A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field
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Transcript of A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field
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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
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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
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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
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Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
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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
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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
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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
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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
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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
![Page 10: A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field](https://reader033.fdocuments.us/reader033/viewer/2022052913/55d0bac8bb61ebcf718b4782/html5/thumbnails/10.jpg)
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
![Page 11: A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field](https://reader033.fdocuments.us/reader033/viewer/2022052913/55d0bac8bb61ebcf718b4782/html5/thumbnails/11.jpg)
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
![Page 12: A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field](https://reader033.fdocuments.us/reader033/viewer/2022052913/55d0bac8bb61ebcf718b4782/html5/thumbnails/12.jpg)
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.
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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
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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.
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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.
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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.
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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.
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Good reduction
If the special fibre X → Spec(k) of such a model X is smooth, then we
say that XK has good reduction.
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Layout
J. Rogelio Perez Buendıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a p-Adic Field
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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 .
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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).
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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).
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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).
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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).
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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.
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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).
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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.
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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.
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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)
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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.
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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 .
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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.
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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.
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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 .
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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 ).
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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 ).
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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 ).
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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 ).
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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 )
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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 )
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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Layout
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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.
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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.
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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.
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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.
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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.
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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 .
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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.
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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.
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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.
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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 ).
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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.
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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)
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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).
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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×.
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Layout
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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.
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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).
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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.
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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.
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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))
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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)).
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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.
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