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arXiv · 2412.00439

Lefschetz principle-type theorems for curve semistable Higgs sheaves and applications to elliptic surfaces

Armando Capasso
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Abstract

I prove ``Lefschetz principle''-type theorems for slope semistable and curve semistable Higgs sheaves on smooth projective varieties, defined over an algebraically closed field of characteristic $0$. These theorems are applied to reduce a conjecture, about curve semistable Higgs bundles, from the previous general setting to the complex case. This conjecture is equivalent to triviality of Chern classes of H-nflat Higgs bundles, which are particular curve semistable Higgs bundles. Where the base variety is a (complex) Jacobian elliptic surfaces, we prove that H-nflat Higgs bundles determine a well-defined subset of moduli spaces of Higgs sheaves: the ``H-nflat locus''. We prove that this is set-wise fixed by a natural action, and at ``limits'' one finds nilpotent, H-nflat Higgs bundles. As consequence, the conjecture for Jacobian elliptic surfaces is reduced to consider nilpotent, H-nflat Higgs bundles; moreover, their Higgs fields vanish, hence the conjecture holds (also) for Jacobian elliptic surfaces.

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Armando Capasso. 2024-11-30. Lefschetz principle-type theorems for curve semistable Higgs sheaves and applications to elliptic surfaces. https://arxiv.org/abs/2412.00439

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