Subbundles of the Hodge bundle over moduli of smooth curves
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For each genus g≥2, does the Hodge bundle over the moduli stack of smooth complex genus-g curves have any nonzero proper subbundle? This is the smooth-curve setting, rather than its stable-curve compactification.
Why it matters and what counts as progress
Why it matters: It tests rigidity of a natural bundle on smooth-curve moduli, where arguments for the compactified or abelian-variety settings do not settle the question.
What counts as progress: Rule out every nonzero proper subbundle on the smooth-curve moduli stack, beyond results for compactifications or abelian-variety moduli.
How the problem developed
- 26 Nov 2020 · Rank construction [1]
Víctor González-Alonso and Sara Torelli Constructed non-isotrivial curve families whose Higgs-kernel subbundle has any rank from 0 to g−1, while bounding the rank of the flat unitary subbundle by (g+1)/2.
- 2021 · Rank comparison [1]
Víctor González-Alonso and Sara Torelli Published the construction and its rank bounds, showing that the inclusion of the flat unitary subbundle in the Higgs kernel can be strict in geometric families.
AI contribution
The Simplicity of the Hodge Bundle
Patel’s revised paper claims that for every genus g≥2, the Hodge bundle on the moduli stack of smooth complex genus-g curves has no nontrivial subbundle. The proof uses traces of involutions and curves with commuting involutions to constrain a hypothetical subbundle’s rank.
Sources
Website timeline
Publication dates and editorial changes.