Existence of nontrivial logarithmic co-Higgs structure on curves

Abstract

We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end we also study the higher dimensional case, specially when the tangent bundle is not semistable.

25 open questions about vector bundles and their moduli

“…Nevertheless, they are interesting varieties as attaining some of the extremal cases for the generalized geometric structures considered by N. Hitchin and M. Gualtieri [Gua,Hi3,Hi4]. The following is a partial list of the papers studying them and some of extensions of this notion, such as logarithmic coHiggs sheaves [Col,BH1,BH2,BH3,BH4,BHM,Co,Gua,Hi2,R1,R2,R3].…”

Toric co-Higgs bundles on toric varieties

“…Co-Higgs bundles have been classified and/or constructed on P 1 [15,4], P 2 [16], P 1 × P 1 [19], and logarithmic curves [1], for example. Over singular varieties, they have been used to some effect towards establishing inequalities related to vectorvalued modular forms [6].…”