Related Compactifications Research Articles

Compactification of level maps of moduli spaces of Drinfeld shtukas

We define Drinfeld level structures for Drinfeld shtukas of any rank and show that their moduli spaces are regular and admit finite flat level maps. In particular, the moduli space of Drinfeld shtukas with Drinfeld Γ0(pn)-level structures provides a good integral model and a relative compactification of the moduli space of shtukas with naive Γ0(pn)-level defined using shtukas for dilated group schemes.

Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-etale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer Z of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification Z‾ by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this relative compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

A resolution of singularities for Drinfeld’s compactification by stable maps

Drinfeld’s relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld’s compactification is universally locally acyclic over the moduli stack of G G -bundles at points sufficiently antidominant relative to their defect.

Hochschild Cohomology of Torus Equivariant D-modules

Abstract We discuss the Hochschild cohomology of the category of D-modules associated to an algebraic stack. In particular we describe the Hochschild cohomology of the category of torus-equivariant D-modules as the cohomology of a D-module on the loop space of the quotient stack. Finally, we give an approach for understanding the Hochschild cohomology of D-modules on general stacks via a relative compactification of the diagonal. This work is motivated by a desire to understand the support theory (in the sense of [6]) of D-modules on stacks.

Degeneration of groups of type E 7 and minimal coupling in supergravity

Abstract We study properties of D = 4 $$ \mathcal{N} \geqslant {2} $$ extended supergravities (and related compactifications of superstring theory) and their consistent truncation to the phenomenologically interesting models of $$ \mathcal{N} = {1} $$ supergravity. This involves a detailed classification of the “degenerations” of the duality groups of type E 7, when the corresponding quartic invariant polynomial built from the symplectic irreducible representation of G4 “degenerates” into a perfect square. With regard to cosmological applications, minimal coupling of vectors in consistent truncation to $$ \mathcal{N} = {1} $$ from higher-dimensional or $$ {\text{higher}} - \mathcal{N} $$ theory is non-generic. On the other hand, non-minimal coupling involving vectors coupled to scalars and axions is generic. These features of supergravity, following from the electric-magnetic duality, may be useful in other applications, like stabilization of moduli, and in studies of non-perturbative black-hole solutions of supergravity/string theory.

Creation of matter in the universe and groups of type E7

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

Almost periodic compactifications of semidirect products of flows

Let \(S\tilde \times T\) be a semidirect product of semitopological semigroups S and T. If S and T act on topological spaces X and Y, respectively, then under suitable conditions there is a natural action of \(S\tilde \times T\) on X × Y. In this paper we characterize the almost periodic and strongly almost periodic compactification of the flow (\(S\tilde \times T\), X × Y) in terms of related compactifications of (S, X) and (T, Y).

Geometric Eisenstein series

The purpose of this of this paper is to develop the theory of Eisenstein series in the framework of geometric Langlands correspondence. Our construction is based on the study of certain relative compactification of the moduli stack of parabolic bundles on a curve suggested by V.Drinfeld. As an application we construct certain automorphic forms for global fields of positive characteristic, whose existence is non-obvious from the point of view of classical tools.

Alternative superstring compactifications will not give new low-energy models

Alternatives to Calabi-Yau and related compactifications, such as orbifolds with nonabelian space groups, could in principle yield low-energy models based on a GUT group different from E 6, SO(10) or SU(5). Such models with chiral fermion generations, trilinear superpotential terms yielding quark and lepton masses as well as Higgs mixing, pertubative gauge couplings, and at least one extra U(1) beyond the standard model are looked for. The only viable model based on an SU(6) GUT group is equivalent to the minimal rank-5 E 6 model studied previously. There are no acceptable models based on the GUT groups SU( N⩾7) or SO(14). When combined with previous arguments disfavouring SU(5) and SO(10) GUT groups, as well aṡ rank-6 E 6 models, the present results point to a unique low-energy SU(3) c×SU(2) L×U(1) Y×U(1) E model inspired by the superstring.

The etale homotopy theory of a geometric fibration

A geometric fibration, f:X→Y, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y,\((X_y )_{\hat et} \), is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(fetr)^. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple F→E→B, the action of πl(F) on the homotopy type of various covering spaces of F extends to an action of πl(E).