Drinfeld Compactification of Bun_P

• Drinfeld compactification of Bun_P is a geometric method for compactifying moduli spaces of P-bundles on curves, capturing degeneration and boundary data.
• It employs reciprocal coordinate rings and explicit stratifications to reveal invariant-theoretic and modular structures underlying the moduli problem.
• The framework supports detailed cohomological computations and desingularization techniques, advancing the applications of the geometric Langlands program.

The Drinfeld compactification of $\operatorname{Bun}_P$, where $P$ is a parabolic subgroup of a reductive group $G$ over a smooth projective curve $X$, is a geometric object that underpins the paper of singularities, intersection cohomology, and moduli-theoretic phenomena at the heart of the geometric Langlands program. It provides a canonical way to compactify moduli spaces (and stacks) of $P$-torsors and related structures, encoding degenerations and boundary phenomena that are invisible in the open locus. The compactification is equipped with rich stratifications, modular interpretations, and relationships to both local and global models, including deep connections with (semi-infinite) intersection cohomology sheaves and factorization categories.

1. Construction and Moduli-Theoretic Framework

Drinfeld's compactification arises as a relative compactification of the moduli stack $\operatorname{Bun}_P$ of $P$-bundles on $X$, fibered over $\operatorname{Bun}_G$. Its geometric model is typically constructed via a closure in a "larger" moduli problem, often parameterizing generalized $P$-reductions or maps from the curve $X$ (and its degenerations) to partial flag varieties associated to $G/P$.

In the case relevant for period domains or Drinfeld modular varieties over finite fields, concrete models arise:

• For vector spaces $V$ over $\mathbb{F}_q$, the Drinfeld period domain $\Omega_V$ (Drinfeld's "open half space") admits two distinguished compactifications:
  • The "classical" compactification by projective space $P_V = \mathbb{P}(V)$.
  • The Drinfeld compactification $Q_V = \operatorname{Proj} R_V$, where $R_V$ is a "reciprocal" coordinate ring generated by $1/v$ for $v\in V\setminus\{0\}$ subject to explicit relations (see below) (Pink et al., 2010).
• Modular analogs with level structures yield compactifications via moduli of $A$-reciprocal maps with corresponding projective coordinate rings (Pink, 2019).

This approach is generalized to arbitrary parabolics $P \subset G$, producing stacks or spaces compactifying $\operatorname{Bun}_P$ by admitting degenerate or limiting "reductions" that fail to be everywhere generic.

2. Coordinate Rings, Generators, and Relations

A key innovation is the use of a "reciprocal" algebra, providing an explicit and coordinate-free presentation of the compactification. For $\dim V = r$, the coordinate ring $R_V$ is generated over $\mathbb{F}_q$ by symbols $1/v$ ($v \neq 0$), with relations:

• $\frac{1}{a v} = a^{-1} \frac{1}{v}$ for $a \in \mathbb{F}_q^{\times}$.
• For $v, v'\in V$ in general position,

$$\frac{1}{v} \cdot \frac{1}{v'} = \frac{1}{v+v'} \left( \frac{1}{v} + \frac{1}{v'} \right).$$

By taking $Q_V = \operatorname{Proj} R_V$, one obtains a "dual" compactification (relative to $P_V$) that is Cohen–Macaulay, normal, and projectively normal, with explicit combinatorial and invariant-theoretic structure—see (Pink et al., 2010), Theorems 1.6 and 1.7.

This algebraic presentation encodes both the boundary stratification and the modular interpretation (see Section 6).

3. Boundary Stratification: Structure and Gluing

The boundary of Drinfeld's compactification is highly stratified. Each stratum corresponds to proper nonzero subspaces $V' \subset V$ (or, more generally, to "defect" indices in the $P$-reduction context):

• The boundary is set-theoretically a disjoint union of locally closed subschemes, each canonically isomorphic to a smaller period domain $Q_{V'}$.
• Intersections of closures of these strata encode intricate incidence relations, organizing the singularity structure and moduli-theoretic degenerations.

This stratification is dual to that appearing in the projective space compactification, and its combinatorics are fundamental for intersection cohomology and for group actions on $Q_V$ [(Pink et al., 2010), Theorem 8.2].

4. Singularities and Desingularization

While the boundary divisors of the Drinfeld compactification are regular in codimension 1, singularities arise along strata of codimension at least 2:

• $Q_V$ is singular precisely along these higher-codimension strata [(Pink et al., 2010), Theorem 8.4].
• A systematic "blowup-along-the-boundary" procedure (as in Section 10 of (Pink et al., 2010)) produces a smooth, projective desingularization $B_V$:
  • $B_V$ is stratified by flags in $V$ (or by partial defect data).
  • The boundary divisor $B_V \setminus Q_V$ exhibits normal crossings.
  • $B_V$ dominates both $Q_V$ and $P_V$ and admits a modular interpretation in terms of "ferns" or flag configurations over the base (Puttick, 2020).

These desingularizations are essential for defining intersection cohomology and for the application of the decomposition theorem in the presence of singularities (Campbell, 2016).

5. Cohomology, Hilbert Functions, and Dualizing Sheaves

The compactification's geometry is further elucidated by explicit computations:

• The graded pieces of $R_V$ (in negative degree or, after shift, positive degree) admit a combinatorial formula:

$\dim (R_V)_{-n} = h_r(n),$

where $h_r(T)$ is an explicit polynomial involving subset combinatorics (see [(Pink et al., 2010), Theorem 1.10]).

• Cohomology of twisting sheaves $H^i(Q_V,\mathcal{O}_{Q_V}(n))$ vanishes outside degree 0 and the top degree, paralleling the projective space model [(Pink et al., 2010), Section 5].
• The dualizing sheaf on $Q_V$ is explicitly described in terms of a line bundle built from the "reciprocal" data [(Pink et al., 2010), Theorem 6.1].

These results support projective normality and control the algebraic invariants relevant for modular forms and their cohomology.

6. Modular Interpretation and Functorial Properties

A fundamental property of the Drinfeld compactification is its modular description:

• Points of $Q_V$ correspond to isomorphism classes of pairs $(\mathcal{L},p)$, where $\mathcal{L}$ is an invertible sheaf and $p:V \to \Gamma(S,\mathcal{L})$ is a reciprocal map satisfying,

  $$p(a v) = a^{-1} p(v), \quad\quad p(v)\,p(v') = p(v+v')(p(v)+p(v')),$$

for all $a\in\mathbb{F}_q^\times$, $v, v'\in V$ [(Pink et al., 2010), Section 7].

• Linear modular data are recovered by setting $p(v) = (X(v))^{-1}$ for injective $X:V \to \Gamma(S,\mathcal{L})$.

This modular viewpoint extends to the desingularization $B_V$ via the moduli of $V$-ferns—stable genus zero curves marked by $V\cup\{\infty\}$ with compatible finite group actions—solidifying the connection to compactified moduli spaces and their application to automorphic forms (Puttick, 2020).

7. Quotients, Invariant Theory, and Symmetries

The compactification and its coordinate rings admit natural actions by finite groups:

• $R_V$ and the classical symmetric algebra $S_V$ admit $GL(V)$, $SL(V)$, and maximal unipotent group $U$ actions.
• Invariant subrings correspond to moduli-theoretic interpretations, yielding quotient varieties that are projective or weighted projective spaces:
  • $Q_V/U \cong \mathbb{P}^{r-1}$,
  • $Q_V/G$ is a weighted projective space with weights determined by $q$-powers.

These symmetries elucidate the structure of the compactification and relate its geometry to the theory of modular forms, including the paper of Hecke operators and representation theory [(Pink et al., 2010), Section 3].

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In summary, the Drinfeld compactification of $\operatorname{Bun}_P$ (and its explicit prototypes for period domains and modular varieties) defines a geometrically natural, stratified, and modular compactification whose boundary encodes rich intersection-theoretic, representation-theoretic, and cohomological information. Its explicit coordinate ring, precise stratification, modular interpretation, and smooth desingularization form a foundational input for advanced work in the geometric Langlands program and beyond, realizing the intricate interplay between algebraic geometry, modular forms, and automorphic representation theory.