Logarithmic Cobordism and Donaldson-Thomas Invariants
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Author(s)
Guzmán, José Luis
Advisor(s)
Maulik, Davesh
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
We construct a cobordism ring ωˡᵒᵍ of pairs (X, D) of a smooth variety X together with an snc (simple normal crossings) divisor D ⊂ X. The relations in this ring come from simple normal crossings degenerations: if (X, D) ⇝ ∪ᵢ(Yᵢ, Dᵢ) is a degeneration of snc pairs then we get the relation (X, D) = X(Yᵢ, Dᵢ) ∈ ωˡᵒᵍ From the perspective of logarithmic enumerative geometry, the use of snc degenerations to get relations is natural. Furthermore, invariants in logarithmic geometry are invariant under logarithmic modifications: if (X ′ , D′ ) is an snc pair obtained from (X, D) by blowing up along a stratum, then the invariants of (X ′ , D′ ) and (X, D) are the same. This motivates a further quotient ωˡᵒᵍ⁺ᵐᵒᵈ of ωˡᵒᵍ in which we impose the relations (X ′ , D′ ) = (X, D). Motivation from logarithmic enumerative geometry predicts that via degeneration and modifications, the invariants of an snc pair (X, D) can be determined by absolute invariants of varieties without boundary. In light of this motivation, we prove that the ring ωˡᵒᵍ⁺ᵐᵒᵈ is generated by varieties with no boundary and is isomorphic to Levine-Pandharipande’s algebraic cobordism ring ωˡᵖ. We also apply our logarithmic cobordism construction to give a proof of a conjecture by MaulikRanganathan in logarithmic Donaldson-Thomas theory and to give a new proof of a theorem of Ellingsrud-Göttsche-Lehn on the cobordism class [S [ⁿ] ] for the hilbert scheme of points S [ⁿ] of a smooth projective surface S.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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