Positivity of vector bundles and Hodge theory

Show simple item record Griffiths, Phillip Green, Mark 2019-02-19T22:44:50Z 2019-02-19T22:44:50Z 2018-10-10
dc.description.abstract 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. en_US
dc.language.iso en_US en_US
dc.subject Algebraic Geometry en_US
dc.subject Complex Variables en_US
dc.subject Differential Geometry en_US
dc.subject Positivity of vector bundles and Hodge theory en_US
dc.subject Hodge Theory en_US
dc.title Positivity of vector bundles and Hodge theory en_US
dc.type Working Paper en_US
dc.identifier.arxiv arXiv:1803.07405v2

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