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Publications of Colleen M Robles    :recent first  alphabetical  combined listing:

%% Papers Published   
@article{fds302461,
   Author = {Bao, D and Robles, C},
   Title = {On Randers spaces of constant flag curvature},
   Journal = {Reports on Mathematical Physics},
   Volume = {51},
   Number = {1},
   Pages = {9-42},
   Publisher = {Elsevier BV},
   Year = {2003},
   Month = {January},
   ISSN = {0034-4877},
   url = {http://dx.doi.org/10.1016/S0034-4877(03)80002-2},
   Abstract = {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 constant flag curvature. For n = 2, we focus
             on those whose flag curvature is a (possibly constant)
             function of position only. We characterize such metrics by
             three efficient conditions. With the help of examples in 2
             and 3 dimensions, we deduce that the Yasuda-Shimada
             classification of Randers space forms actually addresses
             only a special case. The corrected classification for that
             special case is sharp, holds for n ≥ 2, and follows
             readily from our three necessary and sufficient
             conditions.},
   Doi = {10.1016/S0034-4877(03)80002-2},
   Key = {fds302461}
}

@article{fds302462,
   Author = {Bao, D and Robles, C and Shen, Z},
   Title = {Zermelo navigation on riemannian manifolds},
   Journal = {Journal of Differential Geometry},
   Volume = {66},
   Number = {3},
   Pages = {377-435},
   Publisher = {International Press of Boston},
   Year = {2004},
   Month = {January},
   ISSN = {0022-040X},
   url = {http://dx.doi.org/10.4310/jdg/1098137838},
   Abstract = {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 Probability Trust.},
   Doi = {10.4310/jdg/1098137838},
   Key = {fds302462}
}

@article{fds340295,
   Author = {Robles, C},
   Title = {Geodesics in Randers spaces of constant curvature},
   Journal = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL
             SOCIETY},
   Volume = {359},
   Number = {4},
   Pages = {1633-1651},
   Publisher = {AMER MATHEMATICAL SOC},
   Year = {2007},
   Month = {January},
   Key = {fds340295}
}

@article{fds302463,
   Author = {Robles, C},
   Title = {Geodesics in randers spaces of constant curvature},
   Journal = {Transactions of the American Mathematical
             Society},
   Volume = {359},
   Number = {4},
   Pages = {1633-1651},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2007},
   Month = {April},
   ISSN = {0002-9947},
   url = {http://dx.doi.org/10.1090/S0002-9947-06-04051-7},
   Abstract = {Geodesics in Randers spaces of constant curvature are
             classified. © 2006 American Mathematical
             Society.},
   Doi = {10.1090/S0002-9947-06-04051-7},
   Key = {fds302463}
}

@article{fds361674,
   Author = {Landsberg, JM and Robles, C},
   Title = {Lines on hypersurfaces},
   Year = {2008},
   Month = {May},
   Abstract = {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.},
   Key = {fds361674}
}

@article{fds302464,
   Author = {Robles, C},
   Title = {The adjoint variety of SLm + 1 C is rigid to
             order three},
   Journal = {Differential Geometry and its Application},
   Volume = {26},
   Number = {6},
   Pages = {683-696},
   Publisher = {Elsevier BV},
   Year = {2008},
   Month = {December},
   ISSN = {0926-2245},
   url = {http://dx.doi.org/10.1016/j.difgeo.2008.04.017},
   Abstract = {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.},
   Doi = {10.1016/j.difgeo.2008.04.017},
   Key = {fds302464}
}

@article{fds302465,
   Author = {Robles, C and Salur, S},
   Title = {Calibrated associative and Cayley embeddings},
   Journal = {Asian Journal of Mathematics},
   Volume = {13},
   Number = {3},
   Pages = {287-306},
   Publisher = {International Press of Boston},
   Year = {2009},
   Month = {January},
   ISSN = {1093-6106},
   url = {http://dx.doi.org/10.4310/AJM.2009.v13.n3.a1},
   Abstract = {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 interior of a smooth, compact
             4-manifold K, whose double doub(K) has a trivial bundle of
             self-dual 2-forms, may be isometrically embedded into a
             Spin(7)-manifold as a Cayley submanifold. Along the way, we
             also show that Bochner's Theorem on real analytic
             approximation of smooth differential forms, can be obtained
             using real algebraic tools developed by Akbulut and King. ©
             2009 International Press.},
   Doi = {10.4310/AJM.2009.v13.n3.a1},
   Key = {fds302465}
}

@article{fds302466,
   Author = {Landsberg, JM and Robles, C},
   Title = {Fubini's theorem in codimension two},
   Journal = {Journal fur die Reine und Angewandte Mathematik},
   Volume = {2009},
   Number = {631},
   Pages = {221-235},
   Publisher = {WALTER DE GRUYTER GMBH},
   Year = {2009},
   Month = {June},
   ISSN = {0075-4102},
   url = {http://dx.doi.org/10.1515/CRELLE.2009.047},
   Abstract = {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 of the Debarre-de Jong
             conjecture and of varieties whose Fano variety of lines has
             dimension 2n - 4. © Walter de Gruyter Berlin · New York
             2009.},
   Doi = {10.1515/CRELLE.2009.047},
   Key = {fds302466}
}

@article{fds302468,
   Author = {Landsberg, JM and Robles, C},
   Title = {Lines and osculating lines of hypersurfaces},
   Journal = {Journal of the London Mathematical Society},
   Volume = {82},
   Number = {3},
   Pages = {733-746},
   Publisher = {Oxford University Press (OUP)},
   Year = {2010},
   Month = {January},
   ISSN = {0024-6107},
   url = {http://dx.doi.org/10.1112/jlms/jdq051},
   Abstract = {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 Society.},
   Doi = {10.1112/jlms/jdq051},
   Key = {fds302468}
}

@article{fds302469,
   Author = {Robles, C},
   Title = {Parallel calibrations and minimal submanifolds},
   Journal = {Illinois Journal of Mathematics},
   Volume = {56},
   Number = {2},
   Pages = {383-395},
   Year = {2012},
   Month = {January},
   ISSN = {0019-2082},
   url = {http://dx.doi.org/10.1215/ijm/1385129954},
   Abstract = {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)-linear subspace P⊂Ωp(M). In
             particular, the calibrated submanifolds are necessarily
             integral submanifolds of the system. (Examples of parallel
             calibrations include the special Lagrangian calibration on
             Calabi-Yau manifolds, (co)associative calibrations on
             G2-manifolds, and the Cayley calibration on
             Spin(7)-manifolds.) © 2013 University of
             Illinois.},
   Doi = {10.1215/ijm/1385129954},
   Key = {fds302469}
}

@article{fds302471,
   Author = {Landsberg, JM and Robles, C},
   Title = {Fubini-Griffiths-Harris rigidity and lie algebra
             cohomology},
   Journal = {Asian Journal of Mathematics},
   Volume = {16},
   Number = {4},
   Pages = {561-586},
   Publisher = {International Press of Boston},
   Year = {2012},
   Month = {January},
   ISSN = {1093-6106},
   url = {http://dx.doi.org/10.4310/AJM.2012.v16.n4.a1},
   Abstract = {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 = a1). In contrast, we show that the
             adjoint variety of SL3ℂ and the Segre product Seg(ℙ1 ×
             ℙn) are flexible at order two. In the SL3ℂ example we
             discuss the relationship between the extrinsic projective
             geometry and the intrinsic path geometry. We extend
             machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka
             and others to reduce the proof of the general theorem to a
             Lie algebra cohomology calculation. The proofs of the
             flexibility statements use exterior differential systems
             techniques. © 2012 International Press.},
   Doi = {10.4310/AJM.2012.v16.n4.a1},
   Key = {fds302471}
}

@article{fds302467,
   Author = {Robles, C and The, D},
   Title = {Rigid Schubert varieties in compact Hermitian symmetric
             spaces},
   Journal = {Selecta Mathematica, New Series},
   Volume = {18},
   Number = {3},
   Pages = {717-777},
   Publisher = {Springer Nature},
   Year = {2012},
   Month = {August},
   ISSN = {1022-1824},
   url = {http://dx.doi.org/10.1007/s00029-011-0082-y},
   Abstract = {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 the existence of Y. This extends
             (independent) work of M. Walters, R. Bryant and J. Hong. Key
             tools include (i) a new characterization of Schubert
             varieties that generalizes the well-known description of the
             smooth Schubert varieties by connected sub-diagrams of a
             Dynkin diagram and (ii) an algebraic Laplacian (à la
             Kostant), which is used to analyze the Lie algebra
             cohomology group associated with the problem. © 2012
             Springer Basel AG.},
   Doi = {10.1007/s00029-011-0082-y},
   Key = {fds302467}
}

@article{fds328609,
   Author = {Landsberg, JM and Robles, C},
   Title = {Fubini-griffiths-harris rigidity of homogeneous
             varieties},
   Journal = {International Mathematics Research Notices},
   Volume = {2013},
   Number = {7},
   Pages = {1643-1664},
   Publisher = {Oxford University Press (OUP)},
   Year = {2013},
   Month = {January},
   url = {http://dx.doi.org/10.1093/imrn/rns016},
   Abstract = {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.},
   Doi = {10.1093/imrn/rns016},
   Key = {fds328609}
}

@article{fds302472,
   Author = {Hammond, C and Robles, C},
   Title = {Projective invariants of CR-hypersurfaces},
   Journal = {Complex Variables and Elliptic Equations},
   Volume = {58},
   Number = {11},
   Pages = {1493-1516},
   Publisher = {Informa UK Limited},
   Year = {2013},
   Month = {November},
   ISSN = {1747-6933},
   url = {http://dx.doi.org/10.1080/17476933.2011.575464},
   Abstract = {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 hypersurfaces are characterized. © 2013
             Copyright Taylor and Francis Group, LLC.},
   Doi = {10.1080/17476933.2011.575464},
   Key = {fds302472}
}

@article{fds302470,
   Author = {Robles, C},
   Title = {Schur flexibility of cominuscule Schubert
             varieties},
   Journal = {Communications in Analysis and Geometry},
   Volume = {21},
   Number = {5},
   Pages = {979-1013},
   Publisher = {International Press of Boston},
   Year = {2013},
   Month = {December},
   ISSN = {1019-8385},
   url = {http://dx.doi.org/10.4310/CAG.2013.v21.n5.a5},
   Abstract = {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 Schubert varieties. Robles and The identified
             a sufficient condition for ξ to be Schur rigid. In this
             paper, we show that the condition is also
             necessary.},
   Doi = {10.4310/CAG.2013.v21.n5.a5},
   Key = {fds302470}
}

@article{fds302473,
   Author = {Coskun, I and Robles, C},
   Title = {Flexibility of Schubert classes},
   Journal = {Differential Geometry and its Application},
   Volume = {31},
   Number = {6},
   Pages = {759-774},
   Publisher = {Elsevier BV},
   Year = {2013},
   Month = {December},
   ISSN = {0926-2245},
   url = {http://dx.doi.org/10.1016/j.difgeo.2013.09.003},
   Abstract = {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 multiple of an obstructed class in a
             cominuscule homogeneous variety can be represented by an
             irreducible subvariety. © 2013 Elsevier
             B.V.},
   Doi = {10.1016/j.difgeo.2013.09.003},
   Key = {fds302473}
}

@article{fds340124,
   Author = {Robles, C},
   Title = {Principal Hodge representations},
   Pages = {259-283},
   Publisher = {American Mathematical Society},
   Year = {2014},
   url = {http://dx.doi.org/10.1090/conm/608/12183},
   Doi = {10.1090/conm/608/12183},
   Key = {fds340124}
}

@article{fds302475,
   Author = {Robles, C},
   Title = {Schubert varieties as variations of Hodge
             structure},
   Journal = {Selecta Mathematica, New Series},
   Volume = {20},
   Number = {3},
   Pages = {719-768},
   Publisher = {Springer Nature},
   Year = {2014},
   Month = {January},
   ISSN = {1022-1824},
   url = {http://dx.doi.org/10.1007/s00029-014-0148-8},
   Abstract = {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 the Schubert VHS form a basis of
             the invariant characteristic cohomology associated with the
             infinitesimal period relation (a.k.a. Griffiths'
             transversality). © 2014 Springer Basel.},
   Doi = {10.1007/s00029-014-0148-8},
   Key = {fds302475}
}

@article{fds340294,
   Author = {Griffths, P and Robles, C and Toledo, D},
   Title = {Quotients of non-classical flag domains are not
             algebraic},
   Journal = {Algebraic Geometry},
   Volume = {1},
   Number = {1},
   Pages = {1-13},
   Publisher = {Foundation Compositio Mathematica},
   Year = {2014},
   Month = {January},
   url = {http://dx.doi.org/10.14231/AG-2014-001},
   Abstract = {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 discrete subgroup of G, the analytic
             space Γ / D does not have an algebraic structure. We also
             give another proof of the theorem of Huckleberry that any
             two points in a non-classical domain D can be joined by a
             finite chain of compact subvarieties of D.},
   Doi = {10.14231/AG-2014-001},
   Key = {fds340294}
}

@article{fds361673,
   Author = {Green, M and Griffiths, P and Robles, C},
   Title = {Extremal degenerations of polarized Hodge
             structures},
   Booktitle = {Proceedings of Hodge Theory and L2-Cohomology, Johns Hopkins
             U},
   Year = {2014},
   Month = {March},
   url = {http://arxiv.org/abs/1403.0646},
   Abstract = {We describe a Hodge theoretic approach to the question: In
             what ways can a smooth projective variety
             degenerate?},
   Key = {fds361673}
}

@article{fds302474,
   Author = {Robles, C},
   Title = {Singular loci of cominuscule Schubert varieties},
   Journal = {Journal of Pure and Applied Algebra},
   Volume = {218},
   Number = {4},
   Pages = {745-759},
   Publisher = {Elsevier BV},
   Year = {2014},
   Month = {April},
   ISSN = {0022-4049},
   url = {http://dx.doi.org/10.1016/j.jpaa.2013.08.014},
   Abstract = {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⊂ X in terms of
             representation theoretic data. The result is based on a
             recent characterization of the Schubert varieties using an
             integer a≥ 0 and a marked Dynkin diagram. Corollaries
             include: (1) the variety is smooth if and only if a= 0; (2)
             if G is of type ADE, then the singular locus occurs in
             codimension at least 3. © 2013 Elsevier
             B.V.},
   Doi = {10.1016/j.jpaa.2013.08.014},
   Key = {fds302474}
}

@article{fds361672,
   Author = {Robles, C},
   Title = {Nilpotent cones and adjoint orbits},
   Year = {2014},
   Month = {May},
   Abstract = {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.},
   Key = {fds361672}
}

@article{fds320190,
   Author = {Robles, C},
   Title = {Characteristic cohomology of the infinitesimal period
             relation},
   Journal = {Asian Journal of Mathematics},
   Volume = {20},
   Number = {4},
   Pages = {725-758},
   Publisher = {International Press of Boston},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.4310/AJM.2016.v20.n4.a7},
   Abstract = {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 of PDE.},
   Doi = {10.4310/AJM.2016.v20.n4.a7},
   Key = {fds320190}
}

@article{fds361671,
   Author = {Brosnan, P and Pearlstein, G and Robles, C},
   Title = {Nilpotent cones and their representation
             theory},
   Year = {2016},
   Month = {January},
   Abstract = {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 subgroup determined by the limit
             mixed Hodge structure. The possible relations between the
             interior of C its faces are described in terms of signed
             Young diagrams. The second approach is to understand the
             Tannakian category of nilpotent orbits via a category D
             introduced by Deligne in a letter to Cattani and Kaplan. In
             analogy with Hodge theory, there is a functor from D to a
             subcategory of SL(2)-orbits. We prove that these fibers are,
             roughly speaking, algebraic. We also give a correction to a
             result of K. Kato.},
   Key = {fds361671}
}

@article{fds320189,
   Author = {Robles, C},
   Title = {Classification of horizontal SL(2)s},
   Journal = {Compositio Mathematica},
   Volume = {152},
   Number = {5},
   Pages = {918-954},
   Publisher = {Oxford University Press (OUP)},
   Year = {2016},
   Month = {May},
   url = {http://dx.doi.org/10.1112/S0010437X15007691},
   Abstract = {We classify the horizontal s and-split polarized mixed Hodge
             structures on a Mumford-Tate domain.},
   Doi = {10.1112/S0010437X15007691},
   Key = {fds320189}
}

@article{fds340293,
   Author = {Robles, C},
   Title = {Degenerations of Hodge structure},
   Journal = {Proceedings of Symposia in Pure Mathematics},
   Volume = {95},
   Pages = {267-283},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1090/pspum/095/01627},
   Abstract = {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 Hodge-theoretic analogs of these
             questions, and the Hodge-theoretic answers in turn provide
             insight into the motivating algebro-geometric questions,
             sometimes with applications to the study of moduli. Recently
             the Hodge-theoretic questions have been completely answered.
             This is an expository survey of that work.},
   Doi = {10.1090/pspum/095/01627},
   Key = {fds340293}
}

@article{fds327152,
   Author = {Kerr, M and Robles, C},
   Title = {Classification of smooth horizontal Schubert
             varieties},
   Journal = {European Journal of Mathematics},
   Volume = {3},
   Number = {2},
   Pages = {289-310},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s40879-017-0140-x},
   Abstract = {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 the classification of smooth
             Schubert varieties in cominuscule G / P.},
   Doi = {10.1007/s40879-017-0140-x},
   Key = {fds327152}
}

@article{fds326604,
   Author = {Kerr, M and Robles, C},
   Title = {Variations of Hodge structure and orbits in flag
             varieties},
   Journal = {Advances in Mathematics},
   Volume = {315},
   Pages = {27-87},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.aim.2017.05.013},
   Abstract = {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 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 GR-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.},
   Doi = {10.1016/j.aim.2017.05.013},
   Key = {fds326604}
}

@article{fds361649,
   Author = {Green, M and Griffiths, P and Laza, R and Robles,
             C},
   Title = {Period mappings and properties of the augmented Hodge line
             bundle},
   Year = {2017},
   Month = {August},
   Abstract = {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.},
   Key = {fds361649}
}

@article{fds328918,
   Author = {Robles, C},
   Title = {Characterization of Calabi–Yau variations of Hodge
             structure over tube domains by characteristic
             forms},
   Journal = {Mathematische Annalen},
   Volume = {371},
   Number = {3-4},
   Pages = {1229-1253},
   Publisher = {Springer Nature},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.1007/s00208-017-1594-3},
   Abstract = {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.},
   Doi = {10.1007/s00208-017-1594-3},
   Key = {fds328918}
}

@article{fds348482,
   Author = {Kerr, M and Pearlstein, GJ and Robles, C},
   Title = {Polarized relations on horizontal SL(2)'s},
   Journal = {Documenta Mathematica},
   Volume = {24},
   Pages = {1295-1360},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.25537/dm.2019v24.1295-1360},
   Abstract = {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 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. The relation is then
             worked out in several examples and special cases, including
             period domains, Hermitian symmetric domains, and complete
             flag domains. Although the above relation is not in general
             a partial order, it leads (via cubical sets) to a poset of
             equivalence classes of multivariable nilpotent orbits on D.
             The elements of this poset encode the possible degeneracy
             relations amongst the polarized mixed Hodge structures that
             arise in a several-variable degeneration of Hodge structure.
             We conclude with an example illustrating a link to mirror
             symmetry for Calabi-Yau VHS.},
   Doi = {10.25537/dm.2019v24.1295-1360},
   Key = {fds348482}
}

@article{fds361504,
   Author = {Green, M and Kim, Y-J and Laza, R and Robles, C},
   Title = {The LLV decomposition of hyper-Kaehler cohomology},
   Year = {2019},
   Month = {June},
   Abstract = {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.},
   Key = {fds361504}
}

@article{fds361593,
   Author = {Green, M and Griffiths, P and Robles, C},
   Title = {The global asymptotic structure of period
             mappings},
   Year = {2020},
   Month = {October},
   Abstract = {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 topological completion with a compatible
             algebraic structure. This raises questions about: (i) the
             global geometry of the fibres; and (ii) the existence of
             period matrix representations on neighborhoods of such
             fibres over which the restricted extension is still proper.
             The purpose of this paper is to investigate these
             questions.},
   Key = {fds361593}
}

@article{fds353256,
   Author = {Han, X and Robles, C},
   Title = {Hodge Representations},
   Journal = {Experimental Results},
   Volume = {1},
   Publisher = {Cambridge University Press (CUP)},
   Editor = {Clingher, A},
   Year = {2020},
   Month = {November},
   url = {http://dx.doi.org/10.1017/exp.2020.55},
   Abstract = {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 corresponding Mumford-Tate subdomains.
             The procedure is illustrated in two examples: (i) weight two
             Hodge structures with; and (ii) weight three CY-type Hodge
             structures.},
   Doi = {10.1017/exp.2020.55},
   Key = {fds353256}
}

@article{fds361347,
   Author = {Green, M and Griffiths, P and Robles, C},
   Title = {Natural line bundles on completions of period
             mappings},
   Year = {2021},
   Month = {February},
   Abstract = {We give conditions under which natural lines bundles
             associated with completions of period mappings are
             semi-ample and ample.},
   Key = {fds361347}
}

@article{fds358294,
   Author = {Green, M and Kim, YJ and Laza, R and Robles, C},
   Title = {The LLV decomposition of hyper-Kähler cohomology (the known
             cases and the general conjectural behavior)},
   Journal = {Mathematische Annalen},
   Volume = {382},
   Number = {3-4},
   Pages = {1517-1590},
   Year = {2022},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s00208-021-02238-y},
   Abstract = {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 determine the
             Looijenga–Lunts–Verbitsky (LLV) decomposition for all
             known examples of compact hyper-Kähler 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 b2(X) of hyper-Kähler
             manifolds (see Kim and Laza in Bull Soc Math Fr
             148(3):467–480, 2020). Specifically, in the K3 [n] and Kum
             n cases, we give generating series for the formal characters
             of the associated LLV representations, which generalize the
             well-known Göttsche formulas for the Euler numbers, Betti
             numbers, and Hodge numbers for these series of hyper-Kähler
             manifolds. For the two exceptional cases of O’Grady (OG6
             and OG10) 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-Kähler
             manifolds of OG 10 type. In a different direction, for all
             known examples of hyper-Kähler manifolds, we establish the
             so-called Nagai’s conjecture on the monodromy of
             degenerations of hyper-Kähler 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-Kähler manifolds.},
   Doi = {10.1007/s00208-021-02238-y},
   Key = {fds358294}
}

@article{fds369339,
   Author = {Robles, C},
   Title = {Extension of Hodge norms at infinity},
   Year = {2023},
   Month = {February},
   Abstract = {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.},
   Key = {fds369339}
}

@article{fds369338,
   Author = {Robles, C},
   Title = {Pseudoconvexity at infinity in Hodge theory: a codimension
             one example},
   Year = {2023},
   Month = {February},
   Abstract = {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.},
   Key = {fds369338}
}

@article{fds374353,
   Author = {Deng, H and Robles, C},
   Title = {Completion of two-parameter period maps by nilpotent
             orbits},
   Year = {2023},
   Month = {December},
   Abstract = {We show that every two-parameter period map admits a
             Kato--Nakayama--Usui completion to a morphism of log
             manifolds.},
   Key = {fds374353}
}


%% Papers Accepted   
@article{fds292862,
   Author = {Colleen Robles},
   Title = {Classification of horizontal SL(2)s},
   Journal = {Compositio Math.},
   Year = {2015},
   url = {http:/},
   Key = {fds292862}
}


%% Papers Submitted   
@article{fds292863,
   Author = {M. Kerr and C. Robles},
   Title = {Hodge theory and real orbits in flag varieties},
   Year = {2015},
   url = {http://arxiv.org/abs/1407.4507},
   Key = {fds292863}
}

 

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