Methods and Applications of Analysis

Volume 25 (2018)

Number 3

In Memory of Professor John N. Mather: Part 1 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

Analytic invariants of multiple points

Pages: 167 – 204



A. G. Aleksandrov (Institute for Control Sciences, Russian Academy of Sciences, Moscow, Russia)


We develop an original approach in computing analytic invariants of zero-dimensional singularities, which is based essentially on the study of properties of differential forms and the cotangent complex of multiple points. Among other things, we consider a series of specific tasks and problems for zero-dimensional complete intersections, graded and gradient singularities, including the computation of cotangent homology and cohomology for certain types of such singularities. We also examine the unimodular families of gradient zero-dimensional singularities, compile an adjacency diagram and compute the primitive ideals of these families. Finally, we briefly discuss the problem of nonexistence of negative weighted derivations, some relationships between the Milnor and Tjurina numbers and estimates of these invariants in the case of zero-dimensional complete intersections.


fat points, thick points, differential forms, derivations, deformations, cotangent complex, duality, complete intersections, gradient singularities

2010 Mathematics Subject Classification

14F10, 14F40, 32S25, 58K45, 58K70

Received 23 May 2018

Accepted 12 October 2018

Published 1 November 2019