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Fujiki relations and fibrations of irreducible symplectic varieties

Martin Schwald

Abstract

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds.

Fujiki relations and fibrations of irreducible symplectic varieties

Abstract

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds.

Paper Structure

This paper contains 20 sections, 22 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Symplectic varieties
  3. Main results
  4. Outline of the paper
  5. Conventions
  6. Acknowledgment
  7. Methods
  8. Extension Theorem
  9. Terminal models
  10. Integration of singular top classes
  11. Hodge Theory for klt varieties
  12. Properties of symplectic varieties
  13. Lagrangian fibrations
  14. Generalized Beauville-Bogomolov form
  15. Definitions of the Beauville-Bogomolov form
  16. ...and 5 more sections

Key Result

Theorem 2

Let $(X,\omega)$ be a $2n$-dimensional, irreducible symplectic variety. The Beauville-Bogomolov form $q_X$ has the following properties:

Theorems & Definitions (45)

  • Definition 1: Symplectic varieties
  • Theorem 2: Fujiki relations, index of the Beauville-Bogomolov form
  • Theorem 3: Fibrations of irreducible symplectic varieties
  • Theorem 4: Fibrations of primitive symplectic varieties
  • Theorem 5: Pullbacks of reflexive forms, MR3084424
  • Theorem 6: Terminal models, BCHM10
  • Lemma 7: Pullbacks of integrals, Sch17b
  • Theorem 8: Hodge decomposition of $\textup{H}^2(X,\,\mathds{C})$, Sch17b, cf. also Kir15, GKP16
  • Remark 9
  • Corollary 10: Bilinear relations on klt varieties, Sch17b
  • ...and 35 more