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Nef cones of fiber products and an application to the Cone Conjecture

Cécile Gachet, Hsueh-Yung Lin, Long Wang

Abstract

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron--Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher dimensional analogues of the Calabi--Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi--Yau pairs, and in each dimension at least three, there exist Schoen varieties with non-polyhedral nef cone. We prove the Kawamata--Morrison--Totaro Cone Conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.

Nef cones of fiber products and an application to the Cone Conjecture

Abstract

We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron--Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher dimensional analogues of the Calabi--Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi--Yau pairs, and in each dimension at least three, there exist Schoen varieties with non-polyhedral nef cone. We prove the Kawamata--Morrison--Totaro Cone Conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.
Paper Structure (19 sections, 24 theorems, 91 equations)

This paper contains 19 sections, 24 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Cone Conjecture
  3. Nef cones of fiber products
  4. Cone Conjecture for Schoen varieties
  5. Relation to other work
  6. Cone Conjecture
  7. Cone Conjectures for varieties with rational polyhedral nef cones
  8. Fiber product constructions
  9. Cone conjecture for movable cones
  10. Structure of the paper
  11. Preliminaries
  12. Notations
  13. Klt Calabi--Yau pairs
  14. Looijenga's result
  15. The nef cone of a fiber product over a curve
  16. ...and 4 more sections

Key Result

Theorem 1.4

For $i=1,2$, let $\phi_i : W_i \to B$ be a surjective morphism from a projective variety to a projective curve $B$. Assume that Then As a consequence, we also have $p_1^*{\rm Amp}(W_1) + p_2^*{\rm Amp}(W_2)={\rm Amp}(W)$.

Theorems & Definitions (60)

  • Conjecture 1.1: Kawamata--Morrison--Totaro Cone Conjecture
  • Conjecture 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • ...and 50 more