Preprint · December 1, 2023
We show that every two-parameter period map admits a Kato--Nakayama--Usui
completion to a morphism of log manifolds. ...
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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. ...
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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 ...
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Journal ArticleMathematische 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 ...
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Journal ArticleExperimental 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 ...
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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 ...
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Journal ArticleDocumenta 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 ...
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Journal ArticleMathematische 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. ...
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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 ...
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Journal ArticleAdvances 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 ...
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Journal ArticleEuropean 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 ...
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Journal ArticleProceedings 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 ...
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Journal ArticleCompositio Mathematica · May 1, 2016
We classify the horizontal s and-split polarized mixed Hodge structures on a Mumford-Tate domain. ...
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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 ...
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Journal ArticleAsian 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 ...
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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. ...
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Journal ArticleJournal 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 ...
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Journal Article · March 3, 2014
We describe a Hodge theoretic approach to the question: In what ways can a
smooth projective variety degenerate? ...
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Journal ArticleSelecta 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 ...
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Journal ArticleAlgebraic 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 ...
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Journal ArticleDifferential 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleComplex 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 ...
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Journal ArticleSelecta 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 ...
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Journal ArticleAsian 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 ...
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Journal ArticleIllinois 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleAsian 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 ...
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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. ...
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Journal ArticleReports 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 ...
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