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$P=W$ phenomena on abelian varieties

Barbara Bolognese, Alex Küronya, Martin Ulirsch

Abstract

Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product.

$P=W$ phenomena on abelian varieties

Abstract

Let be a complex abelian variety. We prove an analogue of both the (cohomological) conjecture and the geometric conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on and the Betti moduli space of characters of the fundamental group of . The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product.
Paper Structure (24 sections, 14 theorems, 109 equations)

This paper contains 24 sections, 14 theorems, 109 equations.

Table of Contents

  1. The Dolbeault moduli space on a complex abelian variety
  2. Preliminaries on Higgs bundles and (semi-)stability
  3. (Semi-)Stability of Higgs bundles on abelian varieties
  4. The moduli space of semistable Higgs bundles
  5. The Hitchin fibration and its compactification
  6. A factorization of the Hitchin morphism
  7. Compactification of the spectral data morphism
  8. The non-abelian Hodge correspondence
  9. The non-abelian Hodge correspondence in rank one
  10. The non-abelian Hodge correspondence for abelian varieties - made explicit
  11. A geometric $P=W$ phenomenon
  12. Dual complexes of dlt compactifications
  13. A geometric $P=W$ phenomenon - Proof of Theorem \ref{['mainthm_geometricP=W']}
  14. The perverse filtration
  15. The perverse $t$-structure
  16. ...and 9 more sections

Key Result

Theorem 1

Let $X$ be a complex abelian variety of dimension $g\geq 1$. Then, under the identification eq_cohomology induced by the non-abelian Hodge correspondence, we have for all $k\in\mathbb{Z}$.

Theorems & Definitions (35)

  • Theorem 1: P=W for abelian varieties
  • Theorem 2: Curious Poincaré and curious hard Lefschetz
  • Theorem 3: Geometric $P=W$ for abelian varieties
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Proposition 1.4
  • proof : Proof of Proposition \ref{['prop_semistableHiggs']}
  • Proposition 1.5
  • proof : Proof of Proposition \ref{['prop_stableHiggs']}
  • ...and 25 more