Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System

Abstract: We study families of abelian varieties over smooth proper curves with small $l$-adic local system over characteristic $p$. We show that such abelian schemes have a non-nef Hodge bundle and cannot be lifted to $W_2(k)$. We also establish an Arakelov-type inequality for families of abelian varieties over smooth proper curves in characteristic $p$, assuming $W_2(k)$-liftability.

• The paper establishes that families of abelian varieties with small l-adic local systems in positive characteristic do not admit W2(k)-liftings, as evidenced by the negative minimal slope of their Hodge bundles.
• The paper utilizes deformation theory and nonabelian Hodge correspondence, deriving a contradiction through Higgs semistability and trace analysis.
• The paper extends these results to moduli of supersingular abelian varieties and K3 surfaces, deriving an Arakelov-type inequality and revealing key characteristic p phenomena.

Non-Liftability of Families of Abelian Varieties with Small $l$-adic Local System

Overview and Main Theorems

This work investigates the deformation-theoretic properties of families of abelian varieties over smooth proper curves in positive characteristic, focusing on those for which the associated $l$-adic local systems are "small"—that is, with finite monodromy image. The main result establishes that such families do not admit lifts to the ring of Witt vectors of length two, $W_2(k)$, and, crucially, that their Hodge bundles are not nef. This leads to a moduli-theoretic characterization of positive-dimensional subvarieties in characteristic $p$ Siegel moduli with trivial $l$-adic monodromy. The paper further provides an Arakelov-type inequality for families that do admit $W_2(k)$-liftings, and extends the analysis to families of supersingular K3 surfaces via the Kuga--Satake construction.

Moduli-Theoretic Context and Stratification

Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $C$ a smooth proper $k$-curve, and $l$0 an abelian scheme. "Small" $l$1-adic local system refers to the property that the associated $l$2-adic Galois representation has finite (in particular, commutative) image. Oort's structural results imply that up to finite étale base change, such families are isogenous to constant ones. The locus in the Siegel moduli $l$3 parameterizing such families is captured by the isogeny leaves in the Newton stratification. Isogeny leaves, defined in Oort's language via $l$4-schemes, are maximal integral subvarieties along which all fibers are isogenous (of purely inseparable type) to a fixed abelian variety. These leaves are proper, in contrast to the quasi-affine central leaves.

By an analysis of the monodromy group, any subvariety with finite $l$5-adic monodromy must factor through an isogeny leaf—sharpening the moduli-theoretic understanding of the positive-dimensional loci of such families.

Non-Liftability and the Hodge Bundle

The fundamental result is the following:

Let $l$6 be an abelian scheme with small $l$7-adic local system (i.e., factoring through an isogeny leaf). Then:

• $l$8 does not admit a flat lift to $l$9.
• The Hodge bundle $W_2(k)$0 fails to be nef; in fact, its minimal slope is negative.

The proof is via contradiction: assuming $W_2(k)$1-liftability, one invokes the characteristic $W_2(k)$2 nonabelian Hodge theory (Ogus–Vologodsky, Lan–Sheng–Zuo), which yields Higgs semistability for the graded pieces of the first de Rham cohomology associated (Hodge bundle and its dual). This semistability forces all Harder–Narasimhan slopes of $W_2(k)$3 to be non-negative. However, Oort's trace-theoretic analysis, together with a careful study of isogeny leaves and the induced behavior on Hodge bundles, forces $W_2(k)$4, a contradiction.

Moreover, the paper formalizes the interplay with the $W_2(k)$5-trace: any such family, if not isotrivial, admits a non-trivial trace map up to finite étale cover. The kernel of this trace gives rise to non-nefness of $W_2(k)$6, a phenomenon specific to characteristic $W_2(k)$7.

Arakelov-Type Inequality under Liftability

Assume $W_2(k)$8 admits a $W_2(k)$9-lift. Then the classical Arakelov inequality

$p$0

holds. This extends known results over characteristic zero to positive characteristic in the presence of liftings, using Higgs semistability and analysis of the associated connections and Kodaira–Spencer maps.

Implications and Applications

A striking application is to families of supersingular abelian varieties and K3 surfaces:

• Any non-isotrivial family of supersingular abelian varieties (or Kuga–Satake images of supersingular K3s) over a proper smooth curve cannot be lifted to $p$1. The infinitesimal rigidity and deformation space of these purely characteristic $p$2 families is thus fundamentally obstructed.

These statements generalize Moret–Bailly’s example of non-constant families of supersingular abelian surfaces and rational curves in the supersingular locus to higher dimensions and to K3 surfaces.

Positivity Failures as a Characteristic $p$3 Phenomenon

The non-nefness of $p$4 is impossible in characteristic zero (due to Griffiths and the curvature argument), and is tied to the existence of non-trivial trace phenomena and purely inseparable isogenies. The author supports a conjectural direction: that failure of nefness for the Hodge bundle indicates the presence of a factor (up to étale cover) isogenous to a constant abelian scheme, i.e., detectable in the structure of the trace.

Future Directions

The paper raises several natural questions. Notably, the author conjectures a full converse: non-nefness of the Hodge bundle (in positive characteristic, for non-isotrivial families) always arises from the presence of a non-isotrivial trace component after finite étale base change. This connects finer positivity properties of vector bundles on curves to the group-theoretic and Galois-theoretic structure of the family.

Moreover, the appearance of constant subgroup schemes in the $p$5-torsion (as in purely inseparable isogenies) is asked to correspond to the presence of isotrivial subschemes in general.

Conclusion

This work identifies a deep interaction between the deformation-theoretic rigidity of families of abelian varieties with "small" $p$6-adic monodromy, the structure of isogeny leaves in characteristic $p$7, and the positivity of the Hodge bundle. It establishes that such families cannot be lifted even infinitesimally to mixed characteristic, and that characteristic $p$8 phenomena lead to a breakdown of positivity properties from characteristic zero. This yields both new rigidity theorems and motivates new conjectures on the interplay between arithmetic monodromy and positivity, with ramifications for the geometry of Shimura varieties and K3 moduli in positive characteristic.

Reference: "Non-liftability of Families of Abelian Varieties with Small $p$9-adic Local System" (2604.17059)