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Nowhere vanishing 1-forms on varieties admitting a good minimal model

Benjamin Church

Abstract

We prove several conjectures relating the existence of nonvanishing 1- forms to smooth morphisms over abelian varieties, assuming the existence of good minimal models. The proof involves a decomposition result for a family of Calabi-Yau varieties equipped with a surjective map to an abelian scheme. In the uniruled case, supposing the MRC base admits a good minimal model, we also achieve a structure theorem for those varieties admitting nowhere vanishing 1-forms.

Nowhere vanishing 1-forms on varieties admitting a good minimal model

Abstract

We prove several conjectures relating the existence of nonvanishing 1- forms to smooth morphisms over abelian varieties, assuming the existence of good minimal models. The proof involves a decomposition result for a family of Calabi-Yau varieties equipped with a surjective map to an abelian scheme. In the uniruled case, supposing the MRC base admits a good minimal model, we also achieve a structure theorem for those varieties admitting nowhere vanishing 1-forms.

Paper Structure

This paper contains 11 sections, 18 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Decomposition of the Abelian Variety Part
  3. The Albanese and Automorphism Schemes In Families
  4. Application to the Iitaka Fibration
  5. Proofs of the Main Results
  6. Conjecture 1.8 of HS21(1)
  7. Conjecture C of MP21
  8. Conjecture B of CCH23
  9. Extension to the Uniruled Case
  10. Generic Nonvanishing Along Rational Maps
  11. Application to the MRC Fibration

Key Result

Theorem 1

Let $X \to A$ be a morphism from a smooth projective variety $X$ to an abelian variety $A$. Assume $\omega_1, \dots, \omega_g \in H^0(A, \Omega_A)$ are $1$-forms such that $f^* \omega_1, \dots, f^* \omega_g$ are pointwise linearly independent. Assuming $X$ admits a good minimal model then there is a with $X' \to B$ an isotrivialflat with isomorphic fibers, in fact we show that $X' \to A$ is trivia

Theorems & Definitions (41)

  • Theorem 1: = Theorem \ref{['thm:Iitaka_decomposition_PLI']}
  • Example 1.1
  • Conjecture 2
  • Theorem 3: = Theorem \ref{['thm:MP_conjecture_C']}
  • Theorem 4: = Theorem \ref{['thm:PLI_number']}
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 31 more