Curve counting via stable pairs in the derived category
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 theory for stable pairs is not \(2\)-term and does not produce a virtual class, but certain derived category provides an alternative obstruction theory which does. The theory of stable pairs provides the closest rigorous approximation to the heuristic BPS states to date. As an illustration, the authors compute local contributions of nonsingular embedded curves in Fano \(3\)-folds, and derive an analog of the topological vertex for computing invariants of the toric Calabi-Yau \(3\)-folds. The computations support a general conjecture that the new invariants coincide with the reduced Donaldson-Thomas invariants and are equivalent to the Gromov-Witten invariants. The paper also describes, based on ideas of Joyce, a conjectural interpretation of the equivalence as a wall-crossing formula in a derived category.
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