Shifted Lagrange multipliers method

Algebraic Geometry and Number Theory Seminar

Modern enumerative geometry is about integrals of cohomology classes against virtual fundamental classes of moduli spaces. When the perfect obstruction theory of a moduli space admits a cosection, the virtual fundamental class is localized to the zero locus of the cosection. When the cosection can be enhanced to a (-1)-shifted closed 1-form, the zero locus admits a (-2)-shifted symplectic structure and hence a virtual fundamental class due to a recent construction of Oh-Thomas. I will talk about a joint work with Hyeonjun Park where we prove that the two virtual fundamental classes coincide up to sign. When applied to a shifted form of Lagrange multipliers method, the result gives us an immediate proof of the quantum Lefschetz principle by Chang-Li.