Exceptional origin of the principal asymptotic class

Determine whether, when the principal asymptotic class A_X(τ) is defined, there exists a K-class V in K(X) such that [A_X(τ)]=[Γ_X Ch(V)], verify whether χ(V,V)=1, and ascertain whether V comes from an exceptional object in the derived category of coherent sheaves on X.

Background

Motivated by Dubrovin’s conjecture and Gamma conjecture II, the authors expect that principal asymptotic classes are realized by Γ-classes of exceptional objects. They pose a precise set of criteria—existence in K-theory, Euler pairing normalization, and categorical origin—to test this expectation for general τ.

References

Question 6.13. When the principal asymptotic class A (τ)Xis defined, does there exist a K-class V ∈ K (X) of topological vector bundles such that [A (τ)]X= [Γ Ch(V X]? Here Ch(V ) = (2πi) ch (V ) is the (2πi)-modified Chern character. Does V sat- k≥0 k isfy χ(V,V ) = 1? Does it come from an exceptional object in the derived category of coherent sheaves?

— Revisiting Gamma conjecture I: counterexamples and modifications (2405.16979 - Galkin et al., 27 May 2024) in Question 6.13, Section 6.2