Inscribed Banach–Colmez Tangent Spaces

• Inscribed Banach–Colmez tangent spaces are internal tangent objects realized as global sections of vector-bundle modifications on the Fargues–Fontaine curve.
• They bridge analytic and discrete p-adic Hodge-theoretic components by encoding slope data and decomposing into connected and étale parts.
• Their connectedness, determined by the absence of slope 0, is crucial for assessing cohomological smoothness in mixed characteristic moduli.

Inscribed Banach--Colmez tangent spaces are Banach--Colmez-valued tangent objects that arise after enriching diamonds and $v$-sheaves by an inscribed structure. In the present literature, they are attached most explicitly to fixed-determinant moduli of mixed characteristic local shtukas with one leg, where the tangent space at a point $z$ is identified with the Banach--Colmez space $BC(z^*\mathcal E^\circ_{\max})$ of global sections of a vector-bundle modification on the Fargues--Fontaine curve. Their connectedness detects a geometrically distinguished locus, called the non-very-special locus, and is conjecturally equivalent to cohomological smoothness of the underlying diamond; this conjecture is proved in the EL infinite-level Rapoport--Zink case. The notion depends on two antecedents: the reinterpretation of Banach--Colmez spaces in terms of coherent sheaves on the Fargues--Fontaine curve, and the later theory of inscribed $v$-sheaves with internal tangent bundles (Bras, 2018, Howe, 15 Aug 2025, Howe, 15 Aug 2025).

1. Banach--Colmez spaces and the Fargues--Fontaine antecedent

The modern geometric background begins with the sheaf-theoretic definition of the Banach--Colmez category $\mathcal{BC}$. On $\mathrm{Perf}_C$ with the pro-étale topology, one considers the constant sheaf $Q=\underline{\mathbf Q_p}$ and the additive sheaf $\mathbf G_a$, given by $S\mapsto \mathcal O_S(S)$. The category $\mathcal{BC}$ is defined as the smallest abelian subcategory, stable under extensions, containing $z$0 and $z$1. This recovers Colmez’s original category $z$2, in which Banach--Colmez spaces are built from finite-dimensional $z$3-vector spaces and finite-dimensional $z$4-vector spaces by extensions and quotients, with intrinsic numerical invariants $z$5 and $z$6 (Bras, 2018).

The decisive structural result is that Banach--Colmez spaces are not treated merely as ad hoc analytic functors. They are identified with a tilted heart of the derived category of coherent sheaves on the Fargues--Fontaine curve $z$7. Writing

$z$8

the relevant heart is

$z$9

and the comparison theorem is

$BC(z^*\mathcal E^\circ_{\max})$0

This places Banach--Colmez spaces inside slope theory, Harder--Narasimhan formalism, and derived $BC(z^*\mathcal E^\circ_{\max})$1-structure methods (Bras, 2018).

Within that framework, the closest pre-inscription surrogate for tangent data is the positive-slope or vector-bundle contribution. For $BC(z^*\mathcal E^\circ_{\max})$2, the exact triangle

$BC(z^*\mathcal E^\circ_{\max})$3

becomes the short exact sequence

$BC(z^*\mathcal E^\circ_{\max})$4

in the tilted heart. Under the equivalence with $BC(z^*\mathcal E^\circ_{\max})$5, the positive-slope vector-bundle contribution behaves as the linearized part, whereas slope-zero and shifted negative-slope pieces encode the discrete or height contribution. This suggests the later inscribed tangent theory rather than constituting it formally (Bras, 2018).

A particularly important precursor is the universal cover of a $BC(z^*\mathcal E^\circ_{\max})$6-divisible group $BC(z^*\mathcal E^\circ_{\max})$7 over $BC(z^*\mathcal E^\circ_{\max})$8. Its Banach--Colmez realization fits into the exact sequence

$BC(z^*\mathcal E^\circ_{\max})$9

Here $v$0 is the direct tangent-space surrogate, and $v$1 is the rational Tate-module correction. For $v$2, one obtains

$v$3

a prototypical one-dimensional Banach--Colmez extension of a $v$4-line by a $v$5-line (Bras, 2018).

2. Inscription and internal tangent theory

The theory of inscribed $v$6-sheaves introduces the tangent formalism missing from earlier Banach--Colmez work. An inscribed fibered category over a category of locally free nilpotent thickenings is required to satisfy a coproduct-to-product condition: for any base thickening $v$7 and any pair of locally free nilpotent thickenings $v$8, the natural functor

$v$9

is an equivalence. This is the deformation-theoretic axiom from which additivity of tangent directions is derived (Howe, 15 Aug 2025).

The internal tangent bundle is defined by square-zero extension. If $\mathcal{BC}$0 denotes the free rank-one square-zero thickening, then

$\mathcal{BC}$1

For an inscribed presheaf $\mathcal{BC}$2, the tangent bundle $\mathcal{BC}$3 carries a canonical structure of an inscribed $\mathcal{BC}$4-module, where

$\mathcal{BC}$5

Relative tangent and normal objects are defined by

$\mathcal{BC}$6

for a morphism $\mathcal{BC}$7 (Howe, 15 Aug 2025).

In the Fargues--Fontaine setting, tangent bundles often become inscribed Banach--Colmez spaces. For a vector bundle $\mathcal{BC}$8 on the relative thickened Fargues--Fontaine curve, one defines

$\mathcal{BC}$9

and

$\mathrm{Perf}_C$0

as the $\mathrm{Perf}_C$1-sheafification of the presheaf of $\mathrm{Perf}_C$2. These are inscribed $\mathrm{Perf}_C$3-sheaves. The basic cohomological mechanism is the fundamental exact sequence

$\mathrm{Perf}_C$4

This exact sequence is the standard conduit between vector bundles on the curve, de Rham-lattice quotients, and Banach--Colmez tangent objects (Howe, 15 Aug 2025).

The same paper computes derivatives of several inscribed period maps. In particular,

$\mathrm{Perf}_C$5

so the derivative of the inscribed Hodge period map is the Kodaira--Spencer map, and the derivative of the inscribed Liu--Zhu period map is identified on the underlying $\mathrm{Perf}_C$6-sheaf with the geometric Sen morphism or canonical Higgs field. This makes the inscribed tangent formalism a direct receptacle for classical $\mathrm{Perf}_C$7-adic Hodge-theoretic differentials (Howe, 15 Aug 2025).

3. Tangent spaces on moduli of local shtukas with one leg

The most explicit definition of an inscribed Banach--Colmez tangent space occurs for the fixed-determinant fiber

$\mathrm{Perf}_C$8

of the moduli diamond of mixed characteristic local shtukas with one leg. The data consist of a finite extension $\mathrm{Perf}_C$9, a connected reductive group $Q=\underline{\mathbf Q_p}$0, a conjugacy class $Q=\underline{\mathbf Q_p}$1, and an element $Q=\underline{\mathbf Q_p}$2. After passing to the fixed-determinant fiber over a point

$Q=\underline{\mathbf Q_p}$3

one obtains an inscribed $Q=\underline{\mathbf Q_p}$4-sheaf $Q=\underline{\mathbf Q_p}$5 with Hodge and Hodge--Tate period maps (Howe, 15 Aug 2025).

The Lie-theoretic input is encoded in

$Q=\underline{\mathbf Q_p}$6

Using the universal meromorphic isomorphism between the trivial $Q=\underline{\mathbf Q_p}$7-bundle and the bundle associated with $Q=\underline{\mathbf Q_p}$8, one defines two $Q=\underline{\mathbf Q_p}$9-lattices,

$\mathbf G_a$0

and

$\mathbf G_a$1

They define vector-bundle modifications $\mathbf G_a$2 and $\mathbf G_a$3 on the relative Fargues--Fontaine curve (Howe, 15 Aug 2025).

The basic tangent computation is

$\mathbf G_a$4

and pointwise, for

$\mathbf G_a$5

the inscribed Banach--Colmez tangent space is

$\mathbf G_a$6

If $\mathbf G_a$7 denotes the simultaneous fiber of the Hodge and Hodge--Tate period maps through $\mathbf G_a$8, then

$\mathbf G_a$9

Moreover,

$S\mapsto \mathcal O_S(S)$0

so the ambient tangent space and the tangent space to the period-fiber intersection are related by duality (Howe, 15 Aug 2025).

Object	Formula	Role
Ambient tangent bundle	$S\mapsto \mathcal O_S(S)$1	Tangent of fixed-determinant moduli
Pointwise tangent space	$S\mapsto \mathcal O_S(S)$2	Tangent at $S\mapsto \mathcal O_S(S)$3
Fiber tangent bundle	$S\mapsto \mathcal O_S(S)$4	Tangent to simultaneous Hodge/Hodge--Tate fiber
Duality	$S\mapsto \mathcal O_S(S)$5	Converts slope data between max and min bundles

This construction gives the phrase “inscribed Banach--Colmez tangent space” its precise contemporary meaning: the Banach--Colmez realization of the internal tangent object, computed as global sections of a naturally defined vector-bundle modification on the Fargues--Fontaine curve (Howe, 15 Aug 2025).

4. Connectedness, slope zero, and the very special locus

The geometry of these tangent spaces is controlled by Harder--Narasimhan slopes. For every geometric point $S\mapsto \mathcal O_S(S)$6, the bundle

$S\mapsto \mathcal O_S(S)$7

has nonnegative Harder--Narasimhan slopes. Because of this nonnegativity, connectedness of the Banach--Colmez tangent space is equivalent to the absence of slope $S\mapsto \mathcal O_S(S)$8: the paper states that

$S\mapsto \mathcal O_S(S)$9

This identifies connectedness with the exclusion of the étale or locally profinite component (Howe, 15 Aug 2025).

The main equivalence theorem gives six equivalent conditions for a point

$\mathcal{BC}$0

They relate the absence of slope $\mathcal{BC}$1 in $\mathcal{BC}$2 and in $\mathcal{BC}$3, the vanishing

$\mathcal{BC}$4

the vanishing of infinitesimal automorphisms

$\mathcal{BC}$5

the discreteness of the stabilizer

$\mathcal{BC}$6

and the discreteness of the simultaneous period fiber

$\mathcal{BC}$7

inside the orbit $\mathcal{BC}$8. In particular,

$\mathcal{BC}$9

Thus the connectedness of the tangent space is equivalent to geometric discreteness of the intersection of the Hodge and Hodge--Tate fibers (Howe, 15 Aug 2025).

A point is called very special precisely when these equivalent conditions fail. Equivalently,

$z$00

The non-very-special locus is therefore exactly the connected-tangent locus,

$z$01

Since the slopes of $z$02 are nonnegative, the condition that $z$03 occur as a slope is closed by semicontinuity of the Harder--Narasimhan polygon, so

$z$04

is a closed subdiamond (Howe, 15 Aug 2025).

This shows that “connectedness” is not a purely topological qualifier. It is a slope-theoretic condition on a Banach--Colmez tangent object, and simultaneously a group-theoretic condition on stabilizers and a geometric condition on intersections of period fibers. The paper’s point is that these three languages coincide (Howe, 15 Aug 2025).

5. Cohomological smoothness and the Jacobian criterion

The central conjecture states that

$z$05

The tangent-theoretic reason is that, for a vector bundle $z$06 on the Fargues--Fontaine curve with nonnegative slopes, the Banach--Colmez space $z$07 has a connected part coming from positive slopes and a locally profinite part coming from slope $z$08. Thus

$z$09

whereas slope $z$10 contributes the discrete defect. This matches the Fargues--Scholze Jacobian criterion, which governs cohomological smoothness by positivity of slopes in the pulled-back relative tangent bundle (Howe, 15 Aug 2025).

The EL infinite-level Rapoport--Zink case proves the conjecture. In that case,

$z$11

is cohomologically smooth. The proof compares the inscribed moduli $z$12 with a Fargues--Scholze moduli of sections of a smooth quasi-projective scheme $z$13 over the Fargues--Fontaine curve, and shows that for a rank-one point $z$14 corresponding to a section $z$15,

$z$16

Since $z$17 has nonnegative slopes and the Jacobian criterion asks for strictly positive slopes, the non-very-special locus is exactly the Jacobian-smooth locus (Howe, 15 Aug 2025).

In the EL interpretation, very special points correspond to extra endomorphisms. If $z$18 corresponds, via Scholze--Weinstein, to a $z$19-divisible group $z$20 up to isogeny, with

$z$21

and $z$22 the center of $z$23, then

$z$24

The slope-zero part of the tangent space is therefore identified with extra $z$25-linear endomorphisms beyond the center (Howe, 15 Aug 2025).

6. Related frameworks, analogies, and scope

Earlier Banach--Colmez literature does not define an intrinsic tangent functor on the category of Banach--Colmez spaces. Instead, it develops the analytic and exact structures from which the later inscribed theory draws. In the effective setting, an effective Banach--Colmez space is an analytic extension

$z$26

with $z$27 finite-dimensional over $z$28 and $z$29 finite-dimensional over $z$30. The connected--étale decomposition

$z$31

and the vector hull

$z$32

already separate analytic from discrete directions. For $z$33-divisible groups, the exact sequence

$z$34

is an explicit precursor of tangent-like Banach--Colmez geometry (Plût, 2016).

The theory of oblique Banach--Colmez spaces adds Harder--Narasimhan formalism and slope filtration. Stable objects are precisely the $z$35, and every oblique Banach--Colmez space has a unique decreasing filtration with semistable graded pieces

$z$36

noncanonically. This provides a graded linear model by slope, but still not an internal tangent theory in the later inscribed sense (Plût, 2016).

Relative Banach--Colmez geometry over a perfectoid base likewise supplies structural analogies without defining pointwise tangent spaces. For flat coherent sheaves $z$37 on the relative Fargues--Fontaine curve $z$38, there are v-local presentations

$z$39

and, for positive slopes,

$z$40

The fully faithful functor

$z$41

and computations such as

$z$42

supply extension-theoretic data of the type usually associated with infinitesimal deformation theory (Anschütz et al., 2021).

The modern phrase “inscribed Banach--Colmez tangent space” should therefore be read narrowly. It does not mean that the classical Banach--Colmez category has acquired a universal intrinsic tangent-space functor. Rather, it denotes the Banach--Colmez realization of the internal tangent object of an inscribed $z$43-sheaf, most notably in moduli problems built from the Fargues--Fontaine curve. A plausible implication is that the earlier Banach--Colmez decomposition into connected $z$44-linear and discrete $z$45-linear parts is not superseded but internalized: positive-slope analytic directions become genuine tangent directions, and slope-zero pieces become the obstruction to connectedness and cohomological smoothness (Bras, 2018, Howe, 15 Aug 2025, Howe, 15 Aug 2025).