2/26 and 2/27 

John Pardon 

Universally counting curves in Calabi--Yau threefolds

February 26, 5:30-7:30, Math 507 

Abstract: Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety.  There are many such compactifications, hence many different enumerative invariants.  I will propose a "universal" (very tautological) enumerative invariant which takes values in a certain "Grothendieck group of 1-cycles".  It is often the case with such "universal" constructions that the resulting Grothendieck group is essentially uncomputable.  But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves.  This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan--Pandharipande and Okounkov--Pandharipande.

February 27, 4:10-5:25, Math 407 (Special Colloquium) 

Abstract: Statements such as "there is a unique line between any pair of distinct points in the plane" and "there are 27 lines on any cubic surface" have given rise to the modern theory of enumerative geometry.  To define such "curve counts" in a general setting usually involves choosing a particularly nice compactification of the space of smooth embedded curves (one which admits a natural "virtual fundamental class").  I will propose a new perspective on enumerative invariants which is based instead on a certain "Grothendieck group of 1-cycles" and the "universal" curve enumeration invariant taking values in this group.  It turns out that if we restrict to complex threefolds with nef anticanonical bundle, this group has a very simple structure: it is generated by "local curves".  This generation result implies some new cases of the MNOP conjecture relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants of complex threefolds.