De Concini–Procesi Compactification

• De Concini–Procesi compactification is a construction in algebraic geometry providing smooth, projective compactifications with stratified boundary divisors exhibiting normal crossings.
• It embeds an adjoint semisimple group into a projective space via an irreducible representation, where each boundary divisor corresponds to a simple root.
• Applications include analyzing orbit degenerations, extending group actions in representation theory, and studying moduli spaces of principal bundles.

The De Concini–Procesi compactification, often referred to as the "wonderful compactification," is a foundational construction in algebraic geometry that provides a highly structured compactification for adjoint semisimple groups, symmetric varieties, and related moduli spaces. It produces smooth projective varieties with stratified boundary divisors of strict normal crossings, yielding a powerful framework for the study of orbits, degenerations, and representation-theoretic phenomena.

1. Construction and Definition

Let $G$ be a connected semisimple (usually of adjoint type) algebraic group over an algebraically closed field $k$, with rank $\ell$. The classical construction proceeds by selecting a regular dominant weight $\lambda$ of the simply connected covering $\widetilde{G}$ and considering the irreducible representation $V_\lambda$ of highest weight $\lambda$: $$G \hookrightarrow \mathbb{P}(\mathrm{End} V_\lambda),\quad g \mapsto [\rho(g)].$$ The wonderful compactification $\overline{G}$ is the Zariski closure of $G$ in $\mathbb{P}(\mathrm{End} V_\lambda)$ (Ilin et al., 2018, Li, 2023, Chen et al., 22 Dec 2025). This variety is smooth, projective, of dimension $\dim G$, and contains $G$ as a dense open subset. Importantly, the resulting compactification is independent (up to canonical $G \times G$-equivariant isomorphism) of the choice of regular $\lambda$.

The $G \times G$-action extends to $\overline{G}$, and the boundary $\overline{G} \setminus G$ (the divisor at infinity) decomposes into $\ell$ irreducible, smooth, $G \times G$-stable divisors $\{D_1, ..., D_\ell\}$, each labeled by the simple roots of $G$ (Ilin et al., 2018, Fu et al., 2020, Wei, 19 May 2025).

2. Structure of the Boundary and Orbit Stratification

The boundary divisors $D_i$ have simple normal crossings: intersections $\bigcap_{i \in I} D_i$ for $I \subset \{1, ..., \ell\}$ are nonempty, smooth, irreducible, of codimension $|I|$, and each $(G \times G)$-orbit in $\overline{G}$ is exactly such an intersection (Ilin et al., 2018, Wong, 2010). The closure relations follow the inclusion order on the index sets: the closure $\overline{S^I}$ of the $G \times G$-orbit $$S^I = \bigcap_{i \notin I} D_i \setminus \bigcup_{j \in I} D_j$$ is $\bigcup_{J \supset I} S^J$.

The boundary strata correspond precisely to the parabolic data:

• Each $I \subset \{1, \ldots, \ell\}$ labels a standard parabolic $P_I$ and Levi $L_I$.
• Each $G \times G$-orbit is equivariantly isomorphic to $(G \times G) \times_{P_I^- \times P_I} G_I$, where $G_I$ is the derived subgroup of $L_I$ (Ilin et al., 2018, Wei, 19 May 2025).

The unique closed orbit is $\bigcap_{i=1}^\ell D_i \cong G/B^- \times G/B$, and the open dense orbit recovers $G$.

3. Combinatorics, Cohomology, and Normal Crossings

The combinatorial structure of the boundary is governed by the lattice of subsets of the simple roots, and more generally by nested-set combinatorics in the context of arrangements or symmetric spaces (Aguirre et al., 2014, Wong, 2010). The Chow ring and cohomology can be computed explicitly, with degree-two classes corresponding to the irreducible boundary divisors generating the entire cohomology ring.

In generalizations to arrangements (i.e., complements of hyperplane arrangements), the De Concini–Procesi model $Y_\mathcal{B}$ is constructed by successive blow-ups along centers determined by a building set $\mathcal{B}$ in the lattice of flats, with boundary divisors and their intersections indexed by nested sets (Denham, 2013):

• Intersections $D_{X_1} \cap \dots \cap D_{X_k} \neq \emptyset$ if and only if $\{X_1, ..., X_k\}$ is nested.
• All such strata are smooth, and the boundary is a strict normal-crossing divisor.

4. Symmetric Spaces and Spherical Varieties

For symmetric spaces $X = G/H$ with $H$ symmetric (i.e., fixed points of an involution), the wonderful compactification is the closure of $G/H$ in an appropriate Grassmannian or space of Lagrangians, admitting an explicit description via linear equations involving invariant alternating forms (Hivert, 2010). The boundary divisors are indexed by the simple restricted roots of the symmetric pair, and the $G$-orbit stratification follows subset data.

The theory extends to any symmetric space $G/G^\theta$, yielding a unique wonderful compactification with boundary divisor count equal to the rank of $G/G^\theta$ and transverse intersection properties (Corniani et al., 2020). Classical examples include the "complete quadrics" ($SL(n+1)/SO(n+1)$) and their symplectic counterparts ($Sp(2r)/GL(r)$), constructed as sequential blow-ups along secant varieties or rank loci (Corniani et al., 2020).

5. Generalizations: Arrangements, Cartan Subalgebras, and Tropical Geometry

Arrangement Models: For hyperplane arrangements, the wonderful model $Y_\mathcal{B}$ is the closure of the complement in a toric variety dictated by the Bergman/nested sets fan. The boundary has normal crossings, and dominant endomorphisms of the open complement extend uniquely to $Y_\mathcal{B}$ (Kurul et al., 2017, Denham, 2013).

Cartan Subalgebra Compactification: The De Concini–Procesi-style compactification can be constructed for a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, yielding a normal projective variety $\overline{\mathfrak{h}}$ in $({\mathbb{P}}^1)^d$. Boundary components correspond to maximal root subsystems, and the poset of strata matches the intersection lattice for the underlying Coxeter arrangement (Evens et al., 2024).

Tropical and Toric Approaches: The wonderful compactification of an arrangement complement matches the tropical compactification induced by the Bergman fan of the associated matroid, unifying the De Concini–Procesi and tropical perspectives (Kurul et al., 2017, Denham, 2013).

6. Representation Theory, Equivariance, and Applications

The wonderful compactification is equivariant for $(G \times G)$-action, facilitates extension of conjugation and diagonal actions, and supports the parameterization and degeneration of objects central in representation theory. This underpins constructions in:

• Geometric realization of Jacquet functors and perverse sheaf categories (Abe et al., 2011).
• Closures of families of Bethe subalgebras in the Yangian to strata parameterized by Levi subalgebras (Ilin et al., 2018).
• Moduli spaces of principal bundles via universal Hecke modifications, where the compactification provides the natural extension space for meromorphic gauge transformations (Wong, 2010, Wei, 19 May 2025).
• Deligne–Knudsen–Mumford moduli spaces, where the wonderful compactification realizes $\overline{M}_{0,n}$ for the type $A_{n-1}$ arrangement (Denham, 2013).

The Picard group of the wonderful compactification is free abelian of rank $\ell$ generated by the irreducible divisors. The anticanonical bundle is ample, with an explicit expression in terms of the boundary divisors and fundamental weights (Fu et al., 2020).

7. Deformations, Rigidity, and Further Properties

The De Concini–Procesi compactification of an adjoint group is a smooth Fano variety of Picard number $\ell$ and is rigid under Fano deformations: if it appears as a fiber in a regular family of Fano varieties, all fibers are isomorphic to it (Fu et al., 2020). The colored fan structure of the compactification embodies the spherical embedding theory, with faces corresponding to orbits and colored data to boundary divisors.

Further, hd-compactifications in the real case provide real-analytic manifold-with-corners versions whose strata and fiber structures match the complex wonderful case, confirming the universality of the combinatorial and geometric structure across fields (Albin et al., 2019).

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References:

(Ilin et al., 2018, Chen et al., 22 Dec 2025, Wei, 19 May 2025, Evens et al., 2024, Li, 2023, Fu et al., 2020, Corniani et al., 2020, Kurul et al., 2017, Aguirre et al., 2014, Denham, 2013, Wong, 2010, Abe et al., 2011, Gaiffi et al., 2011, Hivert, 2010, Albin et al., 2019)