Dice Question Streamline Icon: https://streamlinehq.com

Validity of the constant-cycle formula for the twisted Beauville–Voisin class in higher-dimensional Bridgeland moduli spaces

Determine whether the formula o_{𝒳} = c_2(F)/rank(F) + (1 − deg(c_2(F))/rank(F)) · o_X, used to define the twisted Beauville–Voisin class via objects lying on constant cycle Lagrangian subvarieties, holds for arbitrary objects F when the Bridgeland moduli space N = M_H(𝒳, w) has dimension greater than 2.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors present a natural construction of o_{𝒳} from the second Chern class of objects lying on constant cycle Lagrangian subvarieties, and show that for N = M_H(𝒳, w) a K3 surface, the construction works without requiring local freeness of the object and yields a specific formula for o_{𝒳}.

They note that for higher-dimensional hyper-Kähler moduli spaces (dim N > 2), whether this same formula holds for arbitrary objects F on constant cycle Lagrangian subvarieties remains unresolved.

References

For $\dim N > 2$, it remains open whether the equation eq:def-twistbv holds for arbitrary $F$ lying on a constant cycle Lagrangian subvariety.

eq:def-twistbv:

$o_X = \frac{c_2(F)}{\rank F} + \left(1 - \frac{\deg(c_2(F))}{\rank F}\right) o_X $

— Filtrations on the derived category of twisted K3 surfaces (2402.13793 - Chen et al., 21 Feb 2024) in Section 4 (Twisted Beauville–Voisin class), Remark following Equation (4.1)