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Geometry of cohomology support loci for local systems
 I. J. Algebraic Geom
, 1997
"... X will stand for a smooth projective variety over C throughout these notes. To simplify some statements, it will be convenient (but not essential) to also assume that (*) H 2 (X, Z) is torsion free. To fix notation recall: The Picard variety P ic 0 (X) is the set of line bundles with trivial first C ..."
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X will stand for a smooth projective variety over C throughout these notes. To simplify some statements, it will be convenient (but not essential) to also assume that (*) H 2 (X, Z) is torsion free. To fix notation recall: The Picard variety P ic 0 (X) is the set of line bundles with trivial first Chern class (the Chern class can be taken H 2 (X, C), since we are assuming that H 2 (X, Z) is torsion free) This is an abelian variety of dimension q(X) = dim H 1 (X, OX). Green and Lazarsfeld have introduced the cohomology support loci S i (X) = {L ∈ P ic 0 (X)  H i (X, L) ̸ = 0} along with certain variants. They proved the following amazing theorem [GL]: Theorem 0.1 (GreenLazarsfeld). S i (X) is a union of translates of abelian subvarieties. Simpson later showed that these are in fact translates by points of finite order. There are now several proofs of this theorem in addition to the original. I would like to explain a couple of “topological ” proofs, due to Simpson [S3] and the speaker [A1]. I also want to say something about the nonabelian version of this in part II. 1. local systems The first step is to find a topological version of S i (X). By a local system, we mean a locally constant sheaf. Let ˜ X → X denote the universal cover. Given a representation ρ: π1(X) → GL(V), we have a diagonal action of π1(X) on V × ˜ X. Then we can form the local system Vρ of locally constant sections of (V × ˜ X)/π1(X) → X. We have the standard fact:
The Symplectic Geometry of Polygons in Euclidean Space
 Journal of Diff. Geometry
, 1998
"... We study the symplectic geometry of moduli spaces M r of polygons with fixed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complexanalytically isomorphic to the weighted quotient of (S 2 ) n constructed by Deligne and Mostow. We ..."
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We study the symplectic geometry of moduli spaces M r of polygons with fixed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complexanalytically isomorphic to the weighted quotient of (S 2 ) n constructed by Deligne and Mostow. We study the Hamiltonian flows on M r obtained by bending the polygon along diagonals and show the group generated by such flows acts transitively on M r . We also relate these flows to the twist flows of Goldman and JeffreyWeitsman. Contents 1 Introduction 2 2 Moduli of polygons and weighted quotients of configuration spaces of points on the sphere 5 3 Bending flows and polygons. 15 4 Actionangle coordinates 21 This research was partially supported by NSF grant DMS9306140 at University of Utah (Kapovich) and NSF grant DMS9205154, the University of Maryland (Millson). 1 5 The connection with gauge theory and the results of Goldman and JeffreyWeitsman 24 6 Transitivity of bending ...
Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
A Darboux theorem for Hamiltonian operators in the formal calculus of variations
 Duke Math. J
"... We prove a formal Darbouxtype theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain a ..."
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Cited by 75 (2 self)
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We prove a formal Darbouxtype theorem for Hamiltonian operators of hydrodynamic type, which arise as dispersionless limits of the Hamiltonian operators in the KdV and similar hierarchies. We prove that the Schouten Lie algebra is a formal differential graded Lie algebra, which allows us to obtain an analogue of the Darboux normal form in this context. We include an exposition of the formal deformation theory of differential graded Lie algebras g concentrated in degrees [−1, ∞); the formal deformations of g are parametrized by a 2groupoid that we call the Deligne 2groupoid of g, and quasiisomorphic differential graded Lie algebras have equivalent Deligne 2groupoids. The Darboux theorem states that all symplectic structures on an affine space are locally isomorphic. Hamiltonian operators are a generalization of symplectic forms introduced by I. Gelfand and I. Dorfman [3] and are important in the study of integrable hierarchies such as the Korteweg–de Vries (KdV) and KodomtsevPetviashvili (KP) equations. It is natural to ask whether an analogue of the Darboux theorem holds for Hamiltonian operators. The problem is considerably simplified by restricting attention to formal deformations of a given Hamiltonian operator H. The study of the moduli space of deformations is then controlled by a differential graded (dg) Lie algebra, the Schouten Lie algebra, with differential ad(H). The study of formal deformations is closely related to the problem of calculating the cohomology of this dg Lie algebra, which was posed by P. Olver [9].
DG coalgebras as formal stacks
 J. Pure Appl. Algebra
"... 1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k) ..."
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1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k)
Topology and geometry of cohomology jump loci
 DUKE MATHEMATICAL JOURNAL
, 2009
"... Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk a ..."
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Cited by 49 (27 self)
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Abstract. We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic, if the group is 1formal; in particular, the tangent cone to Vk at 1 equals Rk. These new obstructions to 1formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasiprojective algebraic variety. This sheds new light on a classical problem of J.P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.
Descent of Deligne groupoids
 Int. Math. Res. Notices
, 1997
"... Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence betwee ..."
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Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence between Σg and the Deligne groupoid corresponding to g. The main result of the paper claims that the functor Σ commutes up to homotopy with the ”total space ” functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman [S1, S2, HS3] which implies that if a deformation problem is described “locally ” by a sheaf of dg Lie algebras g on a topological space X then the global deformation problem is described by the homotopy Lie algebra RΓ(X, g). 1.