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Mapping Class Groups of Simply Connected Kähler Manifolds

Richard Hain

TL;DR

The paper investigates mapping class and Torelli groups for simply connected compact Kahler manifolds, linking these groups to monodromy of moduli spaces and to rational homotopy theory via Johnson-type invariants. It develops a Johnson-type framework using loop-space homology and Adams filtration, and employs Sullivan's minimal models to relate the Torelli group to unipotent completions and to a central distortion group $D_M$, yielding a main theorem that the unipotent completion of $T_M$ maps onto the Torelli data of the rational model with kernel $D_M$. In the hypersurface setting, Carlson–Toledo show a large kernel for the monodromy into the mapping class group and infinite index for the image in certain dimensions, highlighting a substantial deviation from the curve case and motivating open problems about linearity, residual finiteness, and Hodge-theoretic structures on moduli. The work provides explicit Lie-algebraic models in the formal 6-manifold case, demonstrates nontrivial Torelli phenomena via explicit automorphisms, and sets a broad program connecting topology, algebraic geometry, and Hodge theory in higher-dimensional moduli problems.

Abstract

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to stimulate future work. Apart from reviewing general background, the paper focuses on the case of hypersurfaces in projective space. We explain how older results of Carlson--Toledo arXiv:alg-geom/9708002 and recent results of Kreck--Su arXiv:2009.08054 imply that the homomorphism from the fundamental group of the moduli space of hypersurfaces in P^4 to the mapping class group of the underlying manifold has a very large kernel (contains a free group of rank 2) and has image of infinite index. This is in contrast to the case of curves, where the homomorphism is an isomorphism.

Mapping Class Groups of Simply Connected Kähler Manifolds

TL;DR

The paper investigates mapping class and Torelli groups for simply connected compact Kahler manifolds, linking these groups to monodromy of moduli spaces and to rational homotopy theory via Johnson-type invariants. It develops a Johnson-type framework using loop-space homology and Adams filtration, and employs Sullivan's minimal models to relate the Torelli group to unipotent completions and to a central distortion group , yielding a main theorem that the unipotent completion of maps onto the Torelli data of the rational model with kernel . In the hypersurface setting, Carlson–Toledo show a large kernel for the monodromy into the mapping class group and infinite index for the image in certain dimensions, highlighting a substantial deviation from the curve case and motivating open problems about linearity, residual finiteness, and Hodge-theoretic structures on moduli. The work provides explicit Lie-algebraic models in the formal 6-manifold case, demonstrates nontrivial Torelli phenomena via explicit automorphisms, and sets a broad program connecting topology, algebraic geometry, and Hodge theory in higher-dimensional moduli problems.

Abstract

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to stimulate future work. Apart from reviewing general background, the paper focuses on the case of hypersurfaces in projective space. We explain how older results of Carlson--Toledo arXiv:alg-geom/9708002 and recent results of Kreck--Su arXiv:2009.08054 imply that the homomorphism from the fundamental group of the moduli space of hypersurfaces in P^4 to the mapping class group of the underlying manifold has a very large kernel (contains a free group of rank 2) and has image of infinite index. This is in contrast to the case of curves, where the homomorphism is an isomorphism.
Paper Structure (19 sections, 26 theorems, 109 equations)

This paper contains 19 sections, 26 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Quick Review of the Classical Johnson Homomorphism
  3. Some Classical Results from Homotopy Theory
  4. Adams' Spectral Sequence for Loop Space Homology
  5. Johnson homomorphisms for Simply Connected Manifolds
  6. Distortion of the Pontryagin classes
  7. Hypersurfaces and complete intersections
  8. Other Johnson homomorphisms
  9. Sullivan's Results
  10. Cohomology Automorphisms of a Compact Kähler Manifold
  11. Proof of Theorem \ref{['thm:main']}
  12. Lie Algebra Models
  13. The DG-Lie algebra model of a finite complex
  14. The Lie algebra model of a simply connected closed formal 6-manifold
  15. Self Rational Homotopy Equivalences of a Simply Connected Closed Formal 6-manifold
  16. ...and 4 more sections

Key Result

Theorem 1

If $M$ is a simply connected compact Kähler 3-fold, there are $S_M$-invariant surjections where $\operatorname{Sym}^2$ denotes the symmetric square and $\Delta : H_4(M;{\mathbb Q}) \to S^2 H_2(M;{\mathbb Q})$ is the dual of the cup product.

Theorems & Definitions (46)

  • Theorem 1
  • Corollary 2
  • Theorem 3: Kreck--Su
  • Theorem 4
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Proposition 4.4
  • proof
  • Corollary 5.1
  • ...and 36 more