The Quantum Lefschetz Principle for Vector Bundles as a Map Between Givental Cones

Abstract

Givental has defined a Lagrangian cone in a symplectic vector space which encodes all genus-zero Gromov-Witten invariants of a smooth projective variety X. Let Y be the subvariety in X given by the zero locus of a regular section of a convex vector bundle. We review arguments of Iritani, Kim-Kresch-Pantev, and Graber, which give a very simple relationship between the Givental cone for Y and the Givental cone for Euler-twisted Gromov-Witten invariants of X. When the convex vector bundle is the direct sum of nef line bundles, this gives a sharper version of the Quantum Lefschetz Hyperplane Principle.

arXiv comment7 pages

References in corpus (4)

• A Mirror Theorem for Toric Stacks
• The Abelian/Nonabelian Correspondence and Frobenius Manifolds
• Quantum Cohomology and Periods
• The Quantum Lefschetz Hyperplane Principle Can Fail for Positive Orbifold Hypersurfaces

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