%% 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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