Simon Felten, Oxford Univ: The logarithmic smoothness of non-d-semistable normal crossing schemes

Описание: Simon Felten, Oxford University: The logarithmic smoothness of non-d-semistable normal crossing schemes
Let V/ℂ be a normal crossing scheme. When V is d-semistable, i.e., ℰxt¹(Ω¹_V,𝒪_V) ≅ 𝒪_D for the singular locus D = Sing(V), there is a global log smooth structure over the standard log point S₀. When
ℰxt¹(Ω¹_V,𝒪_V) is only globally generated, we can still find a well-behaved log smooth structure over the standard log point S₀; however, it is not globally defined but defined outside a log singular locus Z ⊂ V.
Up to now, we considered the thus-constructed object as a partial log scheme (V, V – Z,ℳ), i.e., a log scheme whose log structure ℳ is defined only on an open subset. This has a number of drawbacks, for example when
studying (classical as well as derived) deformations of or log curves on (V, V – Z, ℳ).
In this talk, I propose a formalism to extend the log smooth structure across Z as a sharp lax log structure. This sharp lax log structure admits a chart, is locally isomorphic to the spectrum of a (sharp) lax log ring, and turns out to have the infinitesimal lifting property in the category of integral sharp lax log schemes—hence we may say that our extension is log smooth.
The same formalism can also be applied to the positive and simple toric log Calabi­­­–Yau spaces of the Gross–Siebert program. In this setting, the infinitesimal lifting property holds in general only up to codimension 3, corresponding to the fact that the general fiber of a toric degeneration can have singularities in codimension 4.

Conference: New Developments in Singularity Theory
Dates: November 10-14, 2025
Location: Ungar Building, Room 528B, University of Miami
Organized by: Helge Ruddat, Nero Budur, Enrique Becerra, Leonardo Cavenaghi
This is an IMSA & ICMS joint event, supported by the Simons Foundation, National Science Foundation and the University of Miami.