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Two-step nilpotent monodromy of local systems on special varieties

Junyan Cao, Ya Deng, Christopher D. Hacon, Mihai Paun

Abstract

Let $X$ be a smooth complex quasi-projective variety that is special in the sense of Campana. We prove that the monodromy group of any complex local system on $X$ is virtually nilpotent of class at most $2$. This result sharply refines a theorem of Cadorel, Yamanoi, and the second author. To establish this result, we develop a deformation theory for certain local systems on quasi-compact Kähler manifolds by constructing universal deformations for such local systems. As a byproduct of our argument, we also show that a general fiber of the quasi-Albanese map of $X$ is special, extending a result of Campana and Claudon from the projective to the quasi-projective setting.

Two-step nilpotent monodromy of local systems on special varieties

Abstract

Let be a smooth complex quasi-projective variety that is special in the sense of Campana. We prove that the monodromy group of any complex local system on is virtually nilpotent of class at most . This result sharply refines a theorem of Cadorel, Yamanoi, and the second author. To establish this result, we develop a deformation theory for certain local systems on quasi-compact Kähler manifolds by constructing universal deformations for such local systems. As a byproduct of our argument, we also show that a general fiber of the quasi-Albanese map of is special, extending a result of Campana and Claudon from the projective to the quasi-projective setting.
Paper Structure (40 sections, 71 theorems, 565 equations, 1 figure)

This paper contains 40 sections, 71 theorems, 565 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Idea of the proof of \ref{['main']}
  4. Technical Preliminaries
  5. Special varieties and orbifolds
  6. Hodge decomposition for currents and $\partial\bar{\partial}$-lemma
  7. A quick tour of Goldman-Millson theory
  8. Transformation groupoids
  9. DGLA
  10. DGLA and deformation functor
  11. Application to the variety of representations
  12. An extension to quasi-compact Kähler manifolds
  13. New perspectives on deformations of local systems on compact Kähler manifolds
  14. Universal connection: analytic construction
  15. Universal connection: universal property
  16. ...and 25 more sections

Key Result

Theorem 1

Let $X$ be a smooth complex quasi-projective variety. Suppose that $X$ is special in the sense of Campana (for example, if its logarithmic Kodaira dimension $\bar{\kappa}(X)=0$). Then for any linear representation $\varrho \colon \pi_1(X)\to \mathop{\mathrm{GL}}\nolimits_N(\mathbb{C})$, the image $\

Figures (1)

  • Figure 1: Logical dependencies between the main theorems

Theorems & Definitions (147)

  • Theorem 1: =\ref{['thm:main']}
  • Definition 1.1: Nilpotent group
  • Definition 1.3: Quasi-compact Kähler
  • Theorem 2: =\ref{['thm:final']}
  • Theorem 3: $\subset$\ref{['main1', 'thm:universal2', 'thm:Step 2']}
  • Theorem 4: =\ref{['thm:zigzag']}
  • Theorem 5: =\ref{['thm:special']}
  • Corollary 6: =\ref{['cor:structure']}
  • Definition 2.1: Multiplicity and Orbifold base
  • Definition 2.2: Orbifold morphism
  • ...and 137 more