Semi-abelian reduction of Albanese varieties

Abstract: Let X_K be a projective geometrically normal variety over the fraction field of a DVR. We show that if X_K admits a projective model whose special fibre has at worst nodes in codimension one, then the Albanese variety Alb_{X_K/K} has semi-abelian reduction over the same base. As an application, we extend the classical nonexistence theorem of Fontaine and Abrashkin for abelian schemes over Z to the singular setting. For instance, if a flat projective Z-scheme X has normal and geometrically connected fibres, then H^1(X_Q, O_{X_Q})=0.

• The paper proves that a projective model with geometrically N1 special fibre gives semi-abelian reduction, while geometrically R1 gives good reduction for the Albanese and reduced Picard variety.
• The paper develops a Bertini-based induction using hypersurface sections and finite-kernel Picard maps, avoiding smoothness assumptions on higher-dimensional relative Picard schemes and allowing nonreduced fibres.
• The paper extends Fontaine–Abrashkin nonexistence results by showing that flat projective schemes over Z with normal geometrically connected R1 fibres have vanishing H1(O) over Q, even with severe higher-codimension singularities.

Overview

This note by Chung (2608.14417) proves a higher-dimensional analogue of the Deligne–Mumford theorem relating stable reduction of curves to semi-abelian reduction of Jacobians. Let $S$ be a Dedekind scheme with fraction field $K$ and closed point $s$, and let $X_K$ be a projective, geometrically normal, geometrically connected $K$-scheme. The main theorem states that if $X_K$ admits a projective model over the DVR $\mathcal{O}_{S,s}$ whose special fibre has at worst nodes in codimension one (geometrically $N_1$), then $\mathrm{Alb}_{X_K/K}$ and $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$ have semi-abelian reduction at $K$0; if the special fibre is smooth in codimension one (geometrically $K$1), they have good reduction. Thus reduction type of the Albanese is governed entirely by codimension-one geometry of the special fibre; singularities in codimension at least two are irrelevant, and the special fibre may even be nonreduced.

The paper's most consequential application is arithmetic: the classical Fontaine–Abrashkin nonexistence theorem for abelian varieties over $K$2 extends to a singular setting. Specifically, if $K$3 is a flat projective $K$4-scheme with normal, geometrically connected fibres and every closed fibre is $K$5, then $K$6. Notably, the fibres are allowed arbitrarily bad singularities in codimension two and above, and may all be nonreduced; the author states that no prior arithmetic nonexistence criterion over $K$7 applies in this generality.

The node condition and Bertini theorems

The technical core is a definition of "node" at a codimension-one point $K$8 over a field $K$9: the residue field $s$0 is separable over $s$1 and, for some separable algebraic closure, the completed strict henselization is isomorphic as a $s$2-algebra to $s$3. A scheme is geometrically $s$4 if every codimension-one point is geometrically regular or a node.

This notion is calibrated to satisfy three requirements: preservation under hypersurface sections, specialization to classical nodes on curves, and agreement with the KSBA notion of node (Kollár's intrinsic node) when the residue field is perfect of odd characteristic. The comparison results include:

• For curves, the definition coincides with the Stacks Project notion of node.
• In any dimension with $s$5, it is equivalent to Kollár's intrinsic node when $s$6 is separable; in characteristic 2 it is strictly stronger, as shown by the Whitney umbrella $s$7, which is an intrinsic node but not a node because $s$8 is not separable over $s$9.
• Being an intrinsic node is intrinsic to the local ring, whereas being a node is a property relative to the base field; a node over $X_K$0 may fail to be a node after viewing $X_K$1 over a subfield $X_K$2.

The key structural tool is a geometric characterization of nodes via étale roofs, proved using Artin approximation: $X_K$3 is a node if and only if there exist pointed étale $X_K$4-morphisms $X_K$5, where $X_K$6 and $X_K$7. The author notes that no explicit reference for this precise formulation exists and includes the proof. The characterization matters because the naive approach of splitting nodes by base change fails in higher dimensions: the obstruction can live in the residue field of the point rather than the base field, as the same Whitney umbrella example demonstrates.

Building on this, the paper proves Bertini theorems for the $X_K$8 condition: over an infinite field, for a general hypersurface $X_K$9 of any degree, $K$0 is geometrically $K$1 whenever $K$2 is (of pure dimension $K$3); over finite fields, the analogous statement holds for all sufficiently large degrees, adapting the density techniques of Poonen and Ghosh–Krishna. These are then lifted to a Bertini theorem over the DVR: given geometrically $K$4 (resp. $K$5) reduction at $K$6, infinitely many hypersurface sections $K$7 inherit the same reduction type, using the specialization map and a flatness argument via avoidance of associated points of the special fibre.

Proof of the main theorem

The proof of the main result proceeds by induction on dimension. The base case reduces to the classical result that a flat model whose geometric fibres are reduced nodal curves has a Jacobian with semi-abelian reduction (BLR, Corollary 9.7/2), using normality of the total space. For $K$8, one takes a hypersurface section $K$9 of sufficiently large degree $X_K$0 with the same reduction type. The inclusion induces $X_K$1, and the kernel is shown to be finite: its tangent space injects into the kernel of $X_K$2, which vanishes for $X_K$3 by the Enriques–Severi–Zariski lemma. The image is then an abelian subvariety of $X_K$4 with semi-abelian reduction, and since the surjection onto the image is an isogeny, which preserves reduction type, $X_K$5 has the claimed reduction; duality transfers the conclusion to $X_K$6.

Why the curve argument does not generalize directly

The paper is explicit about the obstruction to extending the Deligne–Mumford–Raynaud strategy. For curves, Raynaud's theorem identifies the identity component $X_K$7 of the Néron model of $X_K$8 with $X_K$9 of the model itself. In relative dimension at least 2, $\mathcal{O}_{S,s}$0 may fail to be smooth over $\mathcal{O}_{S,s}$1: the obstruction to formal smoothness lies in $\mathcal{O}_{S,s}$2, which vanishes only for curves. Concretely, $\mathcal{O}_{S,s}$3 can fail to be smooth when $\mathcal{O}_{S,s}$4 jumps, or when $\mathcal{O}_{S,s}$5 of a fibre is nonreduced — the latter occurring in characteristic $\mathcal{O}_{S,s}$6 even for smooth projective surfaces over algebraically closed fields (Igusa). The proof circumvents this by working with the Néron model of $\mathcal{O}_{S,s}$7 rather than $\mathcal{O}_{S,s}$8.

Comparison with the cohomological route

A higher-dimensional statement also follows from classical results: if a smooth proper $\mathcal{O}_{S,s}$9 has semi-stable reduction, Grothendieck's nearby/vanishing cycles theory gives unipotent monodromy on $N_1$0, and Grothendieck's criterion for semi-abelian reduction forces semi-abelian reduction of $N_1$1. However, this route requires a regular model, a smooth generic fibre, and a reduced normal crossings special fibre. The Bertini approach requires only projectivity of the model, geometric normality of $N_1$2, and codimension-one control on the special fibre — an advantage in dimensions above three, where resolution of singularities remains open for models in mixed characteristic and varieties in positive characteristic. This flexibility is precisely what enables the nonexistence results for nonreduced fibres.

Arithmetic nonexistence consequences

Combining the main theorem with Fontaine–Abrashkin yields a chain of corollaries over $N_1$3 and related rings:

• A normal projective geometrically connected $N_1$4 with everywhere $N_1$5 reduction has trivial Albanese, i.e. $N_1$6. The same holds over $N_1$7, $N_1$8, $N_1$9 via Fontaine's corollary.
• Consequently, no flat projective family of irregular normal varieties exists over $\mathrm{Alb}_{X_K/K}$0, and there is no nonzero abelian variety over $\mathrm{Alb}_{X_K/K}$1 with everywhere $\mathrm{Alb}_{X_K/K}$2 reduction.
• For $\mathrm{Alb}_{X_K/K}$3, if $\mathrm{Alb}_{X_K/K}$4 has $\mathrm{Alb}_{X_K/K}$5 reduction at all primes $\mathrm{Alb}_{X_K/K}$6 and $\mathrm{Alb}_{X_K/K}$7 reduction at $\mathrm{Alb}_{X_K/K}$8, then $\mathrm{Alb}_{X_K/K}$9 is trivial — generalizing Brumer–Kramer and Schoof. For $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$0, $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$1 is isogenous to a power of $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$2.
• For $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$3, $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$4 at $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$5 may be replaced by semi-log canonical reduction, since slc schemes are regular or nodal at codimension-one points when $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$6.

The moduli interpretation is that Fontaine–Abrashkin forbids $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$7-points of the smooth irregular locus of the KSB stack $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$8 of stable surfaces; the extension covers the normal irregular locus, and the slc corollary shows the absence persists even when the $(\mathrm{Pic}^0_{X_K/K})_{\mathrm{red}}$9-point meets the boundary via a genuine slc degeneration at a specified prime.

Two further structural corollaries deserve mention. First, an abelian variety over $K$00 has good reduction if and only if it has geometrically $K$01 reduction — a geometric restatement of the Néron–Ogg–Shafarevich criterion; in equal characteristic zero, semi-abelian reduction is equivalent to geometrically $K$02 reduction (the converse in odd residue characteristic contingent on a theorem of Kollár holding in that setting). Second, even when $K$03 is trivial, iterating hypersurface sections yields infinitely many successive sections with semi-abelian Albanese, giving a geometric obstruction to the existence of geometrically $K$04 reduction itself.

Sharpness and limitations

The paper is candid about the optimality and scope of the hypotheses. For curves of genus $K$05, the geometrically $K$06 condition is optimal: by Deligne–Mumford, semi-abelian reduction of the Jacobian forces the minimal regular model to be geometrically $K$07. For good reduction, however, the $K$08 condition can be weakened even for curves (BLR, Example 9.2/8), so $K$09 is not sharp in that direction. The equivalence between semi-abelian reduction and geometrically $K$10 reduction of abelian varieties is established only in equal characteristic zero; the extension to perfect residue fields of odd characteristic depends on the validity of Kollár's Theorem 42 in that setting, which the paper flags as an open input. The locally stable reduction corollary likewise requires the residue field to be perfect of characteristic $K$11, a constraint inherited from the comparison between intrinsic nodes and nodes in characteristic 2. Finally, the author notes that the $K$12/$K$13 conditions admit natural higher-codimension extensions via Serre's condition $K$14 and a corresponding $K$15, and that the pursuit of these extensions and their arithmetic applications is ongoing work.

Conclusion

The paper establishes that codimension-one nodal behavior of a single projective model suffices to control semi-abelian and good reduction of Albanese and Picard varieties in all dimensions, via a Bertini-theoretic induction that avoids both the smoothness failure of relative Picard schemes and the regularity hypotheses of the cohomological monodromy approach. The resulting singular generalization of the Fontaine–Abrashkin theorem — including the allowance of nonreduced fibres — extends a body of arithmetic nonexistence results to settings where no prior criteria applied, and frames a concrete open question: whether the equivalence between semi-abelian and geometrically $K$16 reduction of abelian varieties holds over DVRs with perfect residue fields of odd characteristic, pending a positive-characteristic version of Kollár's result on Néron models and minimal models.