The Donaldson-Thomas invariants under blowups and flops.

*(English)*Zbl 1256.14039In the paper under review, the authors study how Donaldson-Thomas invariants change under blow-up of a point, some flops and extremal transitions. The motivation comes from previously known results about the behaviour of Gromov-Witten invariants under blow-ups (studied by J. Hu [Compos. Math. 125, No. 3, 345–352 (2001; Zbl 1023.14029)] and [Math. Z. 233, No. 4, 709–739 (2000; Zbl 0948.53046)]), flops and extremal transitions (by A.-M. Li and Y. Ruan in [Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)]).

If \(X\) is a smooth projective \(3-\)fold, \(n\in\mathbb{Z}\) and \(\beta\in H_{2}(X,\mathbb{Z})\), we write \(I_{n}(X,\beta)\) for the moduli space of closed subschemes \(Y\) of dimension 1 of \(X\) such that \(\chi(\mathcal{O}_{Y})=n\) and \([Y]=\beta\). The virtual dimension of \(I_{n}(X,\beta)\) is \(\int_{\beta}c_{1}(T_{X})\). For every \(k_{1},\dots ,k_{r}\in\mathbb{N}\) and for every real cohomology classes \(\gamma_{l_{1}},\dots ,\gamma_{l_{r}}\), we let \(Z_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}}))_{\beta}:=\sum_{n}<\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}})>_{n,\beta}q^{n}\) be the Donaldson-Thomas partition function with descendent intersection (where \(<\tilde{\tau}_{k_{1}}(\gamma_{l_{1}}),\dots ,\tilde{\tau}_{k_{r}}(\gamma_{l_{r}})>_{n,\beta}\) is the descendent invariant). We write \(Z'_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}}))_{\beta}\) for the reduced partition function (obtained from \(Z_{DT}\) by formally removing the degree 0 contributions).

In section 3, the authors study the behaviour of the relative reduced DT-partition function under the blow-up \(p:\widetilde{X}\to X\) of \(X\) along a smooth point \(x\). The main result they show is that for every \(\gamma_{l_{1}},\dots ,\gamma_{l_{r}}\in H^{*}(X,\mathbb{R})\) we have \(Z'_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{0}(\gamma_{l_{i}}))_{\beta}=Z'_{DT}(\widetilde{X};q|\prod_{i=1}^{r}\tilde{\tau}_{0}(p^{*}\gamma_{l_{i}}))_{\iota(\beta)}\), where \(\iota:H_{2}(X)\to H_{2}(\widetilde{X})\) is the natural injection. The main idea of the proof is to consider the blow-up \(\mathcal{X}\) of \(X\times\mathbb{C}\) along \((x,0)\): then \(\pi:\mathcal{X}\to\mathbb{C}\) is a semistable degeneration of \(X\) whose central fiber \(\mathcal{X}_{0}\) is the union of \(\widetilde{X}\) and \(\mathbb{P}^{3}\) (which is the exceptional divisor of \(\mathcal{X}\). The idea is then to use a degeneration formula of the reduced DT-partition functions showed, for example, by J. Li and B. Wu [Commun. Anal. Geom. 23, No. 4, 841–921 (2015; Zbl 1349.14014)].

In section 4, the authors study the behaviour of the reduced DT-partition function under some flops. More precisely, let \(D\) be an effective divisor, and suppose that \(X\) has an extremal contraction \(\phi:X\to Y\) with respect to \(K_{X}+\epsilon D\) for \(0<\epsilon\ll 1\), with exceptional locus given by finitely many disjoint \((-1,-1)-\)curves \(\Gamma_{2},\dots ,\Gamma_{l}\), and let \(\phi^{f}:X^{f}\to Y\) be the flop of \(\phi\). The authors show that if \(\beta=m[\Gamma]\), then \(Z'_{DT}(X;q)_{\beta}=Z'_{DT}(X^{f};q)_{-\varphi_{*}(\beta)}\), where \(\varphi_{*}:H_{2}(X,\mathbb{Z})\to H_{2}(X^{f},\mathbb{Z})\) is the isomorphism. For more general \(\beta\), the authors show that the relation between the reduced DT-partition functions on \(X\) and \(X^{f}\) are expressed by means of some power series (see Theorem 4.2). Even in this case, the main idea is to use some degeneration (here one blows up \(X\times\mathbb{C}\) along the curves \(\Gamma_{i}\times\mathbb{C}\)) together with degeneration formulas for \(Z'_{DT}\).

Section 5 is devoted to the study of extremal transitions in the case of a small resolution \(\rho:X\to X_{0}\), where \(X\) is a smooth Calabi-Yau \(3-\)fold and \(X_{0}\) is a singular projective \(3-\)fold whose singularities are ordinary double points.

If \(X\) is a smooth projective \(3-\)fold, \(n\in\mathbb{Z}\) and \(\beta\in H_{2}(X,\mathbb{Z})\), we write \(I_{n}(X,\beta)\) for the moduli space of closed subschemes \(Y\) of dimension 1 of \(X\) such that \(\chi(\mathcal{O}_{Y})=n\) and \([Y]=\beta\). The virtual dimension of \(I_{n}(X,\beta)\) is \(\int_{\beta}c_{1}(T_{X})\). For every \(k_{1},\dots ,k_{r}\in\mathbb{N}\) and for every real cohomology classes \(\gamma_{l_{1}},\dots ,\gamma_{l_{r}}\), we let \(Z_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}}))_{\beta}:=\sum_{n}<\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}})>_{n,\beta}q^{n}\) be the Donaldson-Thomas partition function with descendent intersection (where \(<\tilde{\tau}_{k_{1}}(\gamma_{l_{1}}),\dots ,\tilde{\tau}_{k_{r}}(\gamma_{l_{r}})>_{n,\beta}\) is the descendent invariant). We write \(Z'_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{k_{i}}(\gamma_{l_{i}}))_{\beta}\) for the reduced partition function (obtained from \(Z_{DT}\) by formally removing the degree 0 contributions).

In section 3, the authors study the behaviour of the relative reduced DT-partition function under the blow-up \(p:\widetilde{X}\to X\) of \(X\) along a smooth point \(x\). The main result they show is that for every \(\gamma_{l_{1}},\dots ,\gamma_{l_{r}}\in H^{*}(X,\mathbb{R})\) we have \(Z'_{DT}(X;q|\prod_{i=1}^{r}\tilde{\tau}_{0}(\gamma_{l_{i}}))_{\beta}=Z'_{DT}(\widetilde{X};q|\prod_{i=1}^{r}\tilde{\tau}_{0}(p^{*}\gamma_{l_{i}}))_{\iota(\beta)}\), where \(\iota:H_{2}(X)\to H_{2}(\widetilde{X})\) is the natural injection. The main idea of the proof is to consider the blow-up \(\mathcal{X}\) of \(X\times\mathbb{C}\) along \((x,0)\): then \(\pi:\mathcal{X}\to\mathbb{C}\) is a semistable degeneration of \(X\) whose central fiber \(\mathcal{X}_{0}\) is the union of \(\widetilde{X}\) and \(\mathbb{P}^{3}\) (which is the exceptional divisor of \(\mathcal{X}\). The idea is then to use a degeneration formula of the reduced DT-partition functions showed, for example, by J. Li and B. Wu [Commun. Anal. Geom. 23, No. 4, 841–921 (2015; Zbl 1349.14014)].

In section 4, the authors study the behaviour of the reduced DT-partition function under some flops. More precisely, let \(D\) be an effective divisor, and suppose that \(X\) has an extremal contraction \(\phi:X\to Y\) with respect to \(K_{X}+\epsilon D\) for \(0<\epsilon\ll 1\), with exceptional locus given by finitely many disjoint \((-1,-1)-\)curves \(\Gamma_{2},\dots ,\Gamma_{l}\), and let \(\phi^{f}:X^{f}\to Y\) be the flop of \(\phi\). The authors show that if \(\beta=m[\Gamma]\), then \(Z'_{DT}(X;q)_{\beta}=Z'_{DT}(X^{f};q)_{-\varphi_{*}(\beta)}\), where \(\varphi_{*}:H_{2}(X,\mathbb{Z})\to H_{2}(X^{f},\mathbb{Z})\) is the isomorphism. For more general \(\beta\), the authors show that the relation between the reduced DT-partition functions on \(X\) and \(X^{f}\) are expressed by means of some power series (see Theorem 4.2). Even in this case, the main idea is to use some degeneration (here one blows up \(X\times\mathbb{C}\) along the curves \(\Gamma_{i}\times\mathbb{C}\)) together with degeneration formulas for \(Z'_{DT}\).

Section 5 is devoted to the study of extremal transitions in the case of a small resolution \(\rho:X\to X_{0}\), where \(X\) is a smooth Calabi-Yau \(3-\)fold and \(X_{0}\) is a singular projective \(3-\)fold whose singularities are ordinary double points.

Reviewer: Arvid Perego (Nancy)

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |