Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Canonical liftings of Calabi--Yau hypersurfaces: Dwork hypersurfaces

Przemysław Grabowski

Abstract

We explicitly compute canonical liftings modulo $p^2$ in a sense of Achinger--Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo $p^2$.

Canonical liftings of Calabi--Yau hypersurfaces: Dwork hypersurfaces

Abstract

We explicitly compute canonical liftings modulo in a sense of Achinger--Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo .
Paper Structure (9 sections, 21 theorems, 57 equations)

This paper contains 9 sections, 21 theorems, 57 equations.

Table of Contents

  1. Partial computation of the crystalline Frobenius
  2. Hypersurfaces
  3. Dwork hypersurfaces
  4. Symmetries
  5. Smoothness
  6. Hasse--Dwork Polynomials
  7. Ordinarity
  8. Computation on Invariant Elements
  9. Canonical Liftings Modulo $p^2$

Key Result

Lemma 1.1

Let $A$ be a ring in which $p^n A = 0$ and let $F\colon A\to A$ be a lifting of Frobenius. Let $I\subseteq A$ be an ideal such that $F(I) \subseteq I^p + p^m A$ for some $m\leq n$ (which is automatic for $m=1$). Then In particular, if $p^2 A = 0$, then for every ideal $I\subseteq A$ we have $F(I^2)\subseteq I^2$.

Theorems & Definitions (47)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • ...and 37 more