The A-infinity Algebra of a Curve and the J-invariant
Datum
2012
Autor:innen
Zeitschriftentitel
ISSN der Zeitschrift
Bandtitel
Verlag
University of Oregon
Zusammenfassung
We choose a generator G of the derived category of coherent sheaves on a smooth
curve X of genus g which corresponds to a choice of g distinguished points P1, . . . , Pg on X.
We compute the Hochschild cohomology of the algebra B = Ext (G,G) in certain internal
degrees relevant to extending the associative algebra structure on B to an A1-structure, which
demonstrates that A1-structures on B are finitely determined for curves of arbitrary genus.
When the curve is taken over C and g = 1, we amend an explicit A1-structure on B
computed by Polishchuk so that the higher products m6 and m8 become Hochschild cocycles.
We use the cohomology classes of m6 and m8 to recover the j-invariant of the curve. When
g 2, we use Massey products in Db(X) to show that in the A1-structure on B, m3 is
homotopic to 0 if and only if X is hyperelliptic and P1, . . . , Pg are chosen to be Weierstrass
points.
iv
Beschreibung
Schlagwörter
A-infinity, Curve, Elliptic curve, Hochschild cohomology, j-invariant