Download Differential Forms on Singular Varieties: De Rham and Hodge by Vincenzo Ancona PDF

By Vincenzo Ancona

Differential varieties on Singular types: De Rham and Hodge concept Simplified makes use of complexes of differential kinds to offer an entire remedy of the Deligne thought of combined Hodge buildings at the cohomology of singular areas. This ebook gains an method that employs recursive arguments on size and doesn't introduce areas of upper measurement than the preliminary house. It simplifies the idea via simply identifiable and well-defined weight filtrations. It additionally avoids dialogue of cohomological descent conception to take care of accessibility. subject matters comprise classical Hodge concept, differential kinds on complicated areas, and combined Hodge buildings on noncompact areas.

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4) xjb We denote Vm (M ) the vector space of complex tangent vectors at m. 5) Complex manifolds, vector bundles, differential forms 21 Recall that by definition ∂ ∂ 1 ∂ − i 2j 2j−1 2 ∂xa ∂xa ∂ ∂ 1 ∂ . 6) ζaj = ξa2j−1 − iξa2j . 8) ζaj . 9) and T m (M ) the vector space of complex tangent vectors at m of type n ζaj v= j=1 ∂ ∂ z¯aj so that Vm (M ) = Tm (M ) ⊕ T m (M ). 3 Classical Hodge theory Holomorphic functions A holomorphic function f : U → C is a continuous differentiable function such that in U ∩ Ua , f ◦ za−1 is holomorphic in za (U ∩ Ua ).

Zan (m) ∈ Cn is called the complex coordinates of m ∈ Ua in Ua . Let U = (Ua ) be another system of complex coordinates, with coordinates za : Ua → za (Ua ). We say that the system (Ua , za ) is equivalent to the system (Ua , za ) if the changes of coordinates za ◦ za−1 are biholomorphic whenever defined. A complex structure is an equivalence class of systems of complex coordinates, and M is then called a complex manifold. A local chart on M is a couple (U, z) where U is an open set of M and z : U → z(U ) ⊂ Cn is a homeomorphism on an open set of Cn , such that z ◦ za−1 is holomorphic on za (Ua ∩ U ).

40) means that : ⎛ 1 ⎛ 1 ⎞ ⎞ ωb (m) ωa (m) ⎜ . ⎟ ⎜ .. ⎟ ⎝ . ⎠ = γab (m) ⎝ .. ⎠ ωar (m) ωbr (m) for m ∈ Ua ∩ Ub , or in other words r ωaα (m) α γab,β (m)ωbβ (m). 41) β=1 In general, it is not possible to define the exterior differentiation of a form with coefficients in F as a form with coefficients in F . Indeed, one can define for each trivialization the differentials dωa1 , . . , dωar and dωb1 , . . 41) because in general dγab = 0. It is only for locally flat bundles, namely when one can choose constant transition matrices γab , that one can define intrinsically the differential of a k-form with coefficient in F .

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