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Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants

November 13, 2023 @ 10:30 am - 11:30 am

Algebraic Geometry in String Theory Seminar

Speaker: Sheldon Katz, UIUC

Title: Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants

Abstract:  A determinantal octic double solid is the double cover X of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3.  These X are nodal CY threefolds which do not admit a projective small resolution.  B-model techniques can be applied to compute GV invariants up to g \le 32.  This raises the question: what is the geometric meaning of these invariants?

Evidence suggests that these enumerative invariants are associated with moduli stacks of coherent sheaves of modules over a sheaf B of noncommutative algebras on X constructed by Kuznetsov.  One of these moduli stacks is a stacky small resolution X’ of X itself.  This leads to another geometric interpretation of the invariants as being associated with moduli of sheaves on X’ twisted by a Brauer class.  Geometric computations based on these working definitions always agree with the B-model computations.

This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.

 

Details

Date:
November 13, 2023
Time:
10:30 am - 11:30 am
Event Category:

Venue

CMSA Room G10
CMSA, 20 Garden Street
Cambridge, MA 02138 United States
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Phone:
6174967132