Algebraic cobordism revisited

By Marc Levine, R. Pandharipande. 2006

175 citations recorded by the source corpus.

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations.Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/857d5a254fe7d767a4084a389fe0f199720207af

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In Geometry and Topology

1. R. Pandharipande
2. Yukinobu Toda
3. Richard P. Thomas
4. T. Bridgeland
5. A. Okounkov
6. D. Oprea
7. Emanuele Macrì
8. D. Huybrechts

Citations in the corpus

• Curve counting theories via stable objects I. DT/PT correspondence (2009, background): “= Z [In(X,β)vir] 1 ∈ Z. We consider the generating series, DT(X) = X n,β In,βx nyβ. Let DT0(X) be the contributions from 0-dimensional subschemes, DT0(X) = X n In,0x n. This is computed in [3], [22], [21], DT0(X) = M(−x)χ(X), M(x) = Y k≥1 1 (1− xk)k . The reduced Donaldson-Thomas theory is defined by”
• A proof of the Göttsche-Yau-Zaslow formula (2010, background): “f nodal curves, only degenerating surfaces is not sufficient. Instead, we consider the degeneration of pairs [S,L] where S is a smooth projective surface and L is a line bundle on S. In Section 14.4 of [LP09], Levine and Pandharipande defined the algebraic cobordism theory ω2,1 for surfaces and line bu”
• Schubert calculus for algebraic cobordism (2009, background): “In a recent article [15], Levine and Pandharipande show that algebraic cobordism WðXÞ allows a presentation with generators being projective morphisms Y ! X of relative codimension nð:1⁄4 dimðX Þ dimðYÞÞ between smooth varieties and relations given by a refinement of the naive algebraic cobordism re”
• Wall crossing from Boltzmann black hole halos (2010, unknown): “8) where M(q) = ∏∞ k=1(1− qk)−k is the Mac-Mahon function [53, 54, 55, 56].”
• Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, background): “) and DT(X/Y) to be DT(X) := X n,β In,βx nyβ, DT(X/Y) := X n,f∗β=0 In,βx nyβ. The reduced series are defined by DT′(X) := DT(X) DT0(X) , DT′(X/Y) := DT(X/Y) DT0(X) . Here DT0(X) is given by [2], [19], [17], DT0(X) := X n In,0x n = M(−x)χ(X), for the MacMahon function, M(x) = Y k≥1 (1−xk)−k. 2.8 Pandh”
• Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, unknown): “The evaluation of the degree 0 partition function was conjectured in [24, 25] and proven in [ 12 , 14],”
• Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, methodology): “(i) In the case of β = 0, the generating series of In,0 is computed by Li [27], Behrend-Fantechi [5] and Levine-Pandharipande [26], ∑”
• Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics (2019, methodology): “The calculations that we have presented, which may be viewed as the beginning of the study of the virtual Euler characteristics of Quot schemes of surfaces, already show some features of Vafa-Witten theory: the appearance of the Kawai-Yoshioka formula (in 8See [27] for a foundational treatment of al”
• Hall algebras and curve-counting invariants (2010, background): “In the case β = 0 the scheme HilbM (β, n) parameterizes zero-dimensional subschemes of M of length n, and the corresponding generating function DT0(q) is known [2, 19, 20].”
• Curve counting via stable pairs in the derived category (2007, methodology): “unction is defined [43] by dividing out by the degree 0 series, Z′ DT,β(q) = ZDT,β(q) ZDT,0(q) . The degree 0 series is evaluated by the formula ZDT,0(q) = M(−q)χ(X), conjectured in [43] and proved in [8, 34, 38]. Here, M(q) = Y n≥1 (1−qn)−n is the MacMahon function. The reduced series Z′ DT,β(q) is ”
• Tautological and non-tautological cohomology of the moduli space of curves (2011, background): “bject. A discussion of obstruction theories and virtual classes can be found in [4, 5, 46]. 6 We do not know any examples at all of nonsingular projective X where ρ∗[M g,n(X,β)]vir ∈/ RH∗(M g,n). See [45] for a discussion. 8 where Cis a complete, connected, nonsingular curve with marked profiles µ1,.”
• Lectures on K-theoretic computations in enumerative geometry (2015, background): “The path from there to general threefolds is the same as in [50].”