Deformations of representations of fundamental groups of complex varieties

Abstract

We describe locally the representation varieties of fundamental groups for smooth complex manifolds admitting a compactification into a Kähler manifold, at representations coming from the monodromy of a variation of mixed Hodge structure. Given such a manifold X and such a linear representation ρ of its fundamental group π 1 (X, x), we use the theory of Goldman-Millson and pursue our previous work that combines mixed Hodge theory with derived deformation theory to construct a mixed Hodge structure on the formal local ring Oρ to the representation variety of π 1 (X, x) at ρ. Then we show how a weighted-homogeneous presentation of Oρ is induced directly from a splitting of the weight filtration of its mixed Hodge structure. In this way we recover and generalize theorems of Eyssidieux-Simpson (X compact) and of Kapovich-Millson (ρ finite).

Mixed Hodge structures on cohomology jump ideals

“…Such theorems are called homotopy transfer of structures. Furthermore, the deformation functors can also be written for L ∞ algebras, and thus computed with H(L) instead of L. We motivated this intensively and used this in our two previous articles [Lef19b,Lef19a].…”

“…This is not so trivial, the ideas come from rational homotopy theory starting with the work of Morgan [Mor78] for the analogous problem with commutative DG algebras. We largely discussed this in [Lef19a] and can now use it directly in § 6 to deal with the additional data of a module over L.…”

“…With mixed Hodge structure. Now we turn exactly to the setting of [Lef19a]: X is either a smooth complex algebraic varieties, and we work with a compactification X ⊂ X by a normal crossing divisor, or X is the complement of such a divisor inside a compact Kähler manifold X (i.e. X is quasi-Kähler).…”

“…Here we can use directly our previous work which is formulated in terms of lax symmetric monoidal functors. Briefly, this means that in [Lef19a] we defined a functor MHC(X, X, V) from VMHS with unipotent monodromy at infinity to MHC, depending on the compactification, such that for two such VMHS V 1 , V 2 there is a natural map…”

“…A MHS on O ρ by Eyssidieux-Simpson [ES11] when X is compact. We generalized this in [Lef19b] (representations with finite image) and then in [Lef19a] for all smooth varieties and representations that are the monodromy of a variation of mixed Hodge structure. In this case, the cohomology of X with local coefficients also carries MHS: this due to Deligne, Zucker, Steenbrink.…”