Semistable Reduction Research Articles

ℓ-Independence for a system of motivic representations

Let $X$ be an smooth projective algebraic variety over a number field $F \subset \mathbb{C}$. Suppose that the Absolute Hodge(AH) motive $M:= h^i(X)$ is contained in the Tannakian category generated by the AH motives of abelian varieties. For every prime number $\ell$ the Galois group $\Gamma_F:= Gal(\bar{F}/F)$ acts on $H_\ell(M)$, the $\ell $-adic realization of $M$. Over a finite extension of $F$ this action factorizes as $\rho_{M,\ell}:\Gamma_F\rightarrow G_M(\ql)$, where $G_M$ is the Mumford-Tate group of $M$. Fix a valuation $v$ of $F$ and suppose $v(\ell)=0 $. The restriction $ \rho_{M,\ell} \vert _{\Gamma_{F_v}}$ defines a representation ${}'W_v \rightarrow G_{M/\ql}$ of the Weil-Deligne group of $F_v$. J-P Serre and J-M Fontaine (independently) have made conjectures that indicates that ${}'W_v \rightarrow G_{M/\ql}$ should be defined over $\mathbb{Q}$ for $\ell$ fixed and that these representations form a compatible system for variable $\ell $. Under certain additional hypothesis , we answer these questions in affirmative, when $X$ has good reduction or Semi-Stable reduction at $v$.

On the integrality of modular symbols and Kato's Euler system for elliptic curves

Let E/\mathbb Q be an elliptic curve. We investigate the denominator of the modular symbols attached to E . We show that one can change the curve in its isogeny class to make these denominators coprime to any given odd prime of semi-stable reduction. This has applications to the integrality of Kato's Euler system and the main conjecture in Iwasawa theory for elliptic curves.

Decomposition of Direct Images of Logarithmic Differentials

The purpose of this note is to give a short proof of a theorem of Koll\'ar that the derived direct image of the canonical sheaf splits into a sum of its cohomology sheaves. This is deduced from a stronger decomposition theorem for direct images of sheaves of logarithmic differentials, together with the weak semistable reduction theorem.

Semi-stable reduction implies minimality of the resultant

For a dynamical system on Pn over a number field or a function field, we show that semi-stable reduction implies the minimality of the resultant. We use this to show that every such dynamical system over a number field admits a globally minimal presentation.

Curves which do not Become Semi-Stable After any Solvable Extension

We show that there is a field F complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension K\vert F such that there is no solvable Galois extension L\vert F such that the extension KL\vert K is unramified, where KL is the composite of K and L . As an application we deduce that that there is a field F as above and there is a smooth, projective, geometrically irreducible curve over F which does not acquire semi-stable reduction over any solvable extension of F .

The system of representations of the Weil–Deligne group associated to an abelian variety

Fix a number field F⊂ℂ, an abelian variety A∕F and let GA be the Mumford–Tate group of A∕ℂ. After replacing F by finite extension one can assume that, for every prime number l, the action of the absolute Galois group ΓF= Gal(F∕F) on the etale cohomology group Het1(AF,ℚl) factors through a morphism ρl:ΓF→GA(ℚl). Let v be a valuation of F and write ΓFv for the absolute Galois group of the completion Fv. For every l with v(l)=0, the restriction of ρl to ΓFv defines a representation ′WFv→GA∕ℚl of the Weil–Deligne group.It is conjectured that, for every l, this representation of ′WFv is defined over ℚ as a representation with values in GA and that the system above, for variable l, forms a compatible system of representations of ′WFv with values in GA. A somewhat weaker version of this conjecture is proved for the valuations of F, where A has semistable reduction and for which ρl(Frv) is neat.

Moduli of PT-semistable objects II

We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category D b ( X ) D^b(X) of coherent sheaves on a smooth projective three-fold X X . Then we construct the moduli of PT-semistable objects in D b ( X ) D^b(X) as an Artin stack of finite type that is universally closed. In the absence of strictly semistable objects, we construct the moduli as a proper algebraic space of finite type.

Projective varieties with bad semi-stable reduction at 3 only

Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p\ne 2 . We construct integral theory of p -adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p) . This modification of Breuil's theory results in the following application in the spirit of the Shafarevich Conjecture. If Y is a projective algebraic variety over \mathbb Q with good reduction modulo all primes l\ne 3 and semi-stable reduction modulo 3 then for the Hodge numbers of Y_C=Y\otimes _{\mathbb Q}\ C , one has h^2(Y_C)=h^{1,1}(Y_C) .

Covers in p-adic analytic geometry and log covers I: Cospecialization of the (p^{\prime})-tempered fundamental group for a family of curves

The tempered fundamental group of a p-adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between (p ′ ) versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction and we will prove the invariance of the geometric log fundamental group of log smooth log schemes over a log point by change of log point.

A descent map for curves with totally degenerate semi-stable reduction

Let K be a local field of residue characteristic p. Let C be a curve over K whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-p rational torsion subgroup on the Jacobian of C. We also determine divisibility of line bundles on C, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of C.

The semistable reduction problem for the space of morphisms on ℙn

We restate the semistable reduction theorem from geometric invariant theory in the context of spaces of morphisms from Pn to itself. For every complete curve C downstairs, we get a Pn-bundle on an abstract curve D mapping finite-to-one onto C, whose trivializations correspond to not necessarily complete curves upstairs with morphisms corresponding to identifying each fiber with the morphism the point represents. Finding a trivial bundle is equivalent to finding a complete D upstairs mapping finite-to-one onto C; we prove that in every space of morphisms, there exists a curve C for which no such D exists. In the case when D exists, we bound the degree of the map from D to C in terms of C for C rational and contained in the stable space.

Simultaneous semi-stable reduction for curves with ADE singularities

A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family X ! B of ADE curves, we give an explicit and natural resolution of the rational map B 99K Mg. Moreover, we discuss a lifting of this map to the moduli stack Mg, i.e. a simultaneous semi-stable reduction for the family X /B. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of B is not a sufficient base change for a lifting of the map B 99K Mg to Mg.

Big monodromy theorem for abelian varieties over finitely generated fields

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on ℓ-torsion points, for almost all primes ℓ, contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with the endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall.

Geometric criteria for tame ramification

We prove an A’Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the l-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito’s criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito’s criterion to arbitrary dimension.

Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory

We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi-)semi-stable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application, we prove that the reciprocity map introduced for smooth projective varieties over local fields K K by Bloch, Kato and Saito is an isomorphism after ℓ \ell -adic completion, if the variety has good or ordinary quadratic reduction and ℓ ≠ c h a r ( K ) \ell \neq \mathrm {char}(K) .

On a weak variant of the geometric torsion conjecture

A consequence of the geometric torsion conjecture for abelian varieties over function fields is the following. Let k be an algebraically closed field of characteristic 0. For any integers d , g ⩾ 0 there exists an integer N : = N ( k , d , g ) ⩾ 1 such that for any function field L / k with transcendence degree 1 and genus ⩽ g and any d-dimensional abelian variety A → L containing no nontrivial k-isotrivial abelian subvariety, A ( L ) tors ⊂ A [ N ] . In this paper, we deal with a weak variant of this statement, where A → L runs only over abelian varieties obtained from a fixed ( d-dimensional) abelian variety by base change. More precisely, let K / k be a function field with transcendence degree 1 and A → K an abelian variety containing no nontrivial k-isotrivial abelian subvariety. Then we show that if K has genus ⩾1 or if A → K has semistable reduction over all but possibly one place, then, for any integer g ⩾ 0 , there exists an integer N : = N ( A , g ) ⩾ 1 such that for any finite extension L / K with genus ⩽ g, A ( L ) tors ⊂ A [ N ] . Previous works of the authors show that this holds—without any restriction on K—for the ℓ-primary torsion (with ℓ a fixed prime). So, it is enough to prove that there exists an integer N : = N ( A , g ) ⩾ 1 such that for any finite extension L / K with genus ⩽ g, the prime divisors of | A ( L ) tors | are all ⩽ N.

One half log discriminant and division polynomials

Szpiro and Tucker recently proved that, under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C ( ( t ) ) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.

Semistable reduction for overconvergentF-isocrystals, IV: local semistable reduction at nonmonomial valuations

AbstractWe complete our proof that given an overconvergentF-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue forF-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached top-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case

Let $K$ be a field of characteristic zero complete for a discrete valuation, with perfect residue field of characteristic $p>0$, and let $K^+$ be the valuation ring of $K$. We relate the log-crystalline cohomology of the special fibre of certain affine $K^+$-schemes $X=\text{Spec}(R)$ with semi-stable reduction to the Galois cohomology of the fundamental group of the geometric generic fibre $\pi_1(X_{\bar{K}})$ with coefficients in a Fontaine ring constructed from $R$. This is based on Faltings' theory of almost etale extensions.