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Classification of Poor Manifolds in Low dimensions

Pisya Vikash

Abstract

The notion of poor manifolds was introduced by Bandman and Zarhin, who asked for their classification. We study poor compact Kähler manifolds, i.e. those containing no rational curves and no codimension-one analytic subvarieties. We classify such manifolds in dimensions at most $3$, and in arbitrary dimension under the additional assumption that $κ(X)\neq -\infty$. We also describe the locus of poor $K3$ surfaces in the period domain.

Classification of Poor Manifolds in Low dimensions

Abstract

The notion of poor manifolds was introduced by Bandman and Zarhin, who asked for their classification. We study poor compact Kähler manifolds, i.e. those containing no rational curves and no codimension-one analytic subvarieties. We classify such manifolds in dimensions at most , and in arbitrary dimension under the additional assumption that . We also describe the locus of poor surfaces in the period domain.
Paper Structure (7 sections, 22 theorems, 87 equations, 1 figure)

This paper contains 7 sections, 22 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Classification
  3. Preliminaries
  4. General classification
  5. Dimension 2
  6. Preliminaries
  7. Poor $K3$ Surfaces

Key Result

Theorem 1.4

[Beauville-Bogomolov Decomposition, beauville1983varietes] Let $X$ be a compact, connected Kähler manifold with $c_1(X)_{\mathbb{R}} = 0$. Then a finite unramified covering $\tilde{X}$ of $X$ decomposes holomorphically as where $T$ is a complex torus and $X_i$, $Y_j$ are simply connected complex manifolds such that:

Figures (1)

  • Figure 1: Moduli space of poor $K3$ surfaces.

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 1.5: Lemma 4.2 of bandman2023simple
  • Theorem 1.6
  • proof
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • ...and 44 more