Searcharxiv⌕ Search

arXiv subjects

Colleen Robles

Publications and source records attributed to Colleen Robles.

At least 19 recordsLinked to original sources
arXiv2025

Period maps at infinity

Let $\overline{B}$ be a smooth projective varieity, and $Z \subset \overline{B}$ a simple normal crossing divisor. Assume that $B = \overline{B} - Z$ admits a variation of pure, polarized Hodge structure. The divisor $Z$ is naturally stratified, and Schmid's nilpotent orbit theorem defines a family/variation of nilpotent orbits along each strata. We study the rich geometric structure encoded by this family, its relationship to the induced (quotient) variation of pure Hodge structure on the strata, and establish a relationship between the extension data in the nilpotent orbits and the normal bundles of the smooth irreducible components of $Z$.

math.AG↗
arXiv2023

Completion of two-parameter period maps by nilpotent orbits

We show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds, and the map onto the image is a morphism of compact algebraic spaces. This result also applies to the case of mixed period maps and we use it to give a construction of generalized N\`eron models.

math.AG↗
arXiv2023

Pseudoconvexity at infinity in Hodge theory: a codimension one example

The generalization of the Satake--Baily--Borel compactification to arbitrary period maps has been reduced to a certain extension problem on certain "neighborhoods at infinity". Extension problems of this type require that the neighborhood be pseudoconvex. The purpose of this note is to establish the desired pseudoconvexity in one relatively simple, but non-trivial, example: codimension one degenerations of a period map of weight two Hodge structures with first Hodge number $h^{2,0}$ equal to 2.

math.AG↗
arXiv2023

Extension of Hodge norms at infinity

It is a long-standing problem in Hodge theory to generalize the Satake--Baily--Borel (SBB) compactification of a locally Hermitian symmetric space to arbitrary period maps. A proper topological SBB-type completion has been constructed, and the problem of showing that the construction is algebraic has been reduced to showing that the compact fibres A of the completion admit neighborhoods X satisfying certain properties. All but one of those properties has been established; the outstanding problem is to show that holomorphic functions on certain divisors "at infinity" extend to X. Extension theorems of this type require that the complex manifold X be pseudoconvex; that is, admit a plurisubharmonic exhaustion function. The neighborhood X is stratified, and the strata admit Hodge norms which are may be used to produce plurisubharmonic functions on the strata. One would like to extend these norms to X so that they may be used to construct the desired plurisubharmonic exhaustion of X. The purpose of this paper is show that there exists a function that simultaneously extends all the Hodge norms along the strata that intersect the fibre A nontrivially.

math.AG↗
arXiv2020

Analog of Satake-Baily-Borel for period maps

We propose an analog of the Satake--Baily--Borel compactification and Borel's extension theorem for arbitrary period maps. The proposed analog is constructed as a proper topological completion of the period map. It is conjectured that the construction is projective algebraic, and the conjecture is reduced to a certain extension problem.

math.AG↗
arXiv2020

Hodge Representations

Hodge representations were introduced by Green-Griffiths-Kerr to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains of a period domain. The purpose of this article is to provide an exposition of how, given a fixed period domain $\mathcal{D}$, to enumerate the Hodge representations corresponding to Mumford-Tate subdomains $D \subset \mathcal{D}$. After reviewing the well-known classical cases that $\mathcal{D}$ is Hermitian symmetric (weight $n=1$, and weight $n=2$ with $p_g = h^{2,0}=1$), we illustrate this in the case that $\mathcal{D}$ is the period domain parameterizing polarized Hodge structures of (effective) weight two Hodge structures with first Hodge number $p_g = h^{2,0} = 2$. We also classify the Hodge representations of Calabi-Yau type, and enumerate the horizontal representations of CY 3-fold type. (The "horizontal" representations those with the property that corresponding domain $D \subset \mathcal{D}$ satisfies the infinitesimal period relation, a.k.a. Griffiths' transversality, and is therefore Hermitian.)

math.AG↗
arXiv2019

The LLV decomposition of hyper-Kaehler cohomology

Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-K\"ahler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact K\"ahler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-K\"ahler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $b_2(X)$ of hyper-K\"ahler manifolds. Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known G\"ottsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-K\"ahler manifolds. For the two exceptional cases of O'Grady we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-K\"ahler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-K\"ahler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-K\"ahler manifolds. More consequentially, we note that Nagai's conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-K\"ahler manifolds.

math.AG↗
arXiv2017

Period mappings and properties of the augmented Hodge line bundle

Let $P$ be the image of a period map. We discuss progress towards a conjectural Hodge theoretic completion $\overline{P}$, an analogue of the Satake-Baily-Borel compactification in the classical case. The set $\overline{P}$ is defined and given the structure of a compact Hausdorff topological space. We conjecture that it admits the structure of a compact complex analytic variety. We verify this conjecture when $\mathrm{dim} P \le 2$. In general, $\overline{P}$ admits a finite cover $\overline{S}$ (also a compact Hausdorff space, and constructed from Stein factorizations of period maps). Assuming that $\overline{S}$ is a compact complex analytic variety, we show that a lift of the augmented Hodge line bundle $\Lambda$ extends to an ample line bundle, giving $\overline{P}$ the structure of a projective normal variety. Our arguments rely on refined positivity properties of Chern forms associated to various Hodge bundles; properties that might be of independent interest.

math.AG↗
arXiv2017

Polarized relations on horizontal SL(2)s

We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D which is compatible with natural partial orders on the sets of nilpotent orbits in the corresponding Lie algebra and boundary orbits in the compact dual. A generalization of the SL(2)-orbit theorem to such domains leads to an algorithm for computing this relation, which is worked out in several examples and special cases including period domains, Hermitian symmetric domains, and complete flag domains, and used to define a poset of equivalence classes of multivariable nilpotent orbits on D.

math.AG↗
arXiv2016

Classification of smooth horizontal Schubert varieties

We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety $G/P$ are homogeneously embedded cominuscule $G'/P'$, and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule $G/P$.

math.AG↗
arXiv2014

Variations of Hodge structure and orbits in flag varieties

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths--Yukawa coupling to the variety of lines on $G/P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$--orbits in $G/P$, and compute the associated Hodge--theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four.

math.AG↗
arXiv2013

Quotients of non-classical flag domains are not algebraic

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.

math.AG↗
arXiv2013

Flexibility of Schubert classes

In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [R, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.

math.AG↗
arXiv2012

Singular loci of cominuscule Schubert varieties

Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) I give a uniform description (that is, independent of type) of the irreducible components of the singular locus of a Schubert variety Y in X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties by an non-negative integer A and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if A=0; (2) if G of Type ADE, then the singular locus occurs in codimension at least three.

math.AG↗