Hilbert schemes and stable pairs: GIT and derived category wall crossings (Q2880694)

On a smooth complex projective threefold \(X\) there are two curve counting theories, which are conjecturally equivalent: Donaldson-Thomas (DT) invariants, studied by \textit{D. Maulik}, \textit{N. Nekrasov}, \textit{A. Okounkov} and \textit{R. Pandharipande} in [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)], and Pandharipande-Thomas (PT) invariants, studied by \textit{R. Pandharipande} and \textit{R. P. Thomas} in [Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026)].NEWLINENEWLINEIf \(\beta\in H_{2}(X,\mathbb{Z})\) and \(n\in\mathbb{Z}\), let \(I_{n}(X,\beta)\) be the Hilbert scheme of subschemes \(Z\) of \(X\) in the class \([Z]=\beta\) with holomorphic Euler characteristic \(\chi(\mathcal{O}_{Z})=n\), and \(I_{n,\beta}:=e(I_{n}(X,\beta))\) be its Euler characteristic. Moreover, let \(P_{n}(X,\beta)\) be the moduli space of stable pairs \((F,s)\), where \(F\) is a pure sheaf on \(X\) with Chern character \((0,0,\beta,-n+\beta\cdot c_{1}(X)/2)\) and \(s\) is a section of \(F\) with \(0\)-dimensional cokernel, and let \(P_{n,\beta}:=e(P_{n}(X,\beta))\). Letting \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\) be the generating series of the \(I_{n,\beta}\) and of the \(P_{n,\beta}\) respectively, the author prove that \(Z^{P}_{\beta}(X)=Z^{I}_{\beta}(X)/Z^{I}_{0}(X)\), which is a topological version of the DT/PT-correspondence. For Calabi-Yau threefolds, this was obtained by \textit{Y. Toda} in [J. Am. Math. Soc. 23, No. 4, 1119--1157 (2010; Zbl 1207.14020)] and by \textit{T. Bridgeland} in [J. Am. Math. Soc. 24, No. 4, 969--998 (2011; Zbl 1234.14039)] with different methods. The strategy of the proof is the following: let \(C\) be any Cohen-Macaulay curve in \(X\) and \(I_{n,C}\) (resp. \(P_{n,C}\)) the Euler characteristic of the subset of \(I_{n}(X,\beta)\) (resp. \(P_{n}(X,\beta)\)) of subschemes whose underlying Cohen-Macaulay curve is \(C\) (resp. of pairs supported at \(C\)), and \(Z^{I}_{C}(X)\) (resp. \(Z^{P}_{C}(X)\)) their generating series. The authors show that \(Z^{I}_{C}(X)=Z^{P}_{C}(X)\cdot Z^{I}_{0}(X)\): integrating over all \(C\), one gets the previous statement involving \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\). In order to relate \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\), the authors provide a GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\), and they study the relation between the Euler characteristic of the fibers of this wall-crossing using the Ringel-Hall algebra machinery of Joyce.NEWLINENEWLINEIn section 2, the authors provide a GIT construction of \(P_{n}(X,\beta)\): Le Potier's construction of the moduli space of stable coherent systems presents \(P_{n}(X,\beta)\) as a quotient of a Quot scheme \(Q:=\mathrm{Quot}(\mathcal{H},P_{\beta})\), where \(\mathcal{H}=H^{0}(F(m))\otimes\mathcal{O}_{X}(-m)\). Here \(m\) is chosen so that for every pair \((F,s)\) the sheaf \(F(m)\) is globally generated, and we let \(V:=H^{0}(F(m))\).NEWLINENEWLINELet \(R_{m}:=H^{0}(\mathcal{O}_{X}(m))\): for any pair \((F,s)\) there is an inclusion \(H^{0}(F)\subseteq V\otimes R_{m}^{*}\). As \(s\) spans a one-dimensional subspace of \(H^{0}(F)\), the pair \((F,s)\) corresponds to a point of the projective space \(\mathbb{P}(V\otimes R_{m}^{*})\). The authors construct a suitable closed subscheme \(\mathcal{N}\) of \(\mathbb{P}(V\otimes R^{*}_{m})\times Q\) parameterizing stable pairs, together with a natural action of \(\mathrm{SL}(V)\) and two ample \(\mathbb{Q}\)-linearizations \(\mathcal{L}_{0}\) and \(\mathcal{L}_{1}\) for the action of \(\mathrm{SL}(V)\). Using the Hilbert-Mumford criterion, the authors show that \(P_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{1}}\mathrm{SL}(V)\) and \(I_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{0}}\mathrm{SL}(V)\). Letting \(\mathcal{L}_{t}:=(1-t)\mathcal{L}_{0}+t\mathcal{L}_{1}\), the authors show that the quotient \(SS_{n}(X,\beta)=\mathcal{N}^{ss}//_{\mathcal{L}_{t*}}\mathrm{SL}(V)\) (for some \(0<t^{*}<1\)) has a stratification \(SS_{n}(X,\beta)=\coprod_{k=0}^{n}I^{\mathrm{pur}}_{n-k}(X,\beta)\times S^{k}(X)\), where \(I^{\mathrm{pur}}_{n-k}(X,\beta)\) is the locus of the Cohen-Macaulay closed subschemes of \(I_{n-k}(X,\beta)\), and \(S^{k}(X)\) is the \(k\)-th symmetric product of \(X\). Moreover, there are two morphisms \(\varphi_{P}:P_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\) and \(\varphi_{I}:I_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\), which are isomorphism on the subschemes of pairs with surjective sections and pure support respectively. This is the GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\).NEWLINENEWLINENow, letting \(I_{n}(X,C)=\varphi_{I}^{-1}(C,S^{n}(X))\), \(P_{n}(X,C)=\varphi^{-1}_{P}(C,S^{n}(X))\) for any Cohen-Macaulay curve \(C\) in \(X\), and \(I_{n,C}=e(I_{n}(X,C))\), \(P_{n,C}=e(P_{n}(X,C))\), the authors show that \(I_{n,C}=P_{n,C}+e(X)P_{n-1,C}+e(\mathrm{Hilb}^{2}(X))P_{n-2,C}+\dots+e(\mathrm{Hilb}^{n}(X))P_{0,C}\), which implies the relation between the generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\). This is obtained by using the Ringel-Hall algebra machinery of Joyce: if \(\mathcal{T}\) is the stack of \(0\)-dimensional sheaves, Joyce provides a Ringel-Hall algebra \(H(\mathcal{T})\) together with an integration map \(P_{q}:H(\mathcal{T})\longrightarrow\mathbb{Q}(q^{1/2})[t]\). The generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\) are the limit (for \(q\rightarrow 1\)) of the integration map \(P_{q}\) computed over some stacks mapping to \(\mathcal{T}\) (namely: for \(Z^{I}_{C}(X)\) the stack \(\Hom(\mathcal{I}_{C},-)\), whose fiber over \(T\) is \(\Hom(\mathcal{I}_{C},T)\), and for \(Z^{P}_{C}(X)\) the stack \(Ext^{1}(-,\mathcal{O}_{C})\), whose fiber over \(T\) is \(Ext^{1}(T,\mathcal{O}_{C})\)). Using convolutions and relations between these stacks, one concludes with the relation between \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\).