Counting conics on sextic \(4\)-folds (Q2278870)

The author studies rational degree \(2\) curves on a smooth sextic hypersurface \(X\) in \(\mathbb{P}^5\), which is a Calabi-Yau \(4\)-fold. There are two natural ways of counting them. One is the Donaldson-Thomas theory of the manifold of 1D stable sheaves \(M_\beta\) with Chern character \((0,0,0,\beta,1)\) (where \(\beta\in H_2(X,\mathbb{Z})\)) due to Borisov-Joyce and Cao-Leung. The other is the Gromov-Witten theory of the moduli of stable maps \(\overline{M}_{0,1}(X,\beta)\). Thus, for any \(\gamma\in H^4(X,\mathbb{Z})\) we obtain two counting invariants \(DT_4(\beta\,|\,\gamma)\) and \(GW_{0,\beta}(\gamma)\). The main result are the following relations between them when \(l\) is a line class: \begin{align*} & GW_{0,l}(\gamma)=DT_4(l\,|\,\gamma),\\ & GW_{0,2l}(\gamma)=DT_4(2l\,|\,\gamma) +\frac14DT_4(l\,|\,\gamma). \end{align*} The proof is based on studying the Hilbert scheme of conics on \(X\). After deforming to a generic hypersurface (which preserves the invariants), \(M_{2l}\) consists of structure sheaves of smooth conics and pairs of distinct intersecting lines. On the other hand, \(\overline{M}_{0,1}(X,\beta)\) is made of two connected components, one with embeddings of smooth and broken conics, and the other with double covers by \(\mathbb{P}^1\) of lines in \(X\). Hence the two terms in the second formula.