Rational homotopy of complex projective varieties with normal isolated singularities

TL;DR

This paper demonstrates that complex projective varieties with isolated normal singularities are formal topological spaces under certain connectivity conditions, using mixed Hodge theory, with applications to various classes of singularities.

Contribution

It establishes formality of such varieties based on the connectivity of links at singular points, extending to contractions of subvarieties.

Findings

Varieties with (n-2)-connected links are formal spaces

Applicable to normal surface singularities and hypersurfaces with isolated singularities

Provides new insights into the topology of complex projective varieties with singularities

Abstract

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.