## Deformation theory of a birationally commutative surface of Gelfand-Kirillov dimension 4

##### Abstract

Let K be the field of complex numbers. In this thesis we construct new examples of
noncommutative surfaces of GK-dimension 4 using the language of formal and infinitesimal
deformations as introduced by Gerstenhaber. Our approach is to find families of
deformations of a certain well known GK-dimension 4 birationally commutative surface
defined by Zhang and Smith in unpublished work cited in [YZ06], which we call A.
Let B* and K* be respectively the bar and Koszul complexes of a PBW algebra
C = KhV / (R) . We construct a graph whose vertices are elements of the free algebra KhV i
and edges are relations in R. We define a map m2 : B2 ! K2 that extends to a
chain map m* : B* → K*. This map allows the Gerstenhaber bracket structure to be
transferred from the bar complex to the Koszul complex. In particular, m2 provides a
mechanism for algorithmically determining the set of infinitesimal deformations with
vanishing primary obstruction.
Using the computer algebra package 'Sage' [Dev15] and a Python package developed
by the author [Cam], we calculate the degree 2 component of the second Hochschild
cohomology of A. Furthermore, using the map m2 we describe the variety U ⊆ HH2/2 (A)
of infinitesimal deformations with vanishing primary obstruction. We further show that
U decomposes as a union of 3 irreducible subvarieties Vg, Vq and Vu.
More generally, let C be a Koszul algebra with relations R, and let E be a localisation
of C at some (left and right) Ore set. Since R is homogeneous in degree two,
there is an embedding R ,↪ C⊗C and in the following we identify R with its (nonzero)
image under this map. We construct an injective linear map ~⋀ : HH²(C) → HH²(E)
and prove that if f ∈ HH²(E) satisfies f(R) ⊆ C then f ∈ Im(~⋀). In this way we
describe a relationship between infinitesimal deformations of C with those of E.
Rogalski and Sierra [RS12] have previously examined a family of deformations of
A arising from automorphism of the surface P1 X P1. By applying our understanding
of the map ~⋀ we show that these deformations correspond to the variety of infinitesimal
deformations Vg. Furthermore, we show that deformations defined similarly by
automorphisms of other minimal rational surfaces also correspond to infinitesimal deformations
lying in Vg.
We introduce a new family of deformations of A, which we call Aq. We show that
elements of this family have families of deformations arising from certain quantum
analogues of geometric automorphisms of minimal rational surfaces, as defined by Alev
and Dumas. Furthermore, we show that after taking the semi-classical limit
q → 1 we obtain a family of deformations of A whose infinitesimal deformation lies in
Vq.
Finally, we apply a heuristic search method in the space of Hochschild 2-cocycles of
A. This search yields another new family of deformations of A. We show that elements
of this family are non-noetherian PBW noncommutative surfaces with GK-dimension
4. We further show that elements of this family can have as function skew field the
division ring of the quantum plane Kq(u; v), the division ring of the first Weyl algebra
D1(K) or the commutative field K(u; v).