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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The $3$–fold vertex via stable pairs

Rahul Pandharipande and Richard P Thomas

Geometry & Topology 13 (2009) 1835–1876

The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3–folds. We evaluate the equivariant vertex for stable pairs on toric 3–folds in terms of weighted box counting. In the toric Calabi–Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities.

The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.

curve, threefold, Gromov–Witten, toric
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 14M25, 14D20, 14J30
Received: 3 June 2008
Accepted: 25 February 2009
Published: 24 March 2009
Proposed: Jim Bryan
Seconded: Lothar Goettsche, Frances Kirwan
Rahul Pandharipande
Department of Mathematics
Princeton University
Princeton, NJ 08544
Richard P Thomas
Department of Mathematics
Imperial College
London SW7 2AZ