Orbit Closures Research Articles

EQUIVARIANT CHOW CLASSES OF MATRIX ORBIT CLOSURES

Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is in the variety.

GL+(2, ℝ)–orbits in Prym eigenform loci

This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The proof uses several topological properties of Prym eigenforms, which are proved by the authors in a previous work. In particular the tools and the proof are independent of the recent results of Eskin-Mirzakhani-Mohammadi. As an application we obtain a finiteness result for the number of closed GL^+(2,R)-orbits (not necessarily primitive) in the Prym eigenform locus Prym_D(2,2) for any fixed D that is not a square.

Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component {\mathcal H^{\mathrm{odd}}(4)} the only {GL^+(2,\mathbb R)} orbit closures are closed orbits, the Prym locus {\tilde{\mathcal{Q}}(3,-1^3)} , and {\mathcal H^{\mathrm{odd}}(4)} . Together with work of Matheus–Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in \mathcal H^{\mathrm{odd}}(4) outside of the Prym locus.

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective–injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

On quantization of a nilpotent orbit closure in $G_2$

On quantization of a nilpotent orbit closure in $G_2$

POLYNOMIALS FOR SYMMETRIC ORBIT CLOSURES IN THE FLAG VARIETY

In [Wyser-Yong '13] we introduced polynomial representatives of cohomology classes of orbit closures in the flag variety, for the symmetric pair $(GL_{p+q}, GL_p \times GL_q)$. We present analogous results for the remaining symmetric pairs of the form $(GL_n,K)$, i.e., $(GL_n,O_n)$ and $(GL_{2n},Sp_{2n})$. We establish the well-definedness of certain representatives from [Wyser '13]. It is also shown that the representatives have the combinatorial properties of nonnegativity and stability. Moreover, we give some extensions to equivariant $K$-theory.

The Bruhat order on clans

We give an explicit description of the closure containment order (or "Bruhat order") on the set of orbits of $$GL_p \times GL_q$$GLp×GLq on the flag variety $$GL_{p+q}/B$$GLp+q/B, relative to the parametrization of the orbits by combinatorial objects called "clans". This leads to a corresponding description of the closures of such orbits as sets of flags satisfying certain incidence conditions encoded by the parametrizing clans.

JOSEPH IDEALS AND LISSE MINIMAL -ALGEBRAS

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite)$W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Projective normality of model varieties and related results

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M M or over a closed G G -stable subvariety of M M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E 8 \mathsf E_8 .

Jet schemes of the closure of nilpotent orbits

We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Musta\u{t}\c{a} (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of these nilpotent orbit closures.

On the normality of orbit closures which are hypersurfaces

Let N be a quiver representation with non-zero admissible annihilator. In this paper, we prove the normality of the orbit closure \(\bar {\mathcal {O}}_{N}\) when it is a hypersurface. The result thus gives new examples of normal orbit closures of quiver representations.

Asymptotic average shadowing property, almost specification property and distributional chaos

It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property.

The distribution ofS-integral points on SL2-orbit closures of binary forms

We study the distribution of $S$-integral points on $\mathrm{SL}_2$-orbit closures of binary forms and prove an asymptotic formula for the number of $S$-integral points of bounded height on $\mathrm{SL}_2$-orbit closures of binary forms. This extends a result of Duke, Rudnick, and Sarnak. The main ingredients of the proof are the method of mixing developed by Eskin-McMullen and Benoist-Oh, Chambert-Loir-Tschinkel's study of asymptotic volume of height balls, and Hassett-Tschinkel's description of log resolutions of $\mathrm{SL}_2$-orbit closures of binary forms.

Orbit closures and rational surfaces

In this paper we study the Grassmannian of submodules of a given dimension inside the radical of a finitely generated projective module P for a finite dimensional algebra Λ over an algebraically closed field. The orbit of such a submodule C under the action of AutΛ(P) on the Grassmannian encodes information on the top-stable degenerations of P/C. The goal of this article is to begin the study of the global geometry of the closures of such orbits. In dimension one, this geometry is determined by the local rings of singular points. The smallest dimension for which the global geometry is not determined by local data is two, and this case is our main focus. We give several examples to illustrate the interplay between the geometry of the projective surfaces which arise and the corresponding posets of top-stable degenerations.

Affine Manifolds and Zero Lyapunov Exponents in Genus 3

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichm\"uller curve in genus three has two zero Lyapunov exponents in the Kontsevich-Zorich cocycle, then it lies in the principal stratum and has at most quadratic trace field. Moreover, there can be at most finitely many such Teichm\"uller curves.

Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space

We prove results about orbit closures and equidistribution for the SL(2;R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent ows. The proofs of the main theorems rely on the measure classication theorem of the rst two authors and a certain isolation property of closed SL(2;R) invariant manifolds developed in this paper.

Normality of orthogonal and symplectic nilpotent orbit closures in positive characteristic

In this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic. We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure, and conversely if the closure is normal, then any type e minimal irreducible degeneration does not occur in it. Here, the minimal irreducible degenerations of a nilpotent orbit are introduced by W. Hesselink in [7] (or see [11] from which we take Table 1 for the complete list of all minimal irreducible degenerations). Our result is a weak version in positive characteristic of [11, Theorem 16.2(ii)], one of the main results of [11] over complex numbers.

Crepant Resolutions of a Slodowy Slice in a Nilpotent Orbit Closure in $\mathfrak {sl}_N(\mathbb C)$

One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in \mathfrak {sl}_N(\mathbb C) can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.

On orbit closures of groups

We establish some results about orbit closures of permutation groups and their relationship to the automorphism groups of power-set graphs with colorings. One of our theorems proves that an arbitrary group can be realized as the automorphism group of a colored power-set graph.

Translation surfaces and their orbit closures: An introduction for a broad audience

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in the study of various basic dynamical systems. They can also be defined as differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL (2, \mathbb R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming from diverse areas of mathematics. Many areas of mathematics, from algebraic geometry and number theory, to dynamics and topology, can be brought to bear on this topic, and known examples of orbit closures are interesting from all these points of view. This survey is an invitation for mathematicians from different backgrounds to become familiar with the subject. Little background knowledge, beyond the definition of a Riemann surface and its cotangent bundle, is assumed, and top priority is given to presenting a view of the subject that is at once accessible and connected to many areas of mathematics.