Phillip Griffiths
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Phillip Griffiths initiated with his collaborators the theory of variation of Hodge structure, which has come to play a central role in many aspects of algebraic geometry and its uses in modern theoretical physics. In addition to algebraic geometry, he has made contributions to differential and integral geometry, geometric function theory, and the geometry of partial differential equations. A former Director of the Institute (1991–2003), Griffiths chaired the Science Initiative Group, which fosters science in the developing world through programs such as the Carnegie–IAS African Regional Initiative in Science and Education.
Click here for his curriculum vitae and here for his bibliography.
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- Atypical Hodge loci(2023-06)Talk based on the paper [BKU] and related works given in the references in that work, and on extensive discussions with Mark Green and Colleen Robles.
53 56 - Hodge Theory and Moduli(2023-04-07)Hodge theory provides a major tool for the study of moduli. Conversely, moduli have furnished a significant stimulus for the developement of Hodge theory.
86 93 - Shafarevich mappings and period mappingsWe shall show that a smooth, quasi-projective variety X has a holomorphically convex universal covering eX when (i) 1(X) is residually nilpotent and (ii) there is an admissable variation of mixed Hodge structure over X whose monodromy representation has a nite kernel, and where in each case a corresponding period mapping is assumed to be proper.
179 124 - Extended Period MappingsClay Lecture at the INI, June 2022. The lecture is based on joint work with Mark Green and Colleen Robles. Theme of the lecture is global properties of period mappings with applications to the geometry of completions of moduli spaces.
134 80 - Limits in Hodge TheoryAlmost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the literature the proofs of these results frequently use the consequences of the analysis of the singularities acquired in a degenerating family of Hodge structures; that analysis itself is treated as a "black box." In these lectures an attempt will be made to give an informal introduction to the subject of limits of Hodge structures and to explain some of the essential ideas of the proofs. One additional topic not yet in the literature that we will discuss is the geometric interpretation of the extension data in limiting mixed Hodge structures and its use in moduli questions.
185 131 - Extended Period MappingsThis lecture will discuss the global structure of period mappings (variation of Hodge structure) defined over complete, 2-dimensional algebraic varieties. Some applications to moduli of general type algebraic surfaces will also be presented.
155 81 - Differential of a period mapping at a singularity(2021)To Herb Clemens. Lefschetz wrote that he put the harpoon of topology into the whale of algebraic geometry. Herb but the harpoon of topology into the whale of Hodge theory. Period mappings, or equivalently variations of Hodge structure, have been used both to study families of algebraic varieties and as a subject in its own right.
239 217 - Completions of Period Mappings: Progress Report(2021)We give an informal, expository account of a project to construct completions of period maps.
417 291 - Positivity of vector bundles and Hodge theory(2021)From S. S. Chern we learned the importance of curvature in geometry and its special features in the complex case. In this case there are significant geometric and analytic consequences of the curvature having a sign. Both positive and negative curvature have major implications in algebraic geometry and in holomorphic mappings between complex manifolds. The vector bundles (Hodge bundles) and complex manifolds (period domains) that arise in Hodge theory have natural metrics and subsequent curvatures that through the work of very many people over an extended period of time have played a central role in the study of Hodge theory as a subject in its own right and in the applications of Hodge theory to algebraic geometry. Of particular importance are (i) the sign properties of the curvature (positivity of the Hodge bundles and cotangent bundles of period domains); (ii) the result that in the geometric case the non-degeneracy of curvature forms is an algebro-geometric property; (iii) the singularity properties of the curvature, especially that of the Chern forms. Regarding (iii) we note that the essential geometric fact that enables one to control the singularities is a curvature property of the bundles that arise in Hodge theory. The primary purpose of this mainly expository paper is to present some (but not by any means all) of the fundamental concepts and to discuss a few of the basic results in this very active and now vast area of research.
405 283 - Topics in the geometric application of Hodge theoryThe classifcation of algebraic varieties is a central part of algebraic geometry. Among the common tools that are employed are birational geometry, especially using the analysis of singularities, and Hodge theory. The latter is frequently used as a sort of "black box": the cohomology of an algebraic variety has a functorial mixed Hodge structure and the formal properties of the corresponding category have very strong consequences. On the other hand, geometric structures arise naturally in Hodge theory and one aspect of this will be the focus of these lectures. This aspect originates from the fact that mixed Hodge structures have extension data expressed by linear algebra.
302 181 - Towards a maximal completion of a period map(2021-02-11)The motivation behind this work is to construct a "Hodge theoretically maximal" completion of a period map. This is done up to finite data (we work with the Stein factorization of the period map). The image of the extension is a Moishezon variety that compactifies a finite cover of the image of the period map.
313 185 - Natural line bundles on completions of period mappings(2021-02-11)We give conditions under which natural lines bundles associated with completions of period mappings are semi-ample and ample.
276 187 - Completions of Period Mappings(2020-10-13)This work initiates a global study of period mappings at infinity, that is motivated by challenges arising when considering the questions: (i) What are natural completions of a period mapping? (ii) What geometric applications do they have?
334 172 - Hodge theory and ModuliThe theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space $\mathcal{M}$ with a canonical completion $\bar{\mathcal{M}}$ has been proved by Koll\'ar/Shepard-Barron/Alexeev. Aside from the classical case of algebraic curves, very little is known about the structure of $\mathcal{M}$, especially it's boundary $\overline{\mathcal{M}} \diagdown\mathcal{M}$. The period mapping from Hodge theory provides a tool for studying these issues. In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.
369 259 - Some geometric applications of Hodge theory (Talks in Chile and Montreal)Hodge theory is a subject of both interest in it's own right and as a tool for studying geometric questions. This talk will be mainly concerned with some of the history and uses of Hodge theory: how did it originate and what are some of the ways it helps to address problems in geometry, especially algebraic geometry? Our discussion of both the early history and uses of Hodge theory will be selective, not comprehensive.
601 243 - Period Mapping at Infinity(2020-05-06)Hodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include: new global invariants of limiting mixed Hodge structures; a generic local Torelli assumption implies that moduli spaces are log canonical (not just log general type); and extension data and asymptotics of the Ricci curvature; a proposed construction of the toroidal compactification of the image of period mapping. The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.
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