Monodromy of Primitive Vanishing Cycles for Hypersurfaces in \mathbb P⁴

2024/01/14 by Yilong Zhang, Zhang, Yilong
Mathematics · #14D05 #14D06 Primary #14F45 #57N40 Secondary #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2401.07202

openalex publication_date 2024/01/14 · openalex created_date 2024/01/18 · openalex updated_date 2026/07/28

Abstract

Let X be a complex submanifold of projective space. Schnell showed that the middle-dimensional primitive cohomology of X is generated by tube classes, which arise from the monodromy of the vanishing homology on hyperplane sections. Clemens asks if the theorem is still true when we restrict the generating set to the tube classes over the class of a single vanishing sphere of nodal degeneration. We prove this is true for hypersurfaces in CP⁴. The proof is based on the degeneration of a hypersurface to the union of hypersurfaces of lower degrees.

Related

• Primitive cohomology and the tube mapping
• Tube classes over elementary vanishing cycles
• Generalized Thomas hyperplane sections and relations between vanishing cycles
• Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces
• Monodromy of a family of hypersurfaces containing a given subvariety