Wonderful Compactification Research Articles

A wonderful embedding of the loop group

I describe the wonderful compactification of loop groups. These compactifications are obtained by adding normal-crossing boundary divisors to the group LG of loops in a reductive group G (or more accurately, to the semi-direct product C×⋉LG) in a manner equivariant for the left and right C×⋉LG-actions. The analogue for a torus group T is the theory of toric varieties; for an adjoint group G, this is the wonderful compactification of De Concini and Procesi. The loop group analogue is suggested by work of Faltings in relation to the compactification of moduli of G-bundles over nodal curves.

The full automorphism group of [formula omitted

Let G‾ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over C. Let T‾⊂G‾ be the closure of a maximal torus T⊂G. We prove that the group of all automorphisms of the variety T‾ is the semi-direct product NG(T)⋊D, where NG(T) is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if G≠PSL(2,C). Note that if G=PSL(2,C), then T‾=CP1 and so in this case Aut(T‾)=PSL(2,C).

An analogue of Springer fibers in certain wonderful compactifications

We investigate the topological structure of a cellular decomposition of the fixed locus of a unipotent operator of regular Jordan type acting on the wonderful compactification of the variety of complete quadrics and the variety of complete skew forms. The Poincaré polynomial is computed in each case and the poset of cell closures under inclusion is described in the complete quadrics case.

Equivariant -theory of regular compactifications: further developments

We describe the -equivariant -ring of , where is a factorial covering of a connected complex reductive algebraic group , and is a regular compactification of . Furthermore, using the description of , we describe the ordinary -ring as a free module (whose rank is equal to the cardinality of the Weyl group) over the -ring of a toric bundle over whose fibre is equal to the toric variety associated with a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see [1]). We also give an explicit presentation of and as algebras over and respectively, where is the wonderful compactification of the adjoint semisimple group . In the case when is a regular compactification of , we give a geometric interpretation of these presentations in terms of the equivariant and ordinary Grothendieck rings of a canonical toric bundle over .

On a smooth compactification of PSL(n,C)/T

Let T be a maximal torus of PSL(n,C). For n≥4, we construct a smooth compactification of PSL(n,C)/T as a geometric invariant theoretic quotient of the wonderful compactification PSL(n,C)¯ for a suitable choice of T-linearized ample line bundle on PSL(n,C)¯. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of PSL(n,C)/T is PSL(n,C) itself.

Geometry of second adjointness for 𝑝-adic groups

We present a geometric proof of second adjointness for a reductive p p -adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a “cospecialization” map between spaces of functions on various varieties carrying a G × G G\times G action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of endo-functors of the module category lead to the second adjointness. We also get a formula for the “cospecialization” map expressing it as a composition of the orispheric transform and inverse intertwining operator; a parallel result for D D -modules was obtained by Bezrukavnikov, Finkelberg and Ostrik. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra generalizing a result by Opdam.

Compactifications of reductive groups as moduli stacks of bundles

Let$G$be a split reductive group. We introduce the moduli problem ofbundle chainsparametrizing framed principal$G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of$G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple$G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of$GL_{n}$.

Automorphisms of [formula omitted

Let G‾ be the wonderful compactification of a simple affine algebraic group G defined over C such that its center is trivial and G≠PSL(2,C). Take a maximal torus T⊂G, and denote by T‾ its closure in G‾. We prove that T coincides with the connected component, containing the identity element, of the group of automorphisms of the variety T‾.

On equivariant principal bundles over wonderful compactifications

Let G be a simple algebraic group of adjoint type over C, and let M be the wonderful compactification of a symmetric space G/H. Take a G˜-equivariant principal R-bundle E on M, where R is a complex reductive algebraic group and G˜ is the universal cover of G. If the action of the isotropy group H˜ on the fiber of E at the identity coset is irreducible, then we prove that E is polystable with respect to any polarization on M. Further, for wonderful compactification of the quotient of PSL(n,C), n≠4 (respectively, PSL(2n,C), n>1) by the normalizer of the projective orthogonal group (respectively, the projective symplectic group), we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification.

Minimal rational curves on wonderful group compactifications

Consider a simple algebraic group G of adjoint type, and its wonderful compactification X. We show that X admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general point. As an application, we show that X has the target rigidity property when G is not of type A_1 or C.

Equivariant vector bundles on complete symmetric varieties of minimal rank

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by [Formula: see text]. Given a [Formula: see text] equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so.

Algebraic renormalization and Feynman integrals in configuration spaces

This paper continues our previous study of Feynman integrals in configuration spaces and their algebro-geometric and motivic aspects. We consider here both massless and massive Feynman amplitudes, from the point of view of potential theory. We consider a variant of the wonderful compactification of configuration spaces that works simultaneously for all graphs with a given number of vertices and that also accounts for the external structure of Feynman graph. As in our previous work, we consider two version of the Feynman amplitude in configuration space, which we refer to as the real and complex versions. In the real version, we show that we can extend to the massive case a method of evaluating Feynman integrals, based on expansion in Gegenbauer polynomials, that we investigated previously in the massless case. In the complex setting, we show that we can use algebro-geometric methods to renormalize the Feynman amplitudes, so that the renormalized values of the Feynman integrals are given by periods of a mixed Tate motive. The regularization and renormalization procedure is based on pulling back the form to the wonderful compactification and replace it with a cohomologous one with logarithmic poles. A complex of forms with logarithmic poles, endowed with an operator of pole subtraction, determine a Rota-Baxter algebra on the wonderful compactifications. We can then apply the renormalization procedure via Birkhoff factorization, after interpreting the regularization as an algebra homomorphism from the Connes-Kreimer Hopf algebra of Feynman graphs to the Rota-Baxter algebra. We obtain in this setting a description of the renormalization group.We also extend the period interpretation to the case of Dirac fermions and gauge bosons.

Toric and tropical compactifications of hyperplane complements

These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel ' fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.

The Chow Motives of Relative Fulton-Macpherson Space

Suppose that $X$ is a complex nonsingular projective variety and $D$ is a smooth divisor. Compactifications of configuration spaces of distinct and non-distinct $n$ points in $X$ away from $D$ were constructed by the author and B. Kim in "A generalization of Fulton-MacPherson configuration spaces" by using the method of wonderful compactification. In this paper, we give explicit presentations of Chow motives and Chow groups of these configuration spaces.

Stability of the tangent bundle of the wonderful compactification of an adjoint group

Let G be a complex linear algebraic group which is simple of adjoint type. Let \overline G be the wonderful compactification of G . We prove that the tangent bundle of \overline G is stable with respect to every polarization on \overline G .

Equivariant K-theory of flag varieties revisited and related results

In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.

Closures of Orbits Under the Diagonal Action in the Wonderful Compactification of PGL(3)

In this article, we study the orbits under the diagonal action of a semisimple adjoint group G on its wonderful compactification X for the case G = PGL(3) and determine the closure relations between such orbits. Moreover, we show an example in the wonderful compactification of PSp(4) in which the closure of an orbit for the diagonal action consists of infinitely many orbits.

Feynman Integrals and Motives of Configuration Spaces

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of arrangements of subvarieties associated to the subgraphs of a Feynman graph and a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of the variety and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincare' residues. While these residues give mixed Tate periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification, which can be motivically more complicated.

The Reisner–Stanley system and equivariant cohomology for a class of wonderful varieties

In this paper we are going to determine the equivariant cohomology of the wonderful compactification of a symmetric variety G/H and its equivariant ring of conditions under the assumption that rk(H)+rk(G/H)=rk(G).

Semistable locus of a group compactification

In this paper, we consider the diagonal action of a connected semisimple group of adjoint type on its wonderful compactification. We show that the semistable locus is a union of the G G -stable pieces and we calculate the geometric quotient.