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BIMSA AG Seminar
A motivic plumbing formula and orientation theory for surface singularities
A motivic plumbing formula and orientation theory for surface singularities
Organizers
Speaker
Frédéric Déglise
Time
Thursday, May 29, 2025 3:00 PM - 4:00 PM
Venue
A6-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
The classical plumbing construction, introduced by Mumford in the study of surface singularities, reveals deep connections between the topology of links and the geometry of resolutions.
In this talk, I will present a motivic analogue of this picture, in which topological invariants are replaced by algebraic counterparts arising from stable motivic homotopy theory. I will describe joint work with Adrien Dubouloz and Paul Arne Østvær, where we introduce punctured tubular neighborhoods as new invariants of singularities. These are computable and yield a general plumbing formula, where combinatorial data from a normal crossings divisor resolution and geometric data from Thom spaces of normal bundles are intricately combined.
We will illustrate this framework in the setting of surface singularities, focusing on the case of du Val singularities. In this context, orientation theory emerges naturally and can be resolved explicitly, leading to complete computations via quadratic Mumford matrices - refining Mumford’s original work by replacing Chow groups over $\mathbb{C}$ with Chow-Witt groups.
In this talk, I will present a motivic analogue of this picture, in which topological invariants are replaced by algebraic counterparts arising from stable motivic homotopy theory. I will describe joint work with Adrien Dubouloz and Paul Arne Østvær, where we introduce punctured tubular neighborhoods as new invariants of singularities. These are computable and yield a general plumbing formula, where combinatorial data from a normal crossings divisor resolution and geometric data from Thom spaces of normal bundles are intricately combined.
We will illustrate this framework in the setting of surface singularities, focusing on the case of du Val singularities. In this context, orientation theory emerges naturally and can be resolved explicitly, leading to complete computations via quadratic Mumford matrices - refining Mumford’s original work by replacing Chow groups over $\mathbb{C}$ with Chow-Witt groups.