We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.
We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety Crit$^{\ge0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to define a boundary stratification of Crit$^{\ge0}_f$. For the case of the top-dimensional positroid cell, we show that the totally nonnegative critical variety Crit$^{\ge0}_{k,n}$ is homeomorphic to the second hypersimplex $Δ_{2,n}$.
The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we p…
Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible …
These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes.
We consider the convex geometry of the cone of nonnegative quadratics over Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of coordinate planes) are amongst the simplest real projective varieties, so this is potentially a starting point that can generalize to more complicated real projective varieties. This subject has some suprising connections to algebraic topology and category theory, which we exploit heavily in our work. These questions are also valuable in applied math, because they directly translate to questions about positive semidefinite (PSD) matrices. In parti…
We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and…
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
An algebraic subvariety Z of A_g is totally geodesic if it is the image via the natural projection map of some totally geodesic submanifold X of the Siegel space. We say that X is the symmetric space uniformizing Z. In this paper we determine which symmetric space uniformizes each of the low genus counterexamples to the Coleman-Oort conjecture obtained studying Galois covers of curves. It is known that the counterexamples obtained via Galois covers of elliptic curves admit two fibrations in totally geodesic subvarieties. The second result of the paper studies the relationship between these fib…
This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin-Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties…
We consider an arbitrary int-amplified surjective endomorphism $f$ of a normal projective variety $X$ over $\mathbb{C}$ and its $f^{-1}$-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that $X$ has at worst Kawamata log terminal singularities. We prove that the total number of $f^{-1}$-stable prime divisors has an optimal upper bound $\dim X+ρ(X)$, where $ρ(X)$ is the Picard number. Also, we give a sufficient condition for $X$ to be rationally connected and simply connected. Finally, by running…
We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places for $1\leq e <d$. Recently, these cycles were constructed by Kudla and Rosu-Yott and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson-Bloch conjecture on the injectivity of the higher Abel-Jacobi map, the generating serie…
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.
We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinato…
This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1$, examples of manifolds/Shimura varieties with precisely $n$ commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic $3$-manifolds and quaternionic Shimura surfaces, wher…
Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).
A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of $\dim(X)+1$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence c…
We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connecti…
