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Phillip Griffiths

Publications and source records attributed to Phillip Griffiths.

16 recordsLinked to original sources
arXiv2025

Atypical Hodge Loci

In the recent works of a number of people there has emerged a beautiful new perspective on the arithmetic properties of Hodge structures. A central result in that development appears in a paper by Baldi, Klingler, and Ullmo. In this expository work we will explain that result and give a proof. The main conceptual step is to formulate Noether-Lefschetz loci in terms of intersections of period images with Mumford-Tate subdomains of period domains. The main technical step is to use the alignment of the Hodge and root space decompositions of the Lie algebras of the associated groups and from it to use the integrability conditions associated to a Pfaffian PDE system. These integrability conditions explain the generally present excess intersection property associated to the integral varieties of a pair of Pfaffian exterior differential systems.

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arXiv2025

Period maps at infinity

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 $Z$.

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arXiv2025

Lagrangian interpretation of Abel-Jacobi mappings associated to Fano threefolds

Using the general framework due to Donagi-Markman \cite{DM} and Markushevich \cite{M} we shall derive an expression for the differential of Abel-Jacobi mappings on Fano threefolds. This formula involves information normal to the Lagrangian submanifolds constructed in \cite{DM} and \cite{M}. It may be applied to give new proofs of a number of classical results about these varieties.

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arXiv2024

Limiting mixed Hodge structures associated to I-surfaces with simple elliptic singularities

An I-surface $X$ is a surface of general type with $K_X^2 =1$ and $p_g(X) =2$. This paper studies the asymptotic behavior of the period map for I-surfaces acquiring simple elliptic singularities. First we describe the relationship between the deformation theory of such surfaces and their $d$-semistable models. Next we analyze the mixed Hodge structures on the $d$-semistable models, the corresponding limiting mixed Hodge structures, and the monodromy. There are $6$ possible boundary strata for which the relevant limiting mixed Hodge structures satisfy: $\dim W_1 = 4$, and hence $W_2/W_1$ is of pure type $(1,1)$. We show that, in each case, the nilpotent orbit of limiting mixed Hodge structures determines the boundary stratum and prove a global Torelli theorem for one such stratum.

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arXiv2024

Infinitesimal invariants of mixed Hodge structures

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and in more detail in the case when $X$ is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

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arXiv2024

Deformations of I-surfaces with elliptic singularities

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The possible normal Gorenstein examples have been enumerated by work of Franciosi-Pardini-Rollenske. The goal of this paper is to give a more precise description of such surfaces in case their singularities are simple elliptic and/or cusp singularities, and to work out their deformation theory. In particular, under some mild general position assumptions, we show that deformations of the surfaces in question are versal for deformations of the singular points, with two exceptions where the discrepancy is analyzed in detail.

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arXiv2022

Shafarevich mappings and period mappings

We shall show that a smooth, quasi-projective variety $X$ has a holomorphically convex universal covering $\wt X$ when (i) $\pi_1(X)$ is residually nilpotent and (ii) there is an admissable variation of \mhs\ over $X$ whose monodromy representation has a finite kernel, and where in each case a corresponding period mapping is assumed to be proper.

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arXiv2020

Analog of Satake-Baily-Borel for period maps

We propose an analog of the Satake--Baily--Borel compactification and Borel's extension theorem for arbitrary period maps. The proposed analog is constructed as a proper topological completion of the period map. It is conjectured that the construction is projective algebraic, and the conjecture is reduced to a certain extension problem.

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arXiv2018

Positivity of vector bundles and Hodge theory

It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge metrics generally have singularities; what can be said about these and their curvatures, Chern forms etc.? (iii) What are some algebro-geometric applications of positivity of Hodge bundles? The purpose of these partly expository notes is fourfold. One is to summarize some of the general measures and types of positivity that have arisen in the literature. A second is to introduce and give some applications of norm positivity. This is a concept that implies the di_erent notions of metric semi-positivity that are present in many of the standard examples and one that has an algebro-geometric interpretation in these examples. A third purpose is to discuss and compare some of the types of metric singularities that arise in algebraic geometry and in Hodge theory. Finally we shall present some applications of the theory from both the classical and recent literature.

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arXiv2017

Period mappings and properties of the augmented Hodge line bundle

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.

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arXiv2013

Quotients of non-classical flag domains are not algebraic

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 any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.

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arXiv2009

Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.

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