New York Journal of Mathematics
Volume 19 (2013) 925-945


Steven Rayan

Co-Higgs bundles on P1

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Published: November 27, 2013
Keywords: Co-Higgs bundle, Higgs bundle, Hitchin fibration, projective line, stability, moduli space, Betti numbers, holomorphic chain
Subject: 14D20, 14H60, 14D22

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defined equation. For ranks r=2,3,4, we explicitly verify the conjectural Betti numbers emerging from the recent work of Chuang, Diaconescu, Pan, and Mozgovoy on the ADHM formula. We state the result for r=5.


Parts of this work were funded by the Commonwealth Scholarship Plan and the Natural Sciences & Engineering Research Council of Canada.

Author information

Dept. of Mathematics, Univ. of Toronto, 40 St. George St., Toronto, ON, CANADA, M5S 2E4.