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The Donaldson-Thomas invariants under blowups and flops. (English) Zbl 1256.14039
In 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.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Citations:
Zbl 1023.14029; Zbl 0948.53046; Zbl 1062.53073; Zbl 1349.14014
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