School of Mathematics
https://hdl.handle.net/20.500.12111/14
School of Mathematics2019-10-15T06:29:08ZComparisons between the Hodge-theoretic and algebro-geometric approaches to limiting mixed Hodge structures
https://hdl.handle.net/20.500.12111/7825
Comparisons between the Hodge-theoretic and algebro-geometric approaches to limiting mixed Hodge structures
Griffiths, Phillip; Green, Mark
Isolated Hypersurface Singularities
https://hdl.handle.net/20.500.12111/6731
Isolated Hypersurface Singularities
Griffiths, Phillip
Lectures given at the University of Miami during April, 2019
2019-04-20T00:00:00ZPositivity of vector bundles and Hodge theory
https://hdl.handle.net/20.500.12111/6515
Positivity of vector bundles and Hodge theory
Griffiths, Phillip; Green, Mark
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.
2018-10-10T00:00:00ZNew Geometric Invariants Arising from Hodge Theory
https://hdl.handle.net/20.500.12111/6514
New Geometric Invariants Arising from Hodge Theory
Griffiths, Phillip; Green, Mark; Laza, Radu; Robles, Colleen
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.
Talk given at the "Algebraic Geometry in Mexico" conference taking place at Puerta Escondido, December 2 - 7, 2018. Partly based on work in progress with Mark Green, Radu Laza and Colleen Robles.
2018-12-01T00:00:00Z