Anishing Theorems for the Logarithmic DE RHAM Complex of Unitary Local System
Abstract
Let X be a non-singular complex projective variety with dimC X = n. We will prove two cohomology vanishing theorems for unitary vector bundle E on X with flat (integrable) connection ∇, which has at worst logarithmic singularities along some boundary divisor D. We will assume D is a simple normal crossing divisor.Such an vector bundle has a de Rham complex DRX(D, E). One approach for the vanishing theorems is to construct a mixed Hodge theory on the DRX(D, E). Then, we will be able to apply the results from Deligne’s study on abstract Hodge theory. The vanishing theorems are then the consequence spectral sequence degeneration as stated in [1] and [2]. Another approach is to interpret E as a semistable Higgs bundle with trivial Higgs field θ. Then the first vanishing theorem is a consequence of the main result of [3]. We will present both approaches in this work.
Degree
Ph.D.
Advisors
Arapura, Purdue University.
Subject Area
Mathematics
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