed.
The curve C̃ 0 admits a K3 crystal. By [4, Theorem 5.3], lifting the K3 crystal is equivalent
to lift the K3 surface. Therefore C̃ 0 admits a formal family of quasipolarized K3 surfaces.
Via the quasipolarization, we can produce a polarized family of double-point K3 surfaces
which can be algebraized.
2. M ODULI OF K3
In this section, we recall the moduli stack of K3 surfaces and some relative deformation
theory results. We follow the terminologies in [2].
2.1. moduli of K3. Let X be a K3 surface over an algebraically closed field k. Recall a
line bundle L over X is a polarization if it is ample and L is a quasipolarization if it is nef
and big.
Given a scheme S, a morphism f : K −→ S between algebraic spaces is a family of K3
surfaces if each geometric fiber is a K3 surface.
An element ξ ∈ Pic (K/S) is a polarization if for any geometric point s −→ S, ξs is a
polarization on Ks in the above sense. Similarly, one can define (primitive) quasipolarization for K/S.
If K −→ S is a polarized family of K3 surfaces, then there exists an etale cover T −→ S
such that KT −→ T is a morphism between projective schemes.
We can define the groupoid-value functor M2d on schemes by
M2d (S) = {f : K −→ S, ξ ∈ Pic (K/S)|K is a family of K3 surfaces over S
with primitive quasipolarization ξ, deg ξ = 2d}
dp
M2d (S) = {f : K −→ S, ξ ∈ Pic (K/S)|K is a family of double-point K3 surfaces over S
with primitive polarization ξ, deg ξ = 2d}
o
We similarly define the moduli functor M2d
of primitively polarized K3 surfaces. Since
dp
o
o
ampleness is an open condition, M2d ⊂ M2d is an open subfuntor. Both M2d
and M2d are
separated.
Theorem 2.2. [2, Proposition 2.1] The functors are Deligne-Mumford stacks of finite type over
1
].
Spec Z. The functor M2d,Z[ 1 ] is smooth over Spec Z[ 2d
2d
2
dp
Remark 2.3. In 1.1, the family X −→ C is equivalent to a morphism C ,→ M2d,F̄p .
2.4. Period map. Let V be a rational vector space of dimension 21, and ψ a non-degenerate
symmetric form on V of signature (2, 19). Let Ω be the period domain, that is,
Ω = {ω ∈ P(V )|ψ(ω, ω) = 0 and ψ(ω, ω̄) > 0}.
Let Sn be the Shimura variety over Q such that
Sn,C = SO(V, ψ)\SO(2, 19) × Ω/Kn .
o
−→ Sn is an open immersion over Q. Adding the
Then with level structure n, M2d,n
quasipolarized locus, M2d,n −→ Sn is etale and surjective.
2.5. K3 crystals. The definition of K3 crystals follows from 5.1 in [4].
Definition 2.6. Given a smooth scheme Z/k with absolute Frobenius σ : Z −→ Z, a K3
crystal over Z/W (k) consists of the following data
• a crystal E in locally free OZ/W −module of rank 22,
• a morphism F : E σ −→ E,
• a symmetric bilinear form (|) : S 2 E −→ OZ/W ,
• an isotropic local direct factor of rank one Fil −→ EZ ,
such that
(1) the form (|) defines an isomorphism E −→ E ∨ ,
(2) a morphism E −→ E σ so that F ◦ V = V ◦ F = p2 ,
(3) the Ann(σ ∗ (Fil)) is canonically isomorphic to ker FZ .
Since the Hodge-de Rham spectral sequence degenerates for K3 surfaces, the weight 2
crystalline cohomology space (or sheaf) is a K3 crystal.
Lemma 2.7. Eh is a K3 crystal.
Proof. Let S 0 be a lifting of S to W . Then we can realize R2 fcris,∗ (OY ) as a vector bundle
over S 0 with a connection and Frobenius action. Let V be the space of global sections
H 0 (S 0 , R2 fcris,∗ (OY )).
Firstly we prove Vh is self dual under Q. The nondegeneracy of Q induces an isomorphism between R2 fcris,∗ (OY ) and its dual, and hence it also induces an isomorphism
between V and V ∨ . As a subspace of V , Vh admits a canonical morphism to Vh∨ induced
by Q.
Let ξ be the primitive quasipolarization of Y /S. Then ξ ∈ Vh . Choose a point s ∈ S
such that ξs is an ample line bundle. Then for every x ∈ Vh , xs also corresponds to a line
bundle. Note (ξs , xs ) is just the intersection form over Ys . By Hodge Index Theorem, (|)s
is nondegenerate on Vh,s . Thus (|)Vh is nondegenerate and Vh is then self dual.
Since Q is nondegenerate on both of Vh ⊗ OS/W and R2 fcris,∗ (OY ), the quotient is also
self dual. In particular, the quotient is a locally free sheaf over S 0 . Then Eh is a crystal in
vector bundles with nondegenerate symmetric bilinear form, since it is isomorphic to the
quotient.
Analogue to Serre-Tate theory, we have an equivalence between deformation of K3
surfaces and the corresponding K3 crystal, cf. [4]. Given Zn a deformation of Z to Wn (k),
let (K3/Zn ) denote the category of K3 surfaces over Zn and (DK3/Zn ) be the category
3
whose objects are triple (K0 , En , α), where K0 is a K3 over Z, En is a K3 crystal over Zn and
α is an isomorphism between weight 2 crystalline cohomology sheaf associated to K0 and
En|Z .
Since any K3 surface over Zn induces a K3 crystal on Zn , we can produce a functor
(K3/Zn ) −→ (DK3/Zn ).
Proposition 2.8. [4, Theorem 5.3] The above functor is an equivalence.
3. E XAMPLE
We give an example of family of K3 surfaces satisfying 1.1. Zarhin classifies the Hodge
group associated to a VHS of K3 type in [9]. The real Hodge group has the form of
SO(2, m − 2) × SO(0, m)×d−1
and the real representation is the direct sum of the standard representations. This result
basically classifies the possible Shimura subvarieties in the moduli of K3.
Here we construct an example for m = 3. Given a totally real cubic extension F, let D be
a quaternion algebra over F which splits at only one real place. Let τ : D∗ −→ Aut(Lie D∗ )
be the conjugate representation of D∗ . Since τ (Gm ) is trivial, im τ is contained in sl(D∗ ).
Let
ρ := τ ⊗ det : D∗ −→ Aut(sl(D∗ )).
Then the restriction of scalars gives
Res(ρ) : Res(D∗ ) =: H −→ GL9 (Q).
It is easy to check Res(ρ) is an irreducible representation.
Let G = Gm .H der ⊂ H. Then G ⊗ R ∼
= R∗ (SL(2) × SU (2) × SU (2)) and
Res(ρ) ⊗ R : G ⊗ R −→ W1 ⊕ W2 ⊕ W3 .
Here Wi are three dimensional representations of SL(2) or SU (2). In particular, W1 is the
second symmetric product of the standard representation of SL(2).
Define the cocharacter χ : C∗ −→ G ⊗ R to be
cos θ sin θ
r(cos θ + i sin θ) −→ r(
× Id).
− sin θ cos θ
Under this cocharacter, the Hodge filtrations on W1 is (1, 1, 1), others (0, 3, 0).
The Killing form (, ) on Lie D∗ gives a F−invariant symmetric bilinear form on sl(D∗ )
which is also ρ−invariant up to the square of the determinant det2 . The trace map gives a
descent of the symmetric product to Q:
x, y 7→ Trace (α(x, y))
depending on α ∈ F∗ .
Base change to R, each of the descents implies a symmetric product on W1 of signature (2, 1). By strong approximation, we can adjust α such that the definite symmetric
products on Wi , i > 1 all have signature (0, 3). Thus Res(ρ) factors through an orthogonal
similitudes with signature (2, 19).
Using [5, Proposition 3.3], there exists a morphism (G, Q9 ) −→ (SO(V, ψ), V ). Since the
cocharacters are compatible, the proper Shimura curve M := Sh(G, ξ) is contained in the
period space of K3 surfaces.
4
Remark 3.1. We can slightly generalize above construction. Note the symmetric form
involving in the construction of M is TrF/Q ◦ (, ) which has discriminant disc F. Let eF :=
P
22−eF
},
p (ep −1) where ep is the ramification index. By [5, Proposition 3.3], if m < max{4,
3
⊕3m
then there exists an embedding (G, Q
) −→ (SO(V, ψ), V ).
Since End(ρ) = F, the sub-variation K3 Q−Hodge structure over M which is orthogonal
to Neron-Severi group has endomorphism ring F.
Since the period map is quasi-finite and étale, the inverse image of M is a union of
curves. We choose one connected component of the inverse image of M via period map
and still denote it as M .
The Shimura curve M has reflex field F. Let p be a prime where F splits. By Lefschetz
principle, the family of K3 surfaces g : A −→ M can descend to the Witt ring W (p), with
the endomorphism ring F. The special fiber f : Y −→ C is generically ordinary, due to
the splitting of F over p.
The Hodge filtration of V is the symmetric product of a variation of weight 1 Hodge
structures whose underlying local system is a standard SL(2)−representation. Therefore
using [3, Theorem 0.5], the iterated Kodaira-Spencer map is an isomorphism.
Due to the comparison theorem between crystalline and deRham cohomology,
0
dim Hcris
(S, R2 fcris,∗ (OY ))F =p ⊗ Qp ≤ dimC H 0 (M, R2 g∗ (Ω.A/M )) = dim V G = 13.
Meanwhile since the specialization of Neron-Severi group is injective, we have
0
dim Hcris
(C, R2 fcris,∗ (OY ))F =p ⊗ Qp = 13.
The family Y −→ C is defined over a finite field Fq . As objects of the Tannakian category of isocrystals, Wi corresponds to P GL(2)−representation and hence we can view
the Frobenius F as a morphism between P GL(2)×3 −representations. By standard representation theory, the Frobenius F permutes the factor Wi and hence commutes with the
endomorphism ring F (see [6, Lemma 5.3]). So the reduction satisfies the conditions of
1.1.
4. D ECOMPOSITION OF Eh
Since F is unramified over p, there exists a positive integer f such that F ⊗ Qpf ∼
.
= Qm
pf
f
Enlarging f if necessary, as F −isocrystals,
(1)
Eh ∼
= W1 ⊕ W2 · · · ⊕ Wm
where each Wi is irreducible of rank 3.
Lemma 4.1. Each Wi admits a symmetric bilinear form and the isomorphism 1 is compatible with
the symmetric forms.
Proof. Let pi = (0, · · · , 0, 1, 0, · · · , 0) be the idemponents in F ⊗ Qpf . Since F preserves the
symmetric product,
⊕pi
E
E
∨
o
⊕p∨
i
5
/
E
E∨
is commutative. Therefore as a subcrystal of Eh , Wi admits a symmetric product and Q is
the direct sum of the symmetric products on Wi .
Since Eh is generically ordinary, the generic Newton slopes of ⊕Wi is {0, f, · · · , f, 2f }.
Note each Wi is self-dual, and hence the generic slopes of Wi are one {0, f, 2f }, others
{f, f, f }. Assume W1 has slopes {0, f, 2f }.
Proposition 4.2. As a morphism between F f −isocrystals, the Frobenius F on ⊕Wi induced by
E decomposes as F = F1 ⊕ F 0 where F1 : W1σ −→ W1 and F 0 : ⊕i>1 Wiσ −→ ⊕i>1 Wi .
Proof. Let i1 and πj be the inclusion of W1 and projector onto Wj respectively. If the
composition g := πj ◦ F ◦ i1 is nonzero, then due to the irreducibility of W1σ and Wj , g
induces an isomorphism between W1σ and Wj . Comparing the Newton slopes, j has to
be 1, that is, F preserves W1 . Note F is an isomorphism between isocrystals, so F can be
decomposed to a direct sum F1 : W1σ −→ W1 and F 0 : ⊕i>1 Wiσ −→ ⊕i>1 Wi .
By 4.2, as an F −isocrystal, W1 has generic slopes {0, 1, 2}. Since all slopes are nonnegative, we can choose an F −crystal model for each Wi which we still denote as Wi .
Let F1 and V1 be the Frobenius and Vershiebung of W1 . Then Eh and ⊕Wi are isogenous
F −crystals.
5. T HE PROOF OF 1.1
Proposition 5.1. The filtration of Eh as a K3 crystal is induced from the filtration of W1 .
Proof. Let φ : ⊕Wi −→ Eh be the isogeny which is nonzero modulo p. Then we have
/
⊕Wiσ
F1 ⊕F 0
⊕Wi
φ
/
Ehσ
F
Eh .
On S, the image of FS : Ehσ S −→ Eh S has rank 1 and come from the slope zero part.
The image of F1 S has rank 1 as well. Therefore φ induces an isomorphism between the
images of F1 and F . Comparing the kernels, φ induces a morphism between Ann(Filσ1 )
and Ann(Filσ ) and thus φ induces an isomorphism between Fil1 and Fil.
The following proposition is proved in [7, Section 4].
Proposition 5.2. There exists a finite etale covering S̃ −→ S such that the pullback F −crystal
W1 S̃ is the symmetric product of a rank 2 F −crystal V. The symmetric bilinear form on W1 S̃ is
the self product of an alternating bilinear form on V.
Note on W1 S̃ , F1 ◦ V1 = V1 ◦ F1 = p2 . Since W1 S̃ ∼
= S 2 V1 , the composition of Frobenius
and Verschiebung of V is p. So V is a Dieudonne crystal over cris(S/W ).
Proposition 5.3. The F -crystal V corresponds to a versally deformed Barsotti-Tate group over S.
Proof. Using [1, Main Theorem 1], it corresponds to a Barsotti-Tate group, or p−divisible
group. The versal deformation follows from the isomorphic iterated Kodaira-Spencer
map.
6
By [8, main theorem], there exists a unique deformation S̃ 0 of S̃ which admits a deformation of the Barsotti-Tate group, which further admits the deformation of the filtration
of the K3 crystal Eh . By 2.8, we have a formal quasipolarized family {Ỹn } of K3 over {S̃n }.
Proposition 5.4. The quasipolarization on Ỹ −→ S̃ lifts to Ỹn −→ S̃n .
Proof. Consider the sequence
0 −→ (1 + pOỸn )∗ −→ OỸ∗n −→ OỸ∗ −→ 0.
The long exact sequence gives · · · −→ R1 f˜n,∗ ((1+pOỸn )∗ ) −→ R1 f˜n,∗ (OỸ∗n ) −→ R1 f˜∗ (OỸ∗ ) −→
R2 f˜n,∗ ((1 + pO )∗ ) · · · . Note log : (1 + pO )∗ −→ O is an isomorphism. Thus
Ỹn
Ỹn
Ỹn
R f˜n,∗ ((1 + pOỸn ) ) ∼
= R f˜n,∗ (OỸn ) = 0.
1
So
∗
1
0 −→ H 0 (S̃n , R1 f˜n,∗ (OỸ∗n )) −→ H 0 (S̃, R1 f˜∗ (OỸ∗ )) −→ H 0 (S̃n , R2 f˜n,∗ (OỸn ))
is exact.
Since the iterated Kodaira-Spencer map is an isomorphism, R2 f∗ (OY ) has negative degree. Thus the third term in above sequence is also zero. All line bundles lift from Ỹ −→ S̃
to Ỹn −→ S̃n .
Using sections of a sufficiently high power of the quasipolarization (3 is enough), we
can produce a family of polarized double-point K3 surfaces from a quasipolarized family
of smooth K3 surfaces. Since the quasipolarization descends, we actually have a polarized
family of double-point K3. Therefore the formal family {Ỹn −→ S̃n } induces a surjective
map to a polarized formal family of double-point K3 surfaces {X̃n } over {S̃n } via the
above result and the reduced fiber is just X −→ S.
By Grothendieck algebraization theorem, the polarized formal family {X̃n −→ S̃n } of
double-point K3 surfaces can be algebraized to a virtual family X̃ 0 −→ S̃ 0 over W . That
concludes the proof of 1.1.
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