Curve counting via stable pairs in the derived category
type: Item
instance of: scholarly article
title: Curve counting via stable pairs in the derived category
zbMATH Open document ID: 1204.14026
author: Rahul Pandharipande
author: Richard P. Thomas
published in: Inventiones Mathematicae
publication date: 2009-11-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0707.2348
review text: Counting algebraic curves in projective varieties is one of the oldest themes in algebraic geometry. The modern approach involves a compactification of the moduli space of curves carrying a virtual class of the expected dimension, and obtaining curve counts by integrating over this class. Unfortunately, various degeneracies leave all existing approaches incomplete in one way or another. The oldest compactification by stable maps is due to Kontsevich and leads to the Gromov-Witten invariants. However, in general these are rational numbers accounting for non-trivial automorphisms of stable maps, and cannot be straightforward counts. On the other hand, Bogomol'nyi, Prasad and Sommerfield suggested a heuristic way based on considerations from string theory of defining integers interpreted as counts of embedded curves in \(3\)-folds. It was later discovered that these counts, now called BPS states, can be obtained from the Gromov-Witten invariants with clever algebraic manipulations. Finding their direct geometric definition remains an open challenge ever since. One approach leads to the integer-valued Donaldson-Thomas invariants produced from the compactification of the moduli of ideal sheaves. However, their geometric meaning is obscured by extra points required by the definition that can roam all over the \(3\)-fold. An algebraic reduction is again necessary to produce the BPS states. Recently, Honsen suggested another compactification based on Cohen-Macaulay curves which is free of automorphisms and roaming points, but no virtual class is known for it and hence no invariants can be defined. The authors modify Honsen's construction by considering stable pairs \((C,D)\) of embedded Cohen-Macaulay curve \(C\) together with a Cartier divisor \(D\subset C\). When \(D\) is empty one essentially gets the ideal sheaf \(\mathcal{I}_C\) , more generally the curve is decorated with finitely many extra points, which however are restricted to lie on it. The natural obstruction th…
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5627037
zbMATH Keywords: stable pairs
zbMATH Keywords: Cohen-Macaulay curves
zbMATH Keywords: BPS states
zbMATH Keywords: Gromov-Witten invariants
zbMATH Keywords: topological vertex
MaRDI profile type: Publication
OpenAlex ID: W2042792562
arXiv ID: 0707.2348
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DOI: 10.1007/S00222-009-0203-9
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