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Hodge adjacency conditions for singularities

RJ Acuna, Matt Kerr

Abstract

We prove compatibility relations between mixed Hodge numbers of $k$-Du Bois fibers in flat projective families and versal deformations of isolated $k$-Du Bois singularities. These extend the notion of polarized relations in asymptotic Hodge theory beyond the normal-crossing boundary case, and we study combinatorial properties of the resulting weak polarized relations graphs.

Hodge adjacency conditions for singularities

Abstract

We prove compatibility relations between mixed Hodge numbers of -Du Bois fibers in flat projective families and versal deformations of isolated -Du Bois singularities. These extend the notion of polarized relations in asymptotic Hodge theory beyond the normal-crossing boundary case, and we study combinatorial properties of the resulting weak polarized relations graphs.
Paper Structure (9 sections, 8 theorems, 31 equations)

This paper contains 9 sections, 8 theorems, 31 equations.

Table of Contents

  1. Weak polarized relations
  2. 1-parameter degenerations
  3. Multiparameter degenerations: normal-crossing discriminant
  4. Multiparameter degenerations: general discriminant
  5. Deformations of isolated singularities
  6. Computing the relation \ref{['eKL1']}
  7. Degenerations of weight 2 Hodge structures
  8. Degenerations of weight 3 Hodge structures
  9. Saturation for weak polarized relations graphs

Key Result

Proposition 2

If $\mathfrak{X}$ is smooth, with special fiber $X_0$$k$-log-canonical,Since $X_0$ has hypersurface singularities, $k$-log-canonical and $k$-Du Bois are equivalent by JKSY. then $H^*_{\mathrm{van}}$ belongs to $F^{k+1}$. Hence

Theorems & Definitions (25)

  • Conjecture 1
  • Proposition 2: KL3
  • Theorem 3
  • proof : Proof of Theorem \ref{['T1par']}
  • Remark 4
  • Remark 5
  • Definition 6
  • Theorem 7
  • Remark 8
  • Example 9
  • ...and 15 more