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Newton Polyhedrons and Hodge Numbers of Non-degenerate Laurent Polynomials

Haoxu Wang

Abstract

Claude Sabbah has defined the Fourier transform $G$ of the Gauss-Manin system for a non-degenerate and convenient Laurent polynomial and has shown that there exists a polarized mixed Hodge structure on the vanishing cycle of $G$. In this article, we consider certain non-degenerate and convenient Laurent polynomials $f_{P,\mathbf{a}}$, whose Newton polyhedron at infinity is a simplicial polytope $P$. First, we consider the stacky fan $\boldsymbolΣ_P$ given by $P$ and show that for each quotient stacky fan of $\boldsymbolΣ_P$, there is a natural polarized mixed Hodge structure on the ring of conewise polynomial functions on it. Then, we describe the polarized mixed Hodge structure on the vanishing cycle associated to $f_{P,\mathbf{a}}$ using these rings of conewise polynomial functions. In particular, we compute the Hodge diamond of the vanishing cycle. As a further consequence, we can solve the Birkhoff problem of such a Laurent polynomial by using elementary methods.

Newton Polyhedrons and Hodge Numbers of Non-degenerate Laurent Polynomials

Abstract

Claude Sabbah has defined the Fourier transform of the Gauss-Manin system for a non-degenerate and convenient Laurent polynomial and has shown that there exists a polarized mixed Hodge structure on the vanishing cycle of . In this article, we consider certain non-degenerate and convenient Laurent polynomials , whose Newton polyhedron at infinity is a simplicial polytope . First, we consider the stacky fan given by and show that for each quotient stacky fan of , there is a natural polarized mixed Hodge structure on the ring of conewise polynomial functions on it. Then, we describe the polarized mixed Hodge structure on the vanishing cycle associated to using these rings of conewise polynomial functions. In particular, we compute the Hodge diamond of the vanishing cycle. As a further consequence, we can solve the Birkhoff problem of such a Laurent polynomial by using elementary methods.
Paper Structure (10 sections, 13 theorems, 72 equations)

This paper contains 10 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Polytopes and stacky fans
  3. Stacky fans
  4. Conewise polynomial functions
  5. Quotient stacky fans
  6. Gauss-Manin system and Brieskorn lattice
  7. Twisted algebraic de Rham complex
  8. The Newton filtration
  9. The vanishing cycle
  10. The graded Jacobian ring

Key Result

Theorem 2.6

Let $l$ be a strictly convex conewise linear function on a complete simplicial fan $\Sigma$. Consider Then the tuple $\left(H_{\mathbb{Q}}(\Sigma), W_{\bullet}, F^{\bullet}, l, Q\right)$ is a polarized mixed Hodge structure of Hodge-Tate type and with weight $n$. (For the definition of polarized mixed Hodge structures, see e.g. hertling2002frobenius. For the definition of Hodge-Tate type, see e.g

Theorems & Definitions (36)

  • Definition 2.1: borisov2005orbifold
  • Example 2.2
  • Definition 2.3: barthel2002combinatorialbraden2006remarksfleming2010hard
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7
  • Corollary 2.8
  • Lemma 3.1
  • ...and 26 more