Framed sheaves over treefolds and symmetric obstruction theories (Q1946070)

In the paper under review, the author studies moduli spaces of framed sheaves over local Calabi-Yau \(3-\)folds, and computes their Donaldson-Thomas Euler characteristics and topological Euler characteristics of the compactified moduli space of framed modules. If \(S\) is a smooth complex projective surface, we let \(X^{\circ}\) be the total space of the canonical bundle \(K_{S}\) of \(S\), which is an open Calabi-Yau \(3-\)fold, and \(\pi:X\longrightarrow S\) the projectivization of \(K_{S}+\mathcal{O}_{S}\), which is a compact \(3-\)fold carrying two divisors \(S_{\infty}\) (corresponding to the \(0-\)section of \(\mathcal{O}_{S}\)) and \(S_{0}\) (corresponding to the \(0-\)section of \(K_{S}\)), and we have \(X^{\circ}=X\setminus S_{\infty}\). The moduli space \(\mathfrak{M}_{n}\) of semistable framed modules \((E,\phi)\) of rank \(r\), trivial first and second Chern classes, \(\chi(E)=r\chi(\mathcal{O}_{S})-n\) and non-zero framing \(\phi:E\longrightarrow\mathcal{O}_{S_{\infty}}\otimes\mathbb{C}^{r}\) was constructed as a projective variety by \textit{D. Huybrechts} and \textit{M. Lehn} in [Int. J. Math. 6, No. 2, 297--324 (1995; Zbl 0865.14004)]. Here semistability is with respect to the ample divisor \(H=\pi^{*}H_{0}+\epsilon c_{1}(\mathcal{O}_{X}(1))\) (for some ample line bundle \(H_{0}\) on \(S\) and for \(\epsilon\in\mathbb{Q}_{+}\) sufficiently small), and with respect to a polynomial \(\delta\) of degree at most 2. Inside \(\mathfrak{M}_{n}\) there is the open subset \(\mathfrak{M}_{n}^{\circ}\) of framed sheaves, i. e. stable framed modules \((E,\phi)\) such that \(E\) is torsion free, locally free near \(S_{\infty}\) and \(\phi\) is an isomorphism along \(S_{\infty}\). In section 3 the author shows that \(\mathfrak{M}_{n}^{\circ}\) admits a symmetric perfect obstruction theory: an obstruction theory for \(\mathfrak{M}_{n}^{\circ}\) is a complex \(\mathbb{F}\) which can be associated to any universal family \((\mathcal{E},\Phi)\). Grothendieck duality and the fact that \(K_{X}=\mathcal{O}_{X}(-2S_{\infty})\) imply that \(\mathbb{F}\) is symmetric; stability of \((E,\phi)\in\mathfrak{M}_{n}^{\circ}\) implies that \(\Hom(E,E(-S_{\infty}))=0\), and this implies that the obstruction theory is perfect with amplitude in \([-1,0]\). The symmetric perfect obstruction theory has associated a Behrend function \(\nu:\mathfrak{M}^{\circ}_{n}\longrightarrow\mathbb{Z}\) allowing the computation of the Donaldson-Thomas weighted Euler characteristic \(\widetilde{\chi}(\mathfrak{M}_{n}^{\circ})\). The author computes it using equivariant localization. Namely, there is an action of \(\mathbb{C}^{*}\) on \(\mathfrak{M}_{n}\): the fixed point set \((\mathfrak{M}_{n}^{\circ})^{\mathbb{C}^{*}}\) inherits a symmetric perfect obstruction theory, and the corresponding Behrend function is that of \(\mathfrak{M}^{\circ}_{n}\) up to a sign depending on the difference of the dimension of the tangent spaces of \(\mathfrak{M}^{\circ}_{n}\) and \((\mathfrak{M}^{\circ}_{n})^{\mathbb{C}^{*}}\). The author shows that the \(\mathbb{C}^{*}-\)fixed framed modules in \(\mathfrak{M}_{n}\) are direct sums of ideals of \(\mathbb{C}^{*}\)-invariants zero-dimensional subschemes \(Z_{i}\) of \(X\) (for \(i=1,\dots,r\)), and those in \(\mathfrak{M}^{\circ}_{n}\) are those for which \(Z_{i}\subseteq X^{\circ}\). Using this description of the fixed points of the \(\mathbb{C}^{*}-\)action the author shows that the generating series of the Donaldson-Thomas weighted Euler characteristic of the \(\mathfrak{M}^{\circ}_{n}\) is the \(re(S)-\)th power of a MacMahon function. The generating series of the topological Euler characteristic of the \(\mathfrak{M}_{n}\) is the \(r-\)th power of the generating series of the topological Euler characteristic of \(\mathrm{Hilb}^{n}(X)\), which is computed by \textit{J. Cheah} [J. Algebr. Geom. 5, No. 3, 479--511 (1996; Zbl 0889.14001)]. Slightly more complicated cases are considered in the last sections: one is the case of a surface \(S\) containing a \((-1)-\)curve \(C\), and we consider the moduli space of semistable framed modules of rank \(r\), \(c_{1}=0\), \(c_{2}=k[C]\) and \(\chi\) and framing as before. Other cases are those for which we replace the projective bundle of \(K_{S}+\mathcal{O}_{S}\) by more general \(3-\)folds \(X\) with a smooth framing divisor \(S_{\infty}\) such that \(K_{X}=-2S_{\infty}\).