On p-adic vector bundles and local systems on diamonds Annette Werner

On p-adic vector bundles and local systems on diamonds

Annette Werner

Report on joint works with Marvin Anas Hahn (Leipzig) and Lucas Mann (Bonn)

Tropical Geometry, Berkovich Spaces, ArithmeticD-Modules and p-adic Local Systems

December 8-10, 2020

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Goal:Relate p-adic vector bundles top-adic local systems.

On p-adic vector bundles and local systems on diamonds

Annette Werner

Report on joint works with Marvin Anas Hahn (Leipzig) and Lucas Mann (Bonn)

Tropical Geometry, Berkovich Spaces, ArithmeticD-Modules and p-adic Local Systems

December 8-10, 2020

.

Goal:Relate p-adic vector bundles top-adic local systems.

Motivation:

Narasimhan-Seshadri correspondence (1965) on a compact Riemann surfaceX, relating irreducible unitary representations of π^top₁ (X,x) to stable vector bundles onX of degree 0.

Simpson’s correspondence (1992) on smooth projective complex varietiesX, relating finite dimensional complex representations of π^top₁ (X,x) to semistable Higgs bundles onX with vanishing Chern classes.

p-adic results

Deninger/W. (2005):p-adic analog of Narasimhan-Seshadri correspondence relating vector bundles on smooth proper curves overQp with nice (potentially strongly semistable) reductions to continuousp-adic representations of the ´etale fundamental group.

p-adic results

Faltings’p-adic Simpson corrrespondence (2005) on smooth proper curvesX overQp, which provides an equivalence of categories betweenp-adic Higgs bundles on X_Cp and generalized

representations of the ´etale fundamental group.

See also Abbes, Gros, Tsuji (Annals of Math Studies 2016) for an alternative and more general approach.

Xu (2017): Both approaches are compatible in the case of trivial Higgs field.

Liu and Zhu’sp-adic Riemann Hilbert correspondence (2016) associates on a smooth rigid varietyX over finite extension K of Qp to everyp-adic ´etale local system a Higgs bundle on X_Cp with nilpotent Higgs field.

p-adic results

Faltings’p-adic Simpson corrrespondence (2005) on smooth proper curvesX overQp, which provides an equivalence of categories betweenp-adic Higgs bundles on X_Cp and generalized

representations of the ´etale fundamental group.

See also Abbes, Gros, Tsuji (Annals of Math Studies 2016) for an alternative and more general approach.

Xu (2017): Both approaches are compatible in the case of trivial Higgs field.

Liu and Zhu’sp-adic Riemann Hilbert correspondence (2016) associates on a smooth rigid varietyX over finite extension K of Qp to every p-adic ´etale local system a Higgs bundle on X_Cp with nilpotent Higgs field.

Vector bundles with numerically flat reduction

Leto_p⊂Cp with residue fieldk 'Fp.

LetX be a smooth proper connected variety overQp.

Denote byB_X the (full) category of all vector bundles E onX_Cp with numerically flat reduction, i.e. such that

there exists a flat, proper scheme X of finite presentation over Zp with generic fiber X, and

there exists a vector bundleE onX ⊗o_p with generic fiber E such that the special fiber E_k =E ⊗k is numerically flat on X_k.

Vector bundles with numerically flat reduction

We call the bundleE_k onX_k numerically flat if both E_k and its dualE_k^∗ are numerically effective.

Langer: This is equivalent to the fact that for allk-morphisms f :C −→ X_k from a smooth projective curveC the bundle f^∗E_k is semistable of degree 0 onC (Nori).

Note that ifX is projective, then the numerically flat line bundles are precisely the ones in the torsion componentPic^τ(X).

E_k numerically flat implies thatE is numerically flat. If X is projective, this means thatE is semistable with vanishing Chern classes.

Theorem 1 (Deninger / W. 2020)

LetX be a proper, smooth, connected variety overQp.

LetE be a vector bundle onX_Cp. Ifα:Y −→X is finite ´etale surjective withα^∗E inB_Y, i.e. ifα^∗E has numerically flat reduction on some model ofY, thenE admitsp-adic parallel transport.

The functor ofp-adic parallel transport is a continuous functor ρ_E : Π1(X_Cp)−→Vec_Cp

from the ´etale fundamental groupoid onX_Cp to the category of finite dimensionalCp-vector spaces which isx 7→Ex on objects, i.e.Cp-rational points.

In particular,E gives rise to ap-adic ´etale local system onX_Cp.

p-adic parallel transport

The functorE 7→ρ_E from Theorem 1 is functorial inE, exact and well-behaved with respect to tensor products, pullbacks, internal homomorphisms and Galois-conjugation.

Any numerically flat line bundleLsatisfies condition i) of Theorem 1, i.e.α^∗Lhas numerically flat reduction on some model of a finite

´etale coverα:Y →X.

Theorem 2 (Deninger / W. 2005, 2007)

IfX is a curve, then Theorem 1 holds for arbitrary finite coverings α:Y →X (not necessarily ´etale ones).

W¨urthen (2019)

LetX be a proper, connected seminormal rigid analytic variety overQp. Then the results of Theorem 1 hold for all analytic vector bundlesE on X with numerically flat reduction on a proper flat formal model ofX.

This allows us to drop smoothness.

Moreover, this approach works with locally free sheaves on the pro-´etale site. Using adic spaces and their integral structure sheaves makes some cumbersome arguments with models of schemes superfluous.

Fundamental open question

Find a category equivalence between genuineCp-representions of the ´etale fundamental group and a suitable category of Higgs bundles.

In the case of vanishing Higgs field, we know that vector bundles with numerically flat reductions give genuinep-adic

representations.

Assume we are given a smooth projective varietyX overQp and a numerically flat vector bundleE onX. How can we produce a finite ´etale coverα:Y −→X and a numerically flat degeneration ofα^∗E ? We may find many models ofX andE, but how do we control the behaviour of the reductions of these models?

IfX is a curve, an arbitrary finite cover α suffices (which makes our choices even larger), but how do we control the pullbacks of the vector bundle in the special fiber under Frobenius powers?

Concrete example

Letj :C ,→P²_K be the Fermat curvex^d+y^d+z^d = 0 over some finite extensionK ofQp.

Consider the stable, rank 2, degree 0 bundle E =j^∗Syz(x²,y²,z²)(3) onC, whereSyz(x²,y²,z²) =ker(O(−2)^{3 (x}

2,y²,z²)

−→ O) on P²_K.

We have the obvious reduction, given by the Fermat curve over O_K and the pullbackE of the analogous syzygy bundle onP²O_K. Its special fiberE_k is semistable, but not numerically flat: Brenner (2005) shows that certain Frobenius pullbacks ofE_k are no longer semistable.

So we need reductions with “bad” singularities.

Idea: UseMustafin varieties, which are degenerations of projective spacePⁿ_K depending on a choice of finitely many lattices inKⁿ⁺¹. Theorem (Cartwright, H¨abich, Sturmfels, W. 2011)

If the configuration Γ of lattice classes is contained in one apartment of the Bruhat-Tits building associated toPGLn+1,K, then the special fiber of the associated Mustafin varietyM(Γ) is determined by the regular mixed subdivision of the scaled simplex associated to the tropical convex hull of Γ.

RecallE =j^∗Syz(x²,y²,z²)(3) on the Fermat curve j :C ,→P²_K, whereSyz(x²,y²,z²) =ker(O(−2)^{3 (x}

2,y²,z²)

−→ O).

Theorem (Hahn, W. 2019)

For the exampleE on the projectively embedded Fermat curve C we find

a (ramified) degree 2 cover α:D →C

a configuration Γ of three lattices such that the closure Dof D in the Mustafin varietyM(Γ) satisfies: the special fiber consists of irreducible components Ci with eachC_i^red 'P¹_k a vector bundle E onD with generic fiberα^∗E such that E_k|_Cred

i is trivial for all i.

This proves that the bundleE admits p-adic parallel transport.

Local systems on diamonds: joint work with Lucas Mann

LetX be a proper adic space of finite type overCp.

Scholze’s theory of diamonds provides a fascinating tool to disguise characteristic zero objects as characteristicp. Diamonds are certain sheaves on the big pro-´etale site on the category Perf of perfectoid characteristicp spaces, which are quotients of perfectoid spaces by pro-´etale equivalence relations.

There is a functorX 7→X mapping analytic adic spaces over Zp

to (locally spatial) diamonds, such that X_et 'X_et

IfK is a non-archimedean extension ofQp, the diamond functor from seminormal rigid analyticK-spaces to diamonds overSpd(K) is fully faithful

We use thev-topology on Perf: Coverings of Z are given by collections of maps toZ such that every quasi-compact open subset ofZ can be covered by finitely many quasi-compact open subsets of our collection.

Thev-site on a diamond X is then defined as the category of small v-sheaves mapping toX, with coverings given by jointly surjective maps. There is a naturalv-structure sheaf ˇO_X with integral structure sheaf ˇO_X⁺.

For every condensed ring Λ (e.g. every metric ring), there is a notion of the asscoiated constant sheaf on a diamond, and hence a notion of local systems (locally free sheaf of Λ-modules) for the v-topology.

LetX be a proper adic space of finite type overCp. LetM⁺ be the full subcategory of all ˇO⁺_X-modules E onX_v such that for alln the ˇO⁺_X/pⁿ-moduleE/pⁿ becomes trivial on a proper cover ofX. There is a fully faithful functor ∆⁺ from M⁺ to the category of o_p=O_Cp-local systems onX such that for everyE inM⁺ we have

∆⁺(E)⊗_op Oˇ⁺_X=E.

LetMbe the (abelian) category of all ˇO_X-modules of the form E =E[p⁻¹] for someE inM⁺.

Theorem 3 (Mann, W. 2020)

Inverting p, we get a functor ∆ fromM to the category ILoc_Cp(X) of Cp-local systems with an integral model.

The functor ∆ is an equivalence of categories, compatible with direct sums, tensor products, internal homs, exterior powers and pullback.

For every E ∈ Mthere is a natural isomorphism

∆(E)⊗_Cp Oˇ_X =E

Theorem 4 (Mann, W. 2020)

Letf:Y →X be a surjective morphism of finite type proper adic spaces overCp and assume that X is normal. Let E be an

Oˇ_X-module. Iff^∗E ∈ M_Y, thenE ∈ M_X. In other words, the property of having properly trivializable reduction descends alongf.

For adic spaces associated to proper smooth algebraic curves, this generalizes the previously explained result for finite pullbacks.