ar X iv : d g - ga / 9 50 30 15 v 1 3 0 M ar 1 99 5 Projective structures on moduli spaces of compact complex hypersurfaces ∗
نویسندگان
چکیده
It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions. 1. Projective connections. Let M be a complex manifold. Consider the following equivalence relation on the set of affine torsion-free connections on M : two connections Γ̂ and Γ are said to be projectively equivalent if they have the same geodesics, considered as unparameterized paths. In a local coordinate chart {tα}, α = 1, . . . ,dimM on M , where Γ̂ and Γ are represented by Christoffel symbols Γ̂γαβ and Γ γ αβ respectively, this equivalence relation reads [H] Γ̂ ∼ Γ if Γ̂γαβ = Γ γ αβ + bβ δ γ α + bα δ γ β for some 1-form b = bα dt α. An equivalence class of torsion-free affine connections under this relation is called a projective structure or a projective connection. Let M be a complex manifold with a projective structure. A complex submanifold P ⊂ M is called totally geodesic if for each point t ∈ P and each direction tangent to P at t the corresponding geodesic of the projective connection is contained in P at least locally. 2. Moduli spaces of compact complex hypersurfaces. Let X be a compact complex hypersurface in a complex manifold Y with normal line bundle N such that H1(X,N) = 0. According to Kodaira [K-1], such a hypersurface X belongs to the complete analytic family {X t | t ∈ M } of compact complex hypersurfaces X t in Y with the moduli space M being a
منابع مشابه
ar X iv : d g - ga / 9 70 10 11 v 1 2 7 Ja n 19 97 MODULI SPACES OF STABLE POLYGONS AND SYMPLECTIC STRUCTURES ON
In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces Mr,ε of stable polygons are constructed as complex analytic spaces. These new moduli spaces are shown to be projective and isomorphic to the moduli space M0,n of the Deligne-Mumford stable curves of genus 0. Built into the structures of stable poly...
متن کاملar X iv : d g - ga / 9 51 00 06 v 1 2 3 O ct 1 99 5 GROUP SYSTEMS , GROUPOIDS , AND MODULI SPACES OF PARABOLIC BUNDLES
متن کامل
ar X iv : d g - ga / 9 60 50 07 v 1 1 9 M ay 1 99 6 Structure “ Hyper - Lie Poisson ” ∗
The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10] [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra a...
متن کاملar X iv : a lg - g eo m / 9 70 30 04 v 1 5 M ar 1 99 7 Geometry of Moduli Spaces of Flat Bundles on Punctured Surfaces
For a Riemann surface with one puncture we consider moduli spaces of flat connections such that the monodromy transformation around the puncture belongs to a given conjugacy class with the property that a product of its distinct eigenvalues is not equal to 1 unless we take all of them. We prove that these moduli spaces are smooth and their natural closures are normal with rational singularities.
متن کاملar X iv : d g - ga / 9 70 10 11 v 2 2 8 A ug 1 99 8 MODULI SPACES OF STABLE POLYGONS AND SYMPLECTIC STRUCTURES ON
In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces Mr,ε of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space M0,n of the Deligne-Mumford stable curves of genus 0. Further, built into ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1997