Nonlinear descent on moduli of local systems
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NONLINEAR DESCENT ON MODULI OF LOCAL SYSTEMS BY
Junho Peter Whang Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA e-mail: [email protected] ABSTRACT
We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.
Contents
1. Introduction . . . . . . . 2. Background . . . . . . . 3. Systoles of local systems 4. Compactness criterion . 5. Further remarks . . . . References . . . . . . . . . .
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1. Introduction 1.1. This paper initiates our Diophantine study of moduli spaces for local systems on surfaces and their mapping class group dynamics. Let us fix a smooth compact oriented surface Σ of genus g ≥ 0 with n ≥ 0 boundary curves, labeled c1 , . . . , cn , satisfying 3g + n − 3 > 0. For each k = (k1 , . . . , kn ) ∈ Cn , let Xk denote the coarse moduli space of SL2 (C)-local systems on Σ with trace ki along ci . Each Xk is an irreducible complex affine algebraic variety of dimension 6g + 2n − 6, and we showed in [60] that it is log Calabi–Yau if the surface has nonempty boundary. For k ∈ Zn , the variety Xk admits a natural model Received January 15, 2019 and in revised form July 21, 2020
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over Z, its integral points corresponding to those local systems whose monodromy elements all have integer trace. The pure mapping class group of Σ acts on Xk via pullback of local systems, preserving the integral structure. Our main Diophantine result for Xk (Z) is the following. Let us say that an algebraic variety Z over C is parabolic if each C-point of Z is in the image of some nonconstant morphism A1 → Z. A related notion is that of a log uniruled variety, defined in [33]. Let us define a subvariety of Xk to be degenerate if it is contained in a parabolic subvariety of Xk , and nondegenerate otherwise. These definitions and our formulation of Theorem 1.1 reflect considerations of Diophantine geometry for log Calabi–Yau varieties, further discussed in Section 1.3. Theorem 1.1: The nondegenerate integral points in Xk (Z) consist of finitely many mapping class group orbits. There exists a parabolic proper closed subvariety of Xk whose orbit gives precisely the locus of degenerate points on Xk . In particular, Xk (Z) is generated from finitely many proper closed irreducible subvarieties of Xk under the mapping class group. We shall obtain Theorem 1.1 by combining the results of this paper (outlined in Section 1.2) with our work in [61], where the second part of Theorem 1.1 is proved together with a modular characterization of the degenerate points on Xk . The latter is summarized in the following paragraph. Throughout this paper, by an essential curve on Σ we
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