We study rational curves on smooth complex Calabi–Yau
–folds via
noncommutative algebra. By the general theory of derived noncommutative deformations
due to Efimov, Lunts and Orlov, the structure sheaf of a rational curve in a smooth CY
–fold
is pro-represented by a nonpositively graded dg
algebra . The curve
is called
nc rigid if
is
finite-dimensional. When
is contractible,
is isomorphic to the contraction algebra defined by Donovan
and Wemyss. More generally, one can show that there exists a
pro-representing
the (derived) multipointed deformation (defined by Kawamata) of a collection of rational
curves
with
. The collection is called
ncrigid if
is finite-dimensional.
We prove that
is a
homologically smooth bimodule 3–CY algebra. As a consequence, we define a (2–CY) cluster category
for such a collection
of rational curves in .
It has finite-dimensional morphism spaces if and only if the collection is nc rigid. When
is (formally) contractible
by a morphism
, then
is equivalent to the
singularity category of
and thus categorifies the contraction algebra of Donovan and Wemyss. The Calabi–Yau structure
on
determines a
canonical class
(defined up to right equivalence) in the zeroth Hochschild homology of
.
Using our previous work on the noncommutative Mather–Yau theorem and
singular Hochschild cohomology, we prove that the singularities underlying a
–dimensional
smooth flopping contraction are classified by the derived equivalence class of the pair
. We
also give a new necessary condition for contractibility of rational curves in terms of
.
Keywords
contractible curves, cluster categories, noncommutative
deformations, quivers with potentials
