Download - A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field
A Crystalline Criterion for Good Reduction on
Semi-stable K3-Surfaces over a p-Adic Field
Thesis Advisor: Prof. Adrian Iovita
J. Rogelio Perez Buendıa
Concordia University
January 10 2014
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Objective:
To give a criterion for the reduction of an algebraic K3-surface over
a p-adic field in terms of its p-adic etale cohomology.
The desired Criterion
A K3-surface over a Local field with semistable reduction has good
reduction if and only if its second etale cohomology group is crystalline.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Objective:
To give a criterion for the reduction of an algebraic K3-surface over
a p-adic field in terms of its p-adic etale cohomology.
The desired Criterion
A K3-surface over a Local field with semistable reduction has good
reduction if and only if its second etale cohomology group is crystalline.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Notations:
Fix a prime number p and let Qp the field of p-adic numbers. Consider:
1 k be a perfect field of characteristic p.
2 k be a fix algebraic closure.
3 W := W (k) the ring of Witt vectors with coefficients in k
4 K0 = Frac(W ) its field of fractions. It is an unramified extension of
Qp.
5 K = K0 if k is ifninite or K be a finite extension of K0 if
[k : Fp] <∞.
6 OK the ring of integers of K .
7 π be a (fixed) uniformizer. So mK = πOK and
k = OK/πOK = W /pW .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
K3-surfaces
Definition
A K3-surface over K is a smooth proper surface XK −→ Spec(K) such
that
1 q := H1(XK ,OXK ) = 0 and
2 ωXK ' OXK . Equivalently KX = 0.
where ωXK stands for the canonical sheaf and KX its canonical divisor.
This definition is independent of the field K , so we can consider this
definition for K = C and we get complex algebraic K3-surfaces.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Examples:
K3-surfaces were named by Andre Weil in honour of three algebraic
geometers, Kummer, Kahler and Kodaira, and the mountain K2 in
Kashmir.
Figure: A quartic in P3K given by x2y 2 + y 2z2 + z2x2
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Examples
Let S be a non-singular sixtic curve in P2k where k is a field and consider
a double cover i.e., a finite generically etale morphism, π : X → P2k which
is ramified along S . Then X is a K3 surface.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Example
Complete intersections: Let X be a smooth surface which is a complete
intersection of n hypersurfaces of degree d1, . . . , dn in Pn+2 over a field k.
The adjunction formula shows that Ω2X/k∼= OX (d1, . . . , dn − n− 3). So a
necessary condition for X to be a K3 surface is d1 + . . .+ dn = n + 3.
The first possibilities are:
n = 1 d1 = 4
n = 2 d1 = 2, d2 = 3
n = 3 d1 = d2 = d3 = 2.
For a complete intersection M of dimension n one has that
H i (M,OM(m)) = 0 for all m ∈ Z and 1 ≤ i ≤ n − 1. Hence in those
three cases we have H1(X ,OX ) = 0 and therefore X is a K3 surface.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Example
Let A be an abelian surface over a field k of characteristic different from
2. Let A[2] be the kernel of the multiplication by-2-map, let π : A→ A
be the blow up of A[2] and let E be the exceptional divisor. The
automorphism [−1]A lifts to an involution [−1]A on A. Let X be the
quotient variety of A by the group of automorphisms idA, [−1]A and
denote by ι : A→ X the quotient morphism. It is a finite map of degree
2. We have the following diagram of morphisms over k. The variety X is
a K3 surface and it is called the Kummer surface associated to A.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Semistable K3-surfaces
Definition
XK has semistable reduction if it has a semi-stable model:
XK//
X
Spec(K) // Spec(OK)
that is a proper (flat) model X → Spec(OK) whose special fibre X is
smooth over k or etale locally a normal crossing divisor.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Good reduction
If the special fibre X → Spec(k) of such a model X is smooth, then we
say that XK has good reduction.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
p-adic representations
Let GK := Gal(K,K) be the absolute Galois group of K .
Definition
A p-adic representation V of GK is a finite dimensional Qp-vector space
with a continuous action of GK .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Examples of p-adic representations
The main example:
Main example
The etale cohomology of a K3-surface. Indeed, in general we have that if
X is a proper and smooth variety over K , then
H iet(XK ,Qp)
is a p-adic representation of GK .
Tate modules of abelian varieties.
The r -Tate twists of Qp, Qp(r).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Examples of p-adic representations
The main example:
Main example
The etale cohomology of a K3-surface. Indeed, in general we have that if
X is a proper and smooth variety over K , then
H iet(XK ,Qp)
is a p-adic representation of GK .
Tate modules of abelian varieties.
The r -Tate twists of Qp, Qp(r).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Ring of periods
In order to study the p-adic representations, Fontaine defines what we
know as ring of periods, which are topological Qp-algebras B (or B•),
with a continuous linear action of GK and some additional structures
which are compatible with the action of GK (for example the monodromy
operator N, Frobenius, filtrations).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Poincare duality
For a smooth and projective variety X of dimension n over the complex
numbers C, we have the Betti cohomlogy H i (X (C),Z).
By Poincare duality we have that:
H2n−i (X ,C) ' H i (X (C),C).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Complex periods
Also we have a perfect pairing given by the periods:
H idR(X (C)/C)× H2n−i (X (C),C) −→ C
(ω, λ) 7→∫λ
ω.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Comparison isomorphism
We have a natural comparison isomorphism:
H idR(X (C)/C) ' H i (X (C),C).
We remark that in order to have this isomorphism it is very important to
have coefficients in C (for example, this is not an isomorphism over Q).
In this sense C is a ring of periods (it contains all the periods∫λω).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The p-adic case of Cp
We denote by
Cp = ˆK
the p-adic completion of K .
We want analogous comparison isomorphisms in the p-adic cases.
However the situation is not as easy as in the complex case, mainly
because Cp does not have enough periods.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Fontaine’s Idea
The original idea of Fontaine was to construct these ring of periods, in
order to be able to have analogous comparison isomorphism between the
different cohomologies in p-adic settings.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
B•
Examples of this rings are
1 BHT :=⊕
q∈Z Cp(q) B Hodge-Tate. is a graded C p-algebra with
GK -action rescts gradings and BGK
HT = K .
2 BdR : B de Rham is a complete discrete valuation field over K with
residue field Cp. It contains K (but not Cp). It has an action of GK
and a filtration by its valuation, and its graded quotient
gr iBdR = Cp(i)
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Bcris
Bcris : B crys is an algebra over K0 and a GK -stable subring of BdR . It
contains K0 but not K . We have a filtration coming from form BdR , a
σ-semilinear injective GK -equivariant endomorphism φ (Frobenius
endomorphism). BGK
cis = K0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Bst
Bst : B semistable is an algebra over K0 and has a GK -action. It contains
Bcris and KO but not K The Frobenius of Bcris extends to Bst and has a
Bcris -derivation
N : Bst −→ Bst . Nφ = pφN
and
BGKst = K0, BN=0
st = Bcris .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Dieudonne Modules
These rings are such that the BGK -modules
DB(V ) := (B ⊗Qp V )GK
give us (or expected to give us) good invariants for V . For example
comparison isomorphisms for the p-adic etale cohomology and de Rham
cohomologie or crystalline cohomology or nice criterion for good
reduction of varieties.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
B-admissible
Let L = BGK .
Definition
A p-adic representation V is B-admissible, if
dimL DB(V ) = dimQp V .
Definition
A p-adic representation V , is crystalline (semistable, Hodge-Tate,
semistable) if V is Bcris-admissible (B•-admissible).
B-admissibility translates to isomorphisms which are analogous to the
comparison isomorphisms in the complex case.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Fontaine has defined several subcategories of the category of all p-adic
representations, denoted by RepGK .
This categories are formed by the property of being B-admissible objects.
So for any of the period rings B we have a subcategory of the category of
p-adic representations denoted by RepB . These categories satisfy proper
contention relations as follows:
RepBcris⊂ RepBst ⊂ RepBdR
⊂ RepBHT ⊂ RepGK .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
C•-conjectures
Let XK be a proper smooth variety over K .
CHT : The Hodge-Tate conjecture. There exists a canonical
iomorphism, which is compatible with the Galois action.
Cp ⊗Qp Hmet (XK ,Qp) '
⊕0≤i≤m
Cp(−i)⊗K Hm−i (XK ,ΩiXK/K
).
CdR : The de Rham conjecture. There exist a conaonical
isomorphism, which is compatible with Galois action and filtrations.
BdR ⊗Qp Hmet (XK ,Qp) ' BdR ⊗K Hm
dR(XK/K ).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
C•-conjectures
Let XK be a proper smooth variety over K .
CHT : The Hodge-Tate conjecture. There exists a canonical
iomorphism, which is compatible with the Galois action.
Cp ⊗Qp Hmet (XK ,Qp) '
⊕0≤i≤m
Cp(−i)⊗K Hm−i (XK ,ΩiXK/K
).
CdR : The de Rham conjecture. There exist a conaonical
isomorphism, which is compatible with Galois action and filtrations.
BdR ⊗Qp Hmet (XK ,Qp) ' BdR ⊗K Hm
dR(XK/K ).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
C•-conjectures
Ccris : The Crystalline conjecture. Let X be a proper smooth model
of XK over OK . Let X be the special fibre of X . There exist a
canonical isomorphism which is compatible with the Galois action,
and Frobenius endomorphism.
Bcris ⊗Qp Hmet (XK ,Qp) ' Bcris ⊗W Hm
crys(X/W )
Barthelo-Ogus isomorphism:
K ⊗W Hmcrys(X/W ) ' Hm
dR(XK/K ).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
C•-conjectures
Ccris : The Crystalline conjecture. Let X be a proper smooth model
of XK over OK . Let X be the special fibre of X . There exist a
canonical isomorphism which is compatible with the Galois action,
and Frobenius endomorphism.
Bcris ⊗Qp Hmet (XK ,Qp) ' Bcris ⊗W Hm
crys(X/W )
Barthelo-Ogus isomorphism:
K ⊗W Hmcrys(X/W ) ' Hm
dR(XK/K ).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Cst conjecture
1 Cst : The semistable conjecture: Let X be a proper semistable model
of XK over OK . Let Y be the special fiber of X , and MY be a
naural log-structure on Y . There is a canonical isomorphsim,
compatible with Galois action, Frobenius and operator N.
Bst ⊗Qp Hmet (XK ,Qp) ' Bst ⊗W Hm
log−crys((Y ,MY ), (W ,O∗))
2 Hydo-Kato isomorphism
K ⊗W Hmlog−crys((Y ,MY ), (W ,O∗)) ' HdR(XK/K )
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Cst conjecture
1 Cst : The semistable conjecture: Let X be a proper semistable model
of XK over OK . Let Y be the special fiber of X , and MY be a
naural log-structure on Y . There is a canonical isomorphsim,
compatible with Galois action, Frobenius and operator N.
Bst ⊗Qp Hmet (XK ,Qp) ' Bst ⊗W Hm
log−crys((Y ,MY ), (W ,O∗))
2 Hydo-Kato isomorphism
K ⊗W Hmlog−crys((Y ,MY ), (W ,O∗)) ' HdR(XK/K )
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
History
For X/K a proper smooth variety over K .
Fontaine: Proved that Tate modules V = TpA⊗Zp Qp of abelian varieties A are de Rham
and that DdR(V ) ' (HdR(X/K))v .
Fontaine and Messing: Proved the comparison theorem for H iet(XK ,Qp) for i ≤ p − 1 and
K/Qp finite unramified.
Kato, Hydo Tsuji: Extended these results to finally prove that one can recover H idR(X/K)
from the data of V = H iet(XK ,Qp) as a p-adic representation.
Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V
is crystalline if V has good reduction and BdR otherwise.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
History
For X/K a proper smooth variety over K .
Fontaine: Proved that Tate modules V = TpA⊗Zp Qp of abelian varieties A are de Rham
and that DdR(V ) ' (HdR(X/K))v .
Fontaine and Messing: Proved the comparison theorem for H iet(XK ,Qp) for i ≤ p − 1 and
K/Qp finite unramified.
Kato, Hydo Tsuji: Extended these results to finally prove that one can recover H idR(X/K)
from the data of V = H iet(XK ,Qp) as a p-adic representation.
Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V
is crystalline if V has good reduction and BdR otherwise.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
History
For X/K a proper smooth variety over K .
Fontaine: Proved that Tate modules V = TpA⊗Zp Qp of abelian varieties A are de Rham
and that DdR(V ) ' (HdR(X/K))v .
Fontaine and Messing: Proved the comparison theorem for H iet(XK ,Qp) for i ≤ p − 1 and
K/Qp finite unramified.
Kato, Hydo Tsuji: Extended these results to finally prove that one can recover H idR(X/K)
from the data of V = H iet(XK ,Qp) as a p-adic representation.
Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V
is crystalline if V has good reduction and BdR otherwise.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
History
For X/K a proper smooth variety over K .
Fontaine: Proved that Tate modules V = TpA⊗Zp Qp of abelian varieties A are de Rham
and that DdR(V ) ' (HdR(X/K))v .
Fontaine and Messing: Proved the comparison theorem for H iet(XK ,Qp) for i ≤ p − 1 and
K/Qp finite unramified.
Kato, Hydo Tsuji: Extended these results to finally prove that one can recover H idR(X/K)
from the data of V = H iet(XK ,Qp) as a p-adic representation.
Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V
is crystalline if V has good reduction and BdR otherwise.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
For abelian varieties
For Abelian varieties, Bcris and Bst are exactly what it takes to decide:
whether A has good reduction or semistable reduction.
Crystalline criterion for abelian varieties:
Coleman-Iovita Breuil: A has good reduction if and only if V is
crystalline. A has semistable reduction if and only if V is semistable.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
My Thesis problem
Crystalline criterion for K3 surfacess:
Let X be a K3 surface over a p-adic field K with semistable reduction. X
has good reduction (X → Spec(k) is smooth) if and only if
V := H2et(XK ,Qp)
is Crystalline (Bcris-admissible).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
One side is Falting’s result:
Remember that
RepBcris⊂ RepBst
Since X has semistable reduction, then V is Bst-admissible. If X has
good reduction, then by Falting’s result, V is Bcris-admissible.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Results of Y. Matsumoto
Theorem
Let K be a local field with residue characteristic p 6= 2 and X a Kummer
surface over K. Assume that X has at least one K-rational point. If
H2et(XK ,Qp) is crystalline, then XK ′ has good reduction for some finite
unramified extension K ′/K.
Theorem
Let K be a local field with residue characteristic p 6= 2, 3, and Y a K3
surface over K with Shioda-Inose structure of product type. If
H2et(YK ,Qp) is crystalline, then YK ′ has good reduction for some finite
extension K ′/K of ramification index 1, 2, 3, 4 or 6.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Main tool
The main tooll is what we call p-adic logarithmic degenerations of a
K3-surface. These will be p-adic analogous of degeneration of K3
surfaces over the complex numbers constructed via degenerations.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Complex degenerations of K3-surfaces
Definition
Over the complex numbers C, a semistable degeneration of a
K3-surface X is a proper flat and surjective morphism
π : X (C)→ ∆
over the open disc, whose general fibre Xt = π−1(t), for t 6= 0 is a
smooth K3-surface and X0 is reduced with normal crossings.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Modification of a degeneration
Definition
A modification of π : X (C)→ ∆ is a degeneration of surfaces
π′ : X (C)→ ∆ such that there exists a birational map
φ : X (C)→ X ′(C) given an isomorphism form
(X (C)− X0) −→ (X ′(C)− X0) and such that the diagram:
X (C)φ //
π""
X ′(C)
π′||∆
commutes.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Kulikov degenerations
We have the following theorem:
Theorem (Kulikov, Persson, Pinkham)
Let π : X (C)→ ∆ be a semistable degeneration of a K3-surface, then
there exists a modification π′ : X ′(C)→ ∆ such that the canonical
divisor of the total space X ′(C) is trivial.
A degeneration with trivial canonical divisor is called a good
degeneration or a Kulikov degeneration.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Kulikov degenerations
We have the following theorem:
Theorem (Kulikov, Persson, Pinkham)
Let π : X (C)→ ∆ be a semistable degeneration of a K3-surface, then
there exists a modification π′ : X ′(C)→ ∆ such that the canonical
divisor of the total space X ′(C) is trivial.
A degeneration with trivial canonical divisor is called a good
degeneration or a Kulikov degeneration.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Kulikov criterion
Theorem
Let X (C)→ ∆ be a good degeneration of a K3-surface. The degenerate
fibre X0 is one of the following three types:
I. X0 is a nonsingular K3 surface.
II. X0 = ∪ni=1Vi where the Vi are rational surfaces and V2, . . . ,Vn−1 are
elliptic ruled surfaces.
III. X0 = ∪ni=1Vi where all the Vi are rational surfaces.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Kulikov criterion
Theorem
Let X (C)→ ∆ be a good degeneration of a K3-surface. The degenerate
fibre X0 is one of the following three types:
I. X0 is a nonsingular K3 surface.
II. X0 = ∪ni=1Vi where the Vi are rational surfaces and V2, . . . ,Vn−1 are
elliptic ruled surfaces.
III. X0 = ∪ni=1Vi where all the Vi are rational surfaces.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
In therms of monodromy
Moreover, the three cases can be distinguished from each other by means
of the monodromy T acting on H2(Xt ,Z):
For Type I we have N := ln T = 0 that is T = id .
For Type II, N 6= 0 but N2 = 0.
For Type III, N2 6= 0 but N3 = 0
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
In therms of monodromy
Moreover, the three cases can be distinguished from each other by means
of the monodromy T acting on H2(Xt ,Z):
For Type I we have N := ln T = 0 that is T = id .
For Type II, N 6= 0 but N2 = 0.
For Type III, N2 6= 0 but N3 = 0
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The plan:
We need:
A p-adic semistable degeneration of our K3-surface.
A monodromy operator N on the log-crystalline cohomology.
We relate it with the monodromy operator on
Dst(H2et(XK ,Qp)) ' H2
log−cris(XK/W ) appearing on Fontain’s
theory.
By p-adic Hodge theory if H2et(XK ,Qp) is crystalline then N = 0.
Indeed BN=0st = Bcris and Dst(V )N=0 = Dcris(V ) , so if a p-adic
representation is crystalline we most have N = 0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The plan:
We need:
A p-adic semistable degeneration of our K3-surface.
A monodromy operator N on the log-crystalline cohomology.
We relate it with the monodromy operator on
Dst(H2et(XK ,Qp)) ' H2
log−cris(XK/W ) appearing on Fontain’s
theory.
By p-adic Hodge theory if H2et(XK ,Qp) is crystalline then N = 0.
Indeed BN=0st = Bcris and Dst(V )N=0 = Dcris(V ) , so if a p-adic
representation is crystalline we most have N = 0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The plan:
We need:
A p-adic semistable degeneration of our K3-surface.
A monodromy operator N on the log-crystalline cohomology.
We relate it with the monodromy operator on
Dst(H2et(XK ,Qp)) ' H2
log−cris(XK/W ) appearing on Fontain’s
theory.
By p-adic Hodge theory if H2et(XK ,Qp) is crystalline then N = 0.
Indeed BN=0st = Bcris and Dst(V )N=0 = Dcris(V ) , so if a p-adic
representation is crystalline we most have N = 0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The plan:
We need:
A p-adic semistable degeneration of our K3-surface.
A monodromy operator N on the log-crystalline cohomology.
We relate it with the monodromy operator on
Dst(H2et(XK ,Qp)) ' H2
log−cris(XK/W ) appearing on Fontain’s
theory.
By p-adic Hodge theory if H2et(XK ,Qp) is crystalline then N = 0.
Indeed BN=0st = Bcris and Dst(V )N=0 = Dcris(V ) , so if a p-adic
representation is crystalline we most have N = 0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Finally we base change to the complex numbers and use Kulikov’s
classification theorem to deduce that our crystalline K3-surface has good
reduction.
Here we use the Deligne’s work on the Monodromy expressed as the
residue at zero of the GM-conexion.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Logarithmic geometry is concerned with a method of finding and
using “hide smoothness” in singular varieties.
Let X be a nonsingular irreducible complex variety, S a smooth
curve with a point s and f : X → S a dominant morphism smooth
away from s, the fiber Xs := f −1(s) = Y1 ∪ · · · ∪ Yn reduced simple
normal crossing divisor.
ΩX/S = ΩX/f ∗ΩS fails to be locally free at the singular points of f .
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Consider ΩX/S(log(Xs)) the sheaf of differentials with at most
logarithmic poles along the Yi , and similarly ΩS(log(s)), there is an
injective sheaf homomorphism
f ∗ΩX (log(Xs)) −→ ΩS(log(s))
and the quotient sheaf ΩX (log(XS))/f ∗ΩX (log(Xs)) is locally free.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
pre-log.st
Definition
1 Let X be a scheme. A pre-log structure on X , is a sheaf of
monoids MX together with a morphism of sheaves of monoids:
α : MX −→ OX , called the structure morphism.
2 A pre-log structure is called a log structure (log.st for short) if
α−1(O∗X ) ' O∗X via α.
3 The pair (X ,MX ) is called a log scheme and it will be denoted by
X×.
4 Morphisms are morphisms of sheaves which are compatible with the
structure morphism.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Induced log.st
We have the forgetful functor i from the category of log.st of X to the
category of pre-log.st of X by sending a log.st M in X to itself considered
as a pre-log.st i(M).
Vice-versa given a pre-log.st we can construct a log.st M ls out of it in
such a way that ( )ls is left adjoint of i , hence M ls is universal.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Inverse image log.st
Definition
Let f : X → Y be a morphism of schemes. Given a log.st MY on Y we
can define a log.st on X , called the inverse image of MY , to be the log
structure associated to the pre-log.st
f −1(MY )→ f −1(OY )→ OX .
This is denoted by f ∗(MY ).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Morphisms of log-schemes
Definition
By a morphism of log-schemes X ∗ −→ Y ∗ we understand a morphism of
the underlying schemes f : X → Y and a morphism f # : f ∗MY → MX of
log.st on X .
We denote by LSch the category of log.schemes.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
One of the main examples of interest for us is the following:
Example
Let X be a regular scheme (we can take for example a K3-surface over K
or a proper model of it). Let D be a divisor of X . We can define a log.st
M on X associated to the divisor D as
M(U) :=
g ∈ OX (U) : g |U\D ∈ O∗X (U \ D)
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
log.st to algebraizable formal schemes
Let X be a scheme and X is a formal completion of X along a closed
subscheme Y , then we have a morphism of ringed spaces:
Xφ−→ X
for which φ is the inclusion Y → X on topological spaces, and on
sheaves, it is the natural projection
OX −→ lim←−OX/I n = OX
where I is the sheaf of ideals defined by the closed immersion Y → X .
If we have a log.st on X , say M, we can give a log structure on X by
taking the inverse image of the log structure M so that X becomes a
log-formal scheme:
(X , φ∗M).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
For K3-surfaces
1 When X is a proper model of a K3-surface XK , we have that the
special fibre X of X is a closed divisor with normal crossings.
This divisor induces a log.st on X . We denote by X× the log-formal
scheme obtained as in the previews paragraph; that is, by completing
X along X and giving to it the inverse image log.st of X induced by
X .
2 Notice that we have an inclusion of ringed spaces X → X×. We
denote by X× the log-scheme obtained by giving to X the inverse
image log.st of X×.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
p-adic degeneration
Definition
A p-adic degeneration of a K3-surface with semistalbe reduction is a proper, flat morphism of
schemes X −→ Spec(W[[t]]) with geometrically connected fibres, such that:
1 We have an isomorphism of the semistable model X of XK with the fibre Xπ of
X→ Spec(W[[t]]) induced by the ring homomorphism: W [[t]]→ OK ; t −→ π.
2 We have an isomorphism (compatible with the previous one) of the special fibre X of the
semistable model of XK with the fibre X0 induced by the projection
W [[t]]→ W [[t]]/(p, t) ' k = OK/πOk .
3 X→ Spec(W[[t]]) is smooth in the complement of X that is
(X− X ) −→ (Spec(W[[t]])− Spec(k))
is smooth.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
In a diagram
Then we have commutative Cartesian diagrams:
X //
X //
X
Spec(k) // Spec(OK) // Spec(W[[t]])
.
Note that Spec(W[[t]]) is the analogous, in p-adic settings, of the open
unit disc ∆ in the complex numbers, and so we call Spec(W[[t]]) the
p-adic unit disc denoted by D. Then X is a family of surfaces
parametrized by the p-adic unit disc D and removing the special fibre X
smooth over D∗ = D − Spec(k).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
rig functor
1 Let Y be the fibre of f at t = 0, that is the fibre induced by the
morphism
W [[t]]→W ; t 7→ 0.
This is a scheme over Spec(W) whose special fibre is again X .
Moreover Y is a normal crossing divisor (but now in characteristic
zero).
2 Call X = (X×)rig, D = (D×)rig, and f = (f ×)rig the rigid analytic
spaces over K0.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
rigid version of the degeneration
Lemma
Under the previous settings we have:
1 X −→ Spec(K0) is smooth
2 Y := f −1(0) = (Y×)rig is a semistable surface over K0.
3 f |X∗ : X ∗ := (X − Y) −→ D∗ := (D − 0) is smooth.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Complex of relative logarithmic differentials
Consider the complex of sheaves K ·X/D induced by the relative
logarithmic differential:
OXd1X/D−−−→ OX ⊗X/K0
Ω1X/D(log(Y))
d2X/D−−−→ OX ⊗X/K0
Ω2X/D(log(Y))
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
The connection
Denote by Hi the i-th logarithmic relative de Rham cohomology group of
X/D with coefficients in OX , i.e, it is the sheaf Rf∗(K ·X/D). For every i ,
Hi is a free OD-module with an integrable, regular-singular connection
∇i : Hi −→ Hi ⊗OD Ω1D/K0
(log(0)).
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
Monodromy
If s is a point in D, let His be the fibre of H1 at s. We now define the
monodromy Ni as the residue at 0 of this connection. That is
Ni = res0(∇i ).
In our case the only important value is for i = 2, so we define the
monodromy as N := N2.
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
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J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
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J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field