Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

Matt Kerr, Radu Laza

Abstract

We study the weighted spectrum and vanishing cohomology for several classes of isolated hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of singularities arising in KSBA and GIT compactifications and mirror symmetry, including nodes on odd-dimensional hypersurfaces, $k$-log-canonical and $k$-rational singularities, and singularities with Calabi-Yau tail.

Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

Abstract

We study the weighted spectrum and vanishing cohomology for several classes of isolated hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of singularities arising in KSBA and GIT compactifications and mirror symmetry, including nodes on odd-dimensional hypersurfaces, -log-canonical and -rational singularities, and singularities with Calabi-Yau tail.

Paper Structure

This paper contains 11 sections, 28 theorems, 120 equations, 2 tables.

Table of Contents

  1. Mixed spectra of isolated singularities
  2. The quasi-homogeneous case
  3. Weighted Fermat singularities and deformations
  4. Nodes on odd-dimensional hypersurfaces
  5. Singularities with Calabi--Yau tail
  6. Isolated hypersurface singularities: birational invariants
  7. Isolated hypersurface singularities: spectral combinatorics
  8. Toric geometry approach
  9. Brieskorn lattice approach
  10. Sebastiani--Thom formula
  11. Application to $k$-log-canonical singularities

Key Result

Theorem A

The isolated quasi-homogeneous surface singularities with pure $K3$ tail (Def. def2.2) are precisely the 14 Dolgachev singularities, the 6 quadrilateral singularities, and 2 trimodal singularities ($V_{15}$ and $N_{16}$).

Theorems & Definitions (82)

  • Theorem A: cf. Theorem \ref{['TCY']}
  • Theorem B: cf. Corollary \ref{['cor2.4a']}, Theorem \ref{['thm2.5A']}, $\S$\ref{['sec-ex-klog']}
  • Remark 2
  • Remark 3: Recent developments on higher Du Bois singularities
  • Remark 4: Higher rational singularities
  • Theorem C: cf. Proposition \ref{['prop2.4b']}
  • Corollary C: cf. Corollaries \ref{['cor2.5B1']}-\ref{['cor2.5B2']}
  • Theorem D: cf. Theorem \ref{['t:schoen']}
  • Remark 5
  • Theorem 6
  • ...and 72 more