About

Colleen M Robles is a geometer whose current research addresses the complex geometry of period maps and related questions in Hodge theory and its applications to the moduli of algebraic varieties. She has also contributed to the fields of Finsler geometry, calibrated geometry, and complex projective geometry. Her work includes exploring the properties of period maps at infinity, their completions, and the extension problems associated with neighborhoods at infinity in Hodge theory. Robles holds the position of Professor of Mathematics at Duke University, a role she has occupied since 2020, and she is also the Assistant Director of the Rhodes Information Initiative at Duke since 2024. Her academic background includes a Ph.D. from the University of British Columbia in 2003. Her research has been supported by grants from the National Science Foundation, focusing on complex geometric properties of period maps and their Lie theoretic aspects. She has published on topics such as the completion of two-parameter period maps and pseudoconvexity at infinity in Hodge theory, contributing significantly to the understanding of the geometric and analytic aspects of period maps and Hodge structures.

Research topics

• Mathematics
• Pure mathematics
• Mathematical analysis
• Geometry

Selected publications

• Pseudoconvexity at Infinity in Hodge Theory: A Codimension One Example

  Simons symposia · 2025-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Period maps at infinity

  ArXiv.org · 2025-09-10

  preprintOpen accessSenior author

  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 $Z$.

  PublisherOA PDFDOI

• Pseudoconvexity at infinity in Hodge theory: a codimension one example

  arXiv (Cornell University) · 2023-02-09

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Completion of two-parameter period maps by nilpotent orbits

  arXiv (Cornell University) · 2023-12-01

  preprintOpen accessSenior author

  We show that every two-parameter period map admits a Kato--Nakayama--Usui completion to a morphism of log manifolds.

  PublisherOA PDFDOI

• Extension of Hodge norms at infinity

  arXiv (Cornell University) · 2023-02-08

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Natural line bundles on completions of period mappings

  arXiv (Cornell University) · 2021-02-11 · 1 citations

  preprintOpen accessSenior author

  We give conditions under which natural lines bundles associated with completions of period mappings are semi-ample and ample.

  PublisherOA PDFDOI

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

  Mathematische Annalen · 2021 · 8 citations

  Senior authorCorresponding
  • Mathematics
  • Pure mathematics
  • Geometry
  PublisherDOI

• Completions of Period Mappings: Progress Report

  arXiv (Cornell University) · 2021-06-08

  preprintOpen accessSenior author

  We give an informal, expository account of a project to construct completions of period maps.

  Publisher

• Completions of period mappings: progress report

  arXiv (Cornell University) · 2021-06-08

  preprintOpen accessSenior author

  We give an informal, expository account of a project to construct completions of period maps.

  PublisherOA PDFDOI

• Towards a maximal completion of a period map

  arXiv (Cornell University) · 2020-10-13 · 1 citations

  preprintOpen accessSenior author

  The motivation behind this work is to construct a Hodge theoretically maximal completion of a period map. This is done up to finite data (we work with the Stein factorization of the period map). The image of the extension is a Moishezon variety that compactifies a finite cover of the image of the period map.

  Publisher

Recent grants

• Hodge Theory and Representation Theory

  NSF · $67k · 2015–2016

• Hodge Theory and Representation Theory

  NSF · $143k · 2013–2015

• Complex Geometric and Lie Theoretic Aspects of Hodge Theory

  NSF · $182k · 2016–2022

• Complex Geometric and Lie Theoretic Aspects of Hodge Theory

  NSF · $224k · 2019–2023

Frequent coauthors

• David Bao

  19 shared

• Zhongmin Shen

  17 shared

• Matt Kerr

  17 shared

• Phillip Griffiths

  16 shared

• Mark Green

  University of Liverpool

  14 shared

• Gregory Pearlstein

  University of Pisa

  11 shared

• J. M. Landsberg

  9 shared

• Radu Laza

  8 shared

Education

• PhD, Mathematics

  University of British Columbia

  2003

• MS, Mathematics

  University of Washington

  1998

• BA, Mathematics

  Smith College

  1996

Awards & honors

• RTG: Linked via L-functions: training versatile researchers…
• Complex Geometric Properties of Period Maps Research Princip…
• Complex Geometric and Lie Theoretic Aspects of Hodge Theory…