Mathematics

Algebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into; this is partly due to its breadth, as traditionally algebra, topology, analysis, differential geometry, Lie theory and more recently combinatorics, logic,. . .are used to study it. I have tried to make these notes accessible to a general audience by illustrating topics with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject.

This talk will be at the interface of the two topics - moduli and singularities and Hodge theory and degenerations of Hodge structures.

Notes based on the lecture presented at the conference “Geometry at the frontier” held at Pucon, Chile during November 2018.

Clay Lecture, based on joint work with Mark Green, Radu Laza, and Colleen Robles. Some of the lecture draws work of and discussions with Marco Fanciosi, Rita Pardini, and Sonke Rollenske.

Algebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field k. When k = C the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal and illustrative account of some answers to the question in the title.

Talk at UIC (April 5, 2019), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles (GLR). Selected references to works quoted in or related to this talk are given at the end.

Talk at the "Geometry at the Frontier" conference, Pucón, Chile (November 12, 2018), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles.

The use of the Hodge line bundle, especially its positivity properties, in algebraic geometry is classical. The Hodge line bundle only sees the associated graded to the limiting mixed Hodge structures (LMHS's) that arise from the singular members in a family of algebraic varieties. This talk will introduce a new type of geometric object associated to the extension data in LMHS's (and not present for general MHS's) whose positivity properties play a central role in the application of Hodge theory to moduli. Their construction will be discussed and illustrated.

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.

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.

Talk given on November 27, 2019 at the IMSA conference held at IMATE at UNAM, Mexico City.