Derived Hilbert schemes.

*(English)*Zbl 1074.14003Usually, moduli spaces in geometry are singular varieties. In order to avoid the difficulties related to their singular nature, the recently invented “Derived Deformation Theory Program (DDT)” aims at developing appropriate versions of the (non-abelian) derived functor of the respective moduli functor. Rather than ordinary varieties or schemes, the resulting geometric objects are sought to be “dg-schemes”, i.e., geometric objects whose algebras of functions are commutative differential graded algebras, and which are considered up to quasi-isomorphisms. While the DDT program appears to be well-established in the formal case, mainly in view of the recent fundamental work of M. Kontsevich, S. Barannikov, V. Hinich, M. Manetti, and others, the structure of global derived moduli spaces is much less understood.

In this vein, the aim of the paper under review is to provide a comprehensive DDT-type construction of the derived Hilbert scheme. In a foregoing paper [cf.: I. Ciocan-Fontanine and M. Kapranov, Ann. Sci. Ec. Norm. Supér., IV. Sér. 34, 403–440 (2001; Zbl 1050.14042)], the authors have already constructed a derived version of a first global algebro-geometric moduli space, namely of Grothendieck’s wellknown “Quot scheme”. Using a somewhat similar but nevertheless different approach, the authors are now investigating another important global moduli space in the context of the DDT program. While in the usual algebraic geometry, the Hilbert scheme is a particular case of the Quot scheme, the two constructions turn out to diverge considerably when passing to the framework of derived categories.

More precisely, let \(k\) be a field of characteristic zero, \(X\) a smooth projective variety over \(k\), and \({\mathcal O}_X(1)\) a very ample line bundle on \(X\) defining a projective embedding. For a given polynomial \(h\), the authors construct a dg-manifold \(\text{RHilb}^{\text{LCI}}_h(X)\) as the derived version of the classical geometric Hilbert scheme \(\text{Hilb}_h(X)\) of closed subschemes of \(X\) with Hilbert polynomial \(h\) relative to the polarization \({\mathcal O}_X(1)\). However, when the polynomial \(h\) is identically 1, then the derived Hilbert scheme turns out to coincide with the variety \(X\) whereas the derived Quot scheme \(\text{RQuot}({\mathcal O}_X)\) is known to be different from \(X\).

As for applications of these DDT-type constructions, the earlier constructed dg-manifolds RQuot are suitable for describing the derived moduli spaces of vector bundles on a fixed variety \(X\). In contrast, the dg-schemes \(\text{RHilb\,}h(X)\) established here are expected to play a similar rôle with regard to the derived moduli spaces of projective varieties themselves, which the authors corroborate by two striking examples. Namely, they use the explicit structure of the dg-schemes \(\text{RHilb\,}h(X)\) to construct two types of geometric derived moduli spaces:

(1) the derived space of maps \(\text{RMap}(C,Y)\) from a fixed projective scheme \(C\) to a fixed smooth projective variety \(Y\) and

(2) the derived stack of stable degree-\(d\) maps \(R\overline M_{g,n}(Y,d)\) from \(n\)-pointed nodal curves of genus \(g\) to a given smooth projective variety \(Y\). The latter example completes some earlier work of M. Kontsevich [in: The moduli spaces of curves, Prog. Math. 129, 335–368 (1995; Zbl 0885.14028)] and others (Behrend-Manin, Fulton-Pandharipande), thereby contributing to the mathematical theory of Gromov-Witten invariants. All in all, this is a very comprehensive paper of fundamental importance in derived moduli theory. The conceptual ingredients and refined techniques for the construction of derived Hilbert schemes include cotangent complexes, Harrison homology, derived moduli of operad algebras, derived schemes of ideals in finite-dimensional commutative algebras, and the theory of algebraic stacks. In spite of its highly advanced character, the exposition is very detailed, systematic and clear.

In this vein, the aim of the paper under review is to provide a comprehensive DDT-type construction of the derived Hilbert scheme. In a foregoing paper [cf.: I. Ciocan-Fontanine and M. Kapranov, Ann. Sci. Ec. Norm. Supér., IV. Sér. 34, 403–440 (2001; Zbl 1050.14042)], the authors have already constructed a derived version of a first global algebro-geometric moduli space, namely of Grothendieck’s wellknown “Quot scheme”. Using a somewhat similar but nevertheless different approach, the authors are now investigating another important global moduli space in the context of the DDT program. While in the usual algebraic geometry, the Hilbert scheme is a particular case of the Quot scheme, the two constructions turn out to diverge considerably when passing to the framework of derived categories.

More precisely, let \(k\) be a field of characteristic zero, \(X\) a smooth projective variety over \(k\), and \({\mathcal O}_X(1)\) a very ample line bundle on \(X\) defining a projective embedding. For a given polynomial \(h\), the authors construct a dg-manifold \(\text{RHilb}^{\text{LCI}}_h(X)\) as the derived version of the classical geometric Hilbert scheme \(\text{Hilb}_h(X)\) of closed subschemes of \(X\) with Hilbert polynomial \(h\) relative to the polarization \({\mathcal O}_X(1)\). However, when the polynomial \(h\) is identically 1, then the derived Hilbert scheme turns out to coincide with the variety \(X\) whereas the derived Quot scheme \(\text{RQuot}({\mathcal O}_X)\) is known to be different from \(X\).

As for applications of these DDT-type constructions, the earlier constructed dg-manifolds RQuot are suitable for describing the derived moduli spaces of vector bundles on a fixed variety \(X\). In contrast, the dg-schemes \(\text{RHilb\,}h(X)\) established here are expected to play a similar rôle with regard to the derived moduli spaces of projective varieties themselves, which the authors corroborate by two striking examples. Namely, they use the explicit structure of the dg-schemes \(\text{RHilb\,}h(X)\) to construct two types of geometric derived moduli spaces:

(1) the derived space of maps \(\text{RMap}(C,Y)\) from a fixed projective scheme \(C\) to a fixed smooth projective variety \(Y\) and

(2) the derived stack of stable degree-\(d\) maps \(R\overline M_{g,n}(Y,d)\) from \(n\)-pointed nodal curves of genus \(g\) to a given smooth projective variety \(Y\). The latter example completes some earlier work of M. Kontsevich [in: The moduli spaces of curves, Prog. Math. 129, 335–368 (1995; Zbl 0885.14028)] and others (Behrend-Manin, Fulton-Pandharipande), thereby contributing to the mathematical theory of Gromov-Witten invariants. All in all, this is a very comprehensive paper of fundamental importance in derived moduli theory. The conceptual ingredients and refined techniques for the construction of derived Hilbert schemes include cotangent complexes, Harrison homology, derived moduli of operad algebras, derived schemes of ideals in finite-dimensional commutative algebras, and the theory of algebraic stacks. In spite of its highly advanced character, the exposition is very detailed, systematic and clear.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14C05 | Parametrization (Chow and Hilbert schemes) |

14H10 | Families, moduli of curves (algebraic) |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14M30 | Supervarieties |

18E30 | Derived categories, triangulated categories (MSC2010) |

18G50 | Nonabelian homological algebra (category-theoretic aspects) |

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\textit{I. Ciocan-Fontanine} and \textit{M. M. Kapranov}, J. Am. Math. Soc. 15, No. 4, 787--815 (2002; Zbl 1074.14003)

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