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Families of singular algebraic varieties that are rationally elliptic spaces

A. Libgober

TL;DR

The paper investigates rationally elliptic spaces arising from singular algebraic varieties with nef canonical or anti-canonical bundles, and analyzes finiteness of homotopy and deformation types. It builds explicit singular models (e.g., $H(a_0,\dots,a_{n+1})$, $V_n^d$, $W^d_n$) to realize $\mathbb{R}$- or $\mathbb{Q}$-homotopy types of projective spaces or quadrics, using monodromy polynomials $\Phi^d_n(t)$, Milnor numbers, and cohomology ring presentations to establish rational ellipticity. The Appendix proves a finiteness result for smooth rationally elliptic 3-folds with nef canonical data and demonstrates infinite families sharing the same real homotopy type but distinct integral homotopy types, highlighting a separation between real and integral classifications. Overall, the work clarifies when rational ellipticity implies finiteness versus infinitude in the algebraic setting and provides concrete singular and smooth models that connect topology with algebraic geometry.

Abstract

We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti-canonical class. In the appendix we show that such an infinite family of smooth rationally elliptic 3-folds does not exist.

Families of singular algebraic varieties that are rationally elliptic spaces

TL;DR

The paper investigates rationally elliptic spaces arising from singular algebraic varieties with nef canonical or anti-canonical bundles, and analyzes finiteness of homotopy and deformation types. It builds explicit singular models (e.g., , , ) to realize - or -homotopy types of projective spaces or quadrics, using monodromy polynomials , Milnor numbers, and cohomology ring presentations to establish rational ellipticity. The Appendix proves a finiteness result for smooth rationally elliptic 3-folds with nef canonical data and demonstrates infinite families sharing the same real homotopy type but distinct integral homotopy types, highlighting a separation between real and integral classifications. Overall, the work clarifies when rational ellipticity implies finiteness versus infinitude in the algebraic setting and provides concrete singular and smooth models that connect topology with algebraic geometry.

Abstract

We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti-canonical class. In the appendix we show that such an infinite family of smooth rationally elliptic 3-folds does not exist.

Paper Structure

This paper contains 4 sections, 8 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Homotopy projective spaces
  3. Homotopy quadrics
  4. Appendix: Smooth rationally elliptic algebraic varieties in dimensions 2 and 3.

Key Result

Theorem 2.1

Let $a_i \in {\mathbb N}, i=0,\cdots n+1$ where $n \ge 3$ is odd. Following J.Kollár (cf. kollaryau, Sec. 5) denote by $d=a_0\cdots a_{n+1}+(-1)^{n+1}$, the determinant of the following system of $n+2$ equations with unknowns $w_i$ where for $j\in {\mathbb Z}$, $\bar{j}$ is defined by conditions: $0 \le \bar{j} \le n+1$ and $j \equiv \bar{j} \ mod \ n+2$. Assume that where $w_i$ are the positive

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • ...and 13 more