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v3.2.9
In algebraic geometry, one often needs to consider an infinite number of data, for example all the powers of a given line bundle or all the divisors appearing in birational modifications of a given variety. In many of such contexts, the infinite number of data can be encoded in a plurisubharmonic function, a notion from complex analysis. Such a function is usually not differentiable or continuous, but comes with rather complicated singularities, which can be the source of interesting problems and applications.
In the first part of this talk, I will explain the motivation and background of the study of singularities of plurisubharmonic functions (and their global version, plurisubharmonic metrics and currents). Not only such a transcendental object is useful in applications to algebraic geometry (as in the fundamental work of Siu, Demailly and many others), but conversely algebraic geometry can also provide very useful insights, examples and even tools to complex analysis.
As an illustration of such interaction, in the second part, we will explain our recent work on ‘log canonical’ singularities of plurisubharmonic functions (joint with J. Kollár). Log canonical is a key condition both in geometry and in analysis, lying between integrability and non-integrability: it has been intensely studied in algebraic geometry in the case when a plurisubharmonic function has algebraic singularities. However, there is a large gap between that case and the general case. Our methods use the theory of quasimonomial valuations developed by Jonsson-Mustata and Xu, as well as the minimal model program over complex analytic spaces due to Fujino and Lyu-Murayama, in addition to complex analytic tools based on Hörmander's L2 estimates.