Examples Violating Golyshev’s Canonical Strip Hypotheses

Abstract

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.

Keywords:

• geography of Fano manifolds
• geography of Calabi-Yau manifolds
• roots of Hilbert polynomials
• Verlinde formula
• moduli spaces of vector bundles on algebraic curves; toric varieties

MSC codes:

• 14J45; 14J32; 14D20

Acknowledgments

The authors thank the referee for their comments.

Notes

1 A strengthening of the narrow canonical strip hypothesis for Fano varieties involving the index ${\iota }_X$ of X, i.e. with the notation of Definition 2 one asks for ${\alpha }_i\in [-1+1/{\iota }_X,-1/{\iota }_X],$ when $H=-{\mathrm{K}}_X.$

Additional information

Funding

The first and third author were supported by the Max Planck Institute for Mathematics in Bonn. The second author was partially supported by the Hausdorff Center for Mathematics during the trimester program “Periods in Number Theory, Algebraic Geometry and Physics” and by the Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N. 14.641.31.0001.