Semistable Reduction Research Articles

P-groups, p-rank, and semistable reduction of coverings of curves

p-groups, p-rank, and semistable reduction of coverings of curves

Skeletal filtrations of the fundamental group of a non-archimedean curve

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first prove a simultaneous semistable reduction theorem for residually tame coverings, which we then use to construct a tropicalization functor from the category of residually tame coverings of a marked curve (X,D) to the category of tame coverings of a metrized complex Σ associated to (X,D). We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. We use this skeletal interpretation to define the absolute decomposition and inertia group of a curve, which can be seen as the first subgroups in a ramification filtration of the fundamental group of the curve. We prove that the cyclic coverings that arise from the corresponding decomposition and inertia quotients coincide with the coverings that arise from the toric and connected parts of the analytic Jacobian of the curve.

Triangulations of non-archimedean curves, semi-stable reduction, and ramification

Let K be a complete discretely valued field with algebraically closed residue field and let ℭ be a smooth projective and geometrically connected algebraic K-curve of genus g. Assume that g⩾2, so that there exists a minimal finite Galois extension L of K such that ℭ L admits a semi-stable model. In this paper, we study the extension L∣K in terms of the minimal triangulation of C, a distinguished finite subset of the Berkovich analytification C of ℭ. We prove that the least common multiple d of the multiplicities of the points of the minimal triangulation always divides the degree [L:K]. Moreover, in the special case when d is prime to the residue characteristic of K, then we show that d=[L:K], obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito’s theorem to the wildly ramified case.

Vologodsky integration on curves with semi-stable reduction

We prove that the Vologodsky integral of a meromorphic one-form on a curve over a p-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve.

Explicit Vologodsky integration for hyperelliptic curves

Vologodsky’s theory of p p -adic integration plays a central role in computing several interesting invariants in arithmetic geometry [Mosc. Math. J. 3 (2003), pp. 205–247, 260]. In contrast with the theory developed by Coleman [Invent. Math. 69 (1982), pp. 171–208; Duke Math. J. 52 (1985), pp. 765–770; Ann. of Math. (2) 121 (1985), pp. 111–168; Invent. Math. 93 (1988), pp. 239-266], it has the advantage of being insensitive to the reduction type at p p . Building on recent work of Besser and Zerbes [Vologodsky integration on curves with semi-stable reduction, to appear in Israel J. Math], we describe an algorithm for computing Vologodsky integrals on bad reduction hyperelliptic curves. This extends previous joint work with Katz [Int. Math. Res. Not. IMRN 8 (2022), pp. 6038–6106] to all meromorphic differential forms. We illustrate our algorithm with numerical examples computed in Sage.

On the semi‐stability degree for abelian varieties

For an abelian variety A $A$ over a number field we study bounds depending only on the dimension of A $A$ for the minimal degree d ( A ) $d(A)$ of a field extension over which A $A$ acquires semi-stable reduction. We first compute d ( A ) $d(A)$ in terms of the cardinalities of the finite monodromy groups of A $A$ which leads to a bound on d ( A ) $d(A)$ in terms of the classical Minkowski bound. We then show this bound is tight up to its 2-part by considering p $p$ -adic coverings of the local points of a universal abelian scheme.

On the stable reduction of hyperelliptic curves

Let $f: S\to B$ be a surface fibration of genus $g\ge 2$ over ${\mathbb{C}}$. The semistable reduction theorem asserts there is a finite base change $\pi: B'\to B$ such that the fibration $S\times_BB'\to B'$ admits a semistable model. An interesting invariant of $f$, denoted by $N(f)$, is the minimum of $\deg(\pi)$ for all such $\pi$. In an early paper of Xiao, he gives a uniform multiplicative upper bound $N_g$ for $N(f)$ depending only on the fibre genus $g$. However, it is not known whether Xiao's bound is sharp or not. In this paper, we give another uniform upper bound $N'_g$ for $N(f)$ when $f$ is hyperelliptic. Our $N'_g$ is optimal in the sense that for every $g\ge 2$ there is a hyperelliptic fibration $f$ of genus $g$ so that $N(f)=N_g'$. In particular, Xiao's upper bound $N_g$ is optimal when $N_g=N'_g$. We show that this last equation $N_g=N_g'$ holds for infinitely many $g$.

Nearby cycles and semipositivity in positive characteristic

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

Strongly semistable reduction of syzygy bundles on plane curves

We investigate degenerations of syzygy bundles on plane curves over p $p$ -adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fibre consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fibre is a syzygy bundle and which become trivial when restricted to each projective line in the special fibre. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in p $p$ -adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the étale fundamental group of a curve. Faltings' p $p$ -adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a p $p$ -adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.

On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

We study the nilpotent part N′ of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the form Q̃ obtained after killing kerN is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of Q̃ to the absolute homology group of Σ is even whenever the quotient of the Nielsen–Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ=F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. This form Q is determined by the Seifert form but can be much more easily computed. Moreover, the form Q̃ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with Q̃ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of Q̃ to the absolute monodromy of Σ=F is not even.

Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p

Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.

P-adic heights and Vologodsky integration

We extend the result of [5] to the case of curves with semi-stable reduction. In this case, one can replace Coleman integration by Vologodsky integration to extend the Coleman-Gross definition of a p-adic height pairing. We show that this pairing still equals the one defined by Nekovář.

Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

AbstractIn this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

Euler characteristics and their congruences for multisigned Selmer groups

AbstractThe notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.

Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces

Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $\overline{\mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $\overline{\mathcal{M}_g}$.

Mustafin Models of Projective Varieties and Vector Bundles

Abstract Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

On Syntomic Complexes with Modulus for Semi-stable Reduction Cases

In this paper, we define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.

Semistable abelian varieties and maximal torsion 1-crystalline submodules

Let p be a prime, let K be a discretely valued extension of ℚ p , and let A K be an abelian K-variety with semistable reduction. Extending work by Kim and Marshall from the case where p>2 and K/ℚ p is unramified, we prove an l=p complement of a Galois cohomological formula of Grothendieck for the l-primary part of the Néron component group of A K . Our proof involves constructing, for each m∈ℤ ≥0 , a finite flat 𝒪 K -group scheme with generic fiber equal to the maximal 1-crystalline submodule of A K [p m ]. As a corollary, we have a new proof of the Coleman–Iovita monodromy criterion for good reduction of abelian K-varieties.

Stability conditions in families

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

Universal stacky semistable reduction

Given a log smooth morphism f: X → S of toroidal embeddings, we perform a Raynaud-Gruson type operation on f to make it flat and with reduced fibers. We do this by studying the geometry of the associated map of cone complexes C(X) → C(S). As a consequence, we show that the toroidal part of semistable reduction of Abramovich-Karu can be done in a canonical way.