Gregory Pearlstein

On the algebraicity of the zero locus of an admissible normal function

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.

Boundary components of Mumford–Tate domains

We study certain spaces of nilpotent orbits in Hodge domains, and treat a number of examples. More precisely, we compute the Mumford-Tate group of the limit mixed Hodge structure of a generic such orbit. The result is used to present these spaces as iteratively fibered algebraic-group orbits in a minimal way. We conclude with two applications to variations of Hodge structure.

Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Abstract: Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds.

Degenerations of mixed Hodge structure

We continue our work on variations of graded-polarized mixed Hodge structures by defining analogs of the harmonic metric equations for filtered bundles and proving a precise analog of Schmids Nilpotent Orbit Theorem for 1-parameter degenerations of graded-polarized mixed Hodge structure.

Zero loci of admissible normal functions with torsion singularities

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].

Asymptotics of degenerations of mixed Hodge structures

Abstract We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [6] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by Hain [12] , analogs of the norm estimates of [19] and the asymptotics of the naive limit Hodge filtration considered in [21] .

Opposite filtrations, variations of Hodge structure, and Frobenius modules

Let X be a complex manifold. Then, an unpolarized complex variation of Hodge structure (E, ∇, F, Φ) over X consists of a flat, C ∞ complex vector bundle (E, ∇) over X equipped with a decreasing Hodge filtration F and an increasing filtration Φ such that (1.1) F is holomorphic with respect to ∇, and \( \nabla \left({{F^{p}}}\right) \subseteq {F^{{p - 1}}} \otimes \Omega _{X}^{1}\); (1.2) Φ is anti-holomorphic with respect to ∇, and \( \nabla \left({{{\bar{\Phi}}_{q}}}\right)\subseteq {\bar{\Phi}_{{q + 1}}} \otimes \overline {\Omega _{X}^{1}} \); (1.3) \( E = {F^{p}} \oplus {\bar{\Phi }_{{p - 1}}}\) for each index p (i.e. F is opposite to Φ).

Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katzs theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.

Classical Period Domains

We survey the role played by Hermitian symmetric domains in the study of variations of Hodge Structure. These are extended notes based on the lectures given by the first author in Vancouver at the “Advances in Hodge Theory” school (June 2013). Introduction There are two classical situations where the period map plays an essential role for the study of moduli spaces, namely the moduli of principally polarized abelian varieties and the moduli of polarized K3 surfaces. What is common for these two situations is the fact that the period domain is in fact a Hermitian symmetric domain. It is well known that the only cases when a period domain is Hermitian symmetric are weight 1 Hodge structures and weight 2 Hodge structures with h 2,0 = 1. In general, it is difficult to study moduli spaces via period maps. A major difficulty in this direction comes from the Griffiths’ transversality relations. Typically, the image Z of the period map in a period domain D will be a transcendental analytic subvariety of high condimension. The only cases when Z can be described algebraically are when Z is a Hermitian symmetric subdomain of D with a totally geodesic embedding (and satisfying the horizontality relation). This is closely related to the geometric aspect of the theory of Shimura varieties of Deligne. It is also the case of unconstrained period subdomains in the sense of [GGK12]. We call this case classical, in contrast to the “non-classical” case when the Griffiths’ transversality relations are non-trivial. The purpose of this survey is to review the role of Hermitian symmetric domains in the study of variations of Hodge structure. Let us give a brief overview of the content of the paper. In Section 1, we review the basic definitions and properties of Hermitian symmetric domains (Section 1.1) and their classification (Section 1.2) following [Mil04]. The classification is done by reconstructing Hermitian symmetric domains from the associated (semisimple) Shimura data, which are also convenient for the purpose of constructing variations of Hodge structure over Hermitian symmetric domains (Section 1.3). As a digression, we also include the discussion that if the universal family of Hodge structures over a period subdomain satisfies Griffiths transversality then the subdomain must be Hermitian symmetric (i.e. unconstrained ⇒ Hermitian symmetric).

Tate twists of Hodge structures arising from abelian varieties

We consider the category of Hodge substructures of the cohomology of abelian varieties, and ask when a Tate twist of such a Hodge structure belongs to the same category.