Representability of Chern–Weil forms
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In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern–Weil form can be represented by a given form? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.
The author is grateful to Harish Seshadri for useful suggestions, as well as for his support and encouragement. We also thank M.S. Narasimhan for pointing out the existence of approximate Hermite–Einstein metrics on semi-stable bundles, and Mario Garcia–Fernandez for answering questions about his paper. Lastly, our gratitude extends copiously to the anonymous referee for a careful reading of this work.
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