Punctual noncommutative Hilbert schemes

TL;DR

The work introduces and analyzes punctual noncommutative Hilbert schemes ${}^0{\rm Hilb}^{(m)}(V)$, projective moduli of finite-codimensional left ideals in noncommutative formal power series rings. It develops invariant-theoretic descriptions, Grassmannian embeddings, and affine pavings indexed by $m$-ary trees, together with a Springer-type small resolution $Z^{(m)}(V)\to {}^0{\rm Hilb}^{(m)}(V)$, enabling explicit computation of motives and the intersection-cohomology Poincaré polynomial. A Harder-Narasimhan stratification yields a functional equation for the motive generating series ${}^0F^{(m)}(t)$, and a second Motives calculation provides a closed formula for the Poincaré polynomial in intersection homology, $\sum_i \dim{\rm IH}^i({}^0{\rm Hilb}^{(m)}(V),\mathbb{Q}) q^{i/2}=\prod_{i=0}^{d-1}\frac{q^{(m-1)i+1}-1}{q-1}$. The results give a precise, motive-theoretic description of punctual NC Hilbert schemes and their small resolutions, with conjectured refinements via alternative affine pavings indexed by ordered $m$-ary trees.

Abstract

Punctual noncommutative Hilbert schemes are projective varieties parametrizing finite codimensional left ideals in noncommutative formal power series rings. We determine their motives and intersection cohomology, by constructing affine pavings and small resolutions of singularities.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 2.1
• Definition 3.1
• Example 3.2
• Theorem 4.1
• Corollary 4.2
• proof
• Theorem 5.1
• proof
• Proposition 6.1