On the derived category of the classical Godeaux surface

TL;DR

This paper constructs a maximal-length exceptional sequence on the classical Godeaux surface, providing a counterexample to Kuznetsov's Nonvanishing Conjecture and revealing symmetries related to the E_8 root lattice.

Contribution

It explicitly constructs the longest possible exceptional sequence on the Godeaux surface and demonstrates its implications for Hochschild homology and conjectures.

Findings

Maximal exceptional sequence length of 11 on the Godeaux surface

Counterexample to Kuznetsov's Nonvanishing Conjecture

Symmetry related to the E_8 root lattice

Abstract

We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z^11+Z/5. In particular, the result answers Kuznetsov's Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E_8. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.