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On k-rational and k-Du Bois local complete intersections

Mircea Mustata, Mihnea Popa

Abstract

We show that k-rational singularities of local complete intersections are k-Du Bois. For hypersurfaces, we characterize k-rationality in terms of the minimal exponent. We also establish some local vanishing results for k-rational and k-Du Bois singularities. Some of these results have been independently obtained in [FL2].

On k-rational and k-Du Bois local complete intersections

Abstract

We show that k-rational singularities of local complete intersections are k-Du Bois. For hypersurfaces, we characterize k-rationality in terms of the minimal exponent. We also establish some local vanishing results for k-rational and k-Du Bois singularities. Some of these results have been independently obtained in [FL2].
Paper Structure (13 sections, 20 theorems, 136 equations)

This paper contains 13 sections, 20 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Definitions and background
  3. Log resolutions
  4. $k$-rational singularities
  5. Brief review of filtered $\mathscr{D}$-modules
  6. The Du Bois complex and $k$-Du Bois singularities
  7. Results for local complete intersections
  8. Injectivity and hierarchy of singularities
  9. Duality
  10. Local vanishing for $k$-Du Bois singularities
  11. Results for hypersurfaces
  12. Minimal exponent $>k+1$ implies $k$-rational singularities
  13. $k$-rational hypersurface singularities have minimal exponent $>k+1$

Key Result

Theorem 1

Let $Z$ be an algebraic variety which is locally a complete intersection, and let $k$ be a nonnegative integer such that $Z$ has $(k-1)$-Du Bois singularities. Then the morphism in the derived category of coherent sheaves on $Z$, obtained by dualizing the canonical morphism $\Omega_Z^k \to \underline{\Omega}_Z^k$, is injective at the level of cohomology.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Proposition 1.1
  • proof
  • Lemma 1.6
  • ...and 39 more