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On Hodge level of weighted complete intersections of general type

Victor Przyjalkowski

Abstract

We show that smooth varieties of general type which are well formed weighted complete intersections of Cartier divisors have maximal Hodge level, that is, their the rightmost middle Hodge numbers do not vanish. We show that this does not hold in the quasi-smooth case.

On Hodge level of weighted complete intersections of general type

Abstract

We show that smooth varieties of general type which are well formed weighted complete intersections of Cartier divisors have maximal Hodge level, that is, their the rightmost middle Hodge numbers do not vanish. We show that this does not hold in the quasi-smooth case.

Paper Structure

This paper contains 4 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted intersections of Cartier divisors of general type
  4. Hodge level for quasi-smooth weighted complete intersections of general type

Key Result

Theorem 1.1

Let $X\subset\mathbb{P}(a_0,\ldots,a_N)$ be a smooth well formed weighted complete intersection of multidegree $(d_1,\ldots,d_k)$. Put $n=N-k=\dim (X)>0$ and $i_X=\sum d_u-\sum a_l$. Then

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1: cf. ChenChenChen
  • Proposition 2.2: RoTe12 or the proof of Do82
  • Definition 2.3: cf. PST17
  • Remark 2.4
  • Theorem 2.5: cf. PSh20b
  • Corollary 2.6
  • proof
  • Theorem 2.7: see Di95, Gr69, Na97, Ma99
  • ...and 20 more