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Colleen M Robles

Professor of Mathematics
Mathematics
Office hours Spring 2024: Tue + Thu, 1:00 - 2:30 PM  

Selected Publications


Completion of two-parameter period maps by nilpotent orbits

Preprint · December 1, 2023 We show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds. ... Open Access Link to item Cite

Pseudoconvexity at infinity in Hodge theory: a codimension one example

Preprint · February 9, 2023 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. ... Link to item Cite

Extension of Hodge norms at infinity

Preprint · February 8, 2023 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 s ... Link to item Cite

The LLV decomposition of hyper-Kähler cohomology (the known cases and the general conjectural behavior)

Journal Article Mathematische Annalen · April 1, 2022 Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold X admits a natural action by the Lie algebra so(4 , b2(X) - 2) , generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we deter ... Full text Cite

Hodge Representations

Journal Article Experimental Results · November 17, 2020 Green-Griffiths-Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains. We summarize how, given a fixed period domain, to enumerate the Hodge representations and corre ... Full text Cite

The global asymptotic structure of period mappings

Journal Article · October 13, 2020 This work is part of a project to construct completions of period mappings. A proper topological SBB-esque completion is constructed. The fibres of are projective varieties, and the image is a union of quasi-projective varieties; one wants to endow the top ... Link to item Cite

Polarized relations on horizontal SL(2)'s

Journal Article Documenta Mathematica · January 1, 2019 We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D. The relation answers the question when is one ℝ-split polarized mixed Hodge structure more singular/degenerate than another? The relation is compatible with natur ... Full text Cite

Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms

Journal Article Mathematische Annalen · August 1, 2018 Sheng and Zuo’s characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross’s canonical variations of Hodge structure of Calabi–Yau type over (Hermitian symmetric) tube domains. ... Full text Cite

Period mappings and properties of the augmented Hodge line bundle

Journal Article · August 30, 2017 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 structu ... Link to item Cite

Variations of Hodge structure and orbits in flag varieties

Journal Article Advances in Mathematics · July 31, 2017 Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford–Tate domains, arise as open GR-orbits in flag varieties G/P. We investigate Hodge-theoretic aspects of the geometry and representation theory associat ... Full text Cite

Classification of smooth horizontal Schubert varieties

Journal Article European Journal of Mathematics · June 1, 2017 We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety G / P are homogeneously embedded cominuscule [InlineEquation not available: see fulltext.], and are classified by subdiagrams of a Dynkin diagram. This generalizes t ... Full text Cite

Degenerations of Hodge structure

Journal Article Proceedings of Symposia in Pure Mathematics · January 1, 2017 Two interesting questions in algebraic geometry are: (i) how can a smooth projective variety degenerate? and (ii) given two such degenerations, when can we say that one is “more singular/degenerate“ than the other? Schmid's Nilpotent Orbit Theorem yields H ... Full text Cite

Classification of horizontal SL(2)s

Journal Article Compositio Mathematica · May 1, 2016 We classify the horizontal s and-split polarized mixed Hodge structures on a Mumford-Tate domain. ... Full text Cite

Nilpotent cones and their representation theory

Journal Article · January 31, 2016 We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subg ... Link to item Cite

Characteristic cohomology of the infinitesimal period relation

Journal Article Asian Journal of Mathematics · January 1, 2016 The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system ... Full text Cite

Nilpotent cones and adjoint orbits

Journal Article · May 13, 2014 A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain. ... Link to item Cite

Singular loci of cominuscule Schubert varieties

Journal Article Journal of Pure and Applied Algebra · April 1, 2014 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 Sc ... Full text Cite

Extremal degenerations of polarized Hodge structures

Journal Article · March 3, 2014 We describe a Hodge theoretic approach to the question: In what ways can a smooth projective variety degenerate? ... Link to item Cite

Completion of two-parameter period maps by nilpotent orbits

Preprint · December 1, 2023 We show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds. ... Open Access Link to item Cite

Pseudoconvexity at infinity in Hodge theory: a codimension one example

Preprint · February 9, 2023 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. ... Link to item Cite

Extension of Hodge norms at infinity

Preprint · February 8, 2023 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 s ... Link to item Cite

The LLV decomposition of hyper-Kähler cohomology (the known cases and the general conjectural behavior)

Journal Article Mathematische Annalen · April 1, 2022 Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold X admits a natural action by the Lie algebra so(4 , b2(X) - 2) , generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we deter ... Full text Cite

Hodge Representations

Journal Article Experimental Results · November 17, 2020 Green-Griffiths-Kerr introduced Hodge representations to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains. We summarize how, given a fixed period domain, to enumerate the Hodge representations and corre ... Full text Cite

The global asymptotic structure of period mappings

Journal Article · October 13, 2020 This work is part of a project to construct completions of period mappings. A proper topological SBB-esque completion is constructed. The fibres of are projective varieties, and the image is a union of quasi-projective varieties; one wants to endow the top ... Link to item Cite

Polarized relations on horizontal SL(2)'s

Journal Article Documenta Mathematica · January 1, 2019 We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D. The relation answers the question when is one ℝ-split polarized mixed Hodge structure more singular/degenerate than another? The relation is compatible with natur ... Full text Cite

Characterization of Calabi–Yau variations of Hodge structure over tube domains by characteristic forms

Journal Article Mathematische Annalen · August 1, 2018 Sheng and Zuo’s characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross’s canonical variations of Hodge structure of Calabi–Yau type over (Hermitian symmetric) tube domains. ... Full text Cite

Period mappings and properties of the augmented Hodge line bundle

Journal Article · August 30, 2017 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 structu ... Link to item Cite

Variations of Hodge structure and orbits in flag varieties

Journal Article Advances in Mathematics · July 31, 2017 Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford–Tate domains, arise as open GR-orbits in flag varieties G/P. We investigate Hodge-theoretic aspects of the geometry and representation theory associat ... Full text Cite

Classification of smooth horizontal Schubert varieties

Journal Article European Journal of Mathematics · June 1, 2017 We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety G / P are homogeneously embedded cominuscule [InlineEquation not available: see fulltext.], and are classified by subdiagrams of a Dynkin diagram. This generalizes t ... Full text Cite

Degenerations of Hodge structure

Journal Article Proceedings of Symposia in Pure Mathematics · January 1, 2017 Two interesting questions in algebraic geometry are: (i) how can a smooth projective variety degenerate? and (ii) given two such degenerations, when can we say that one is “more singular/degenerate“ than the other? Schmid's Nilpotent Orbit Theorem yields H ... Full text Cite

Classification of horizontal SL(2)s

Journal Article Compositio Mathematica · May 1, 2016 We classify the horizontal s and-split polarized mixed Hodge structures on a Mumford-Tate domain. ... Full text Cite

Nilpotent cones and their representation theory

Journal Article · January 31, 2016 We describe two approaches to classifying the possible monodromy cones C arising from nilpotent orbits in Hodge theory. The first is based upon the observation that C is contained in the open orbit of any interior point N in C under an associated Levi subg ... Link to item Cite

Characteristic cohomology of the infinitesimal period relation

Journal Article Asian Journal of Mathematics · January 1, 2016 The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system ... Full text Cite

Nilpotent cones and adjoint orbits

Journal Article · May 13, 2014 A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain. ... Link to item Cite

Singular loci of cominuscule Schubert varieties

Journal Article Journal of Pure and Applied Algebra · April 1, 2014 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 Sc ... Full text Cite

Extremal degenerations of polarized Hodge structures

Journal Article · March 3, 2014 We describe a Hodge theoretic approach to the question: In what ways can a smooth projective variety degenerate? ... Link to item Cite

Schubert varieties as variations of Hodge structure

Journal Article Selecta Mathematica, New Series · January 1, 2014 We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS 'span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to th ... Full text Cite

Quotients of non-classical flag domains are not algebraic

Journal Article Algebraic Geometry · January 1, 2014 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 for Γ an infinite, finitely generated d ... Full text Cite

Flexibility of Schubert classes

Journal Article Differential Geometry and its Application · December 1, 2013 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 [22, Theorem 3.1] by proving that every positive mul ... Full text Cite

Schur flexibility of cominuscule Schubert varieties

Journal Article Communications in Analysis and Geometry · December 1, 2013 Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class ξ is Schur rigid if the only irreducible subvarieties Y X with homology class [Y] ε Zξ are Sch ... Full text Cite

Projective invariants of CR-hypersurfaces

Journal Article Complex Variables and Elliptic Equations · November 1, 2013 We study the equivalence problem under projective transformation for CR-hypersurfaces of complex projective space. A complete set of projective differential invariants for analytic hypersurfaces is given. The self-dual strongly ℂ-linearly convex hypersurfa ... Full text Cite

Fubini-griffiths-harris rigidity of homogeneous varieties

Journal Article International Mathematics Research Notices · January 1, 2013 Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given. © 2012 The Author(s) 2012. ... Full text Cite

Rigid Schubert varieties in compact Hermitian symmetric spaces

Journal Article Selecta Mathematica, New Series · August 1, 2012 Given a singular Schubert variety X w in a compact Hermitian symmetric space X, it is a long-standing question to determine when X w is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to ... Full text Cite

Fubini-Griffiths-Harris rigidity and lie algebra cohomology

Journal Article Asian Journal of Mathematics · January 1, 2012 We prove a rigidity theorem for represented semi-simple Lie groups. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in ℙg) is extrinsically rigid to third order (with the exception of g = a ... Full text Cite

Parallel calibrations and minimal submanifolds

Journal Article Illinois Journal of Mathematics · January 1, 2012 Given a parallel calibration φ ∈ Ωp(M) on a Riemannian manifold M, I prove that the φ-critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the φ-critical submanifolds are precisely the integral manifolds of a C∞(M)-l ... Full text Cite

Lines and osculating lines of hypersurfaces

Journal Article Journal of the London Mathematical Society · January 1, 2010 We define systems of partial differential equations that govern the infinitesimal variation of lines, and osculating lines, through a point of a hypersurface in projective space. The work answers questions posed by J.-M. Hwang. © 2010 London Mathematical S ... Full text Cite

Fubini's theorem in codimension two

Journal Article Journal fur die Reine und Angewandte Mathematik · June 1, 2009 We classify codimension two complex submanifolds of projective space X n ⊂ having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study o ... Full text Cite

Calibrated associative and Cayley embeddings

Journal Article Asian Journal of Mathematics · January 1, 2009 Using the Cartan-Kähler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G2-manifold as an associative submanifold. Second, the inter ... Full text Cite

The adjoint variety of SLm + 1 C is rigid to order three

Journal Article Differential Geometry and its Application · December 1, 2008 I prove that the adjoint variety of SLm + 1 C in P (slm + 1 C) is rigid to order three. © 2008 Elsevier B.V. All rights reserved. ... Full text Cite

Lines on hypersurfaces

Journal Article · May 16, 2008 This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a system of partial differential equations which we describe explicitly. ... Link to item Cite

Geodesics in randers spaces of constant curvature

Journal Article Transactions of the American Mathematical Society · April 1, 2007 Geodesics in Randers spaces of constant curvature are classified. © 2006 American Mathematical Society. ... Full text Cite

Geodesics in Randers spaces of constant curvature

Journal Article TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY · January 1, 2007 Link to item Cite

Zermelo navigation on riemannian manifolds

Journal Article Journal of Differential Geometry · January 1, 2004 In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry, namely the complete classification of strongly convex Randers metrics of constant flag curvature. © 2003 Applied Probabili ... Full text Cite

On Randers spaces of constant flag curvature

Journal Article Reports on Mathematical Physics · January 1, 2003 This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constan ... Full text Cite