Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Derived Hilbert schemes

Authors: Ionut Ciocan-Fontanine and Mikhail M. Kapranov
Journal: J. Amer. Math. Soc. 15 (2002), 787-815
MSC (2000): Primary 14M30; Secondary 18G50
Published electronically: June 21, 2002
MathSciNet review: 1915819
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme $X$ with given Hilbert polynomial $h$. This is a dg-manifold (smooth dg-scheme) $RHilb_h(X)$ which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is given by truncations of the homogeneous coordinate rings of subschemes in $X$. In particular, $RHilb_h(X)$ differs from $RQuot_n({\mathcal O_X})$, the derived Quot scheme constructed in our previous paper, which carries only a family of $A_\infty$-modules over the coordinate algebra of $X$.

As an application, we construct the derived version of the moduli stack of stable maps of algebraic curves to a given projective variety $Y$, thus realizing the original suggestion of M. Kontsevich.

References [Enhancements On Off] (What's this?)

  • [B] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601-617. MR 98i:14015
  • [BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. MR 98e:14022
  • [BK] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Research Notices 4 (1998), 201-215. MR 99b:14009
  • [BM] K. Behrend and Y. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1-60. MR 98i:14014
  • [CK] I. Ciocan-Fontanine and M. Kapranov, Derived Quot schemes, Ann. Sci. ENS (4) 34 (2001), 403-440.
  • [FP] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Proc. Sympos. Pure Math. 62 (1997), Pt. 2, p. 45-96. MR 98m:14025
  • [GJ] E. Getzler and J.D.S. Jones, Operads, homotopy algebras and iterated integrals for double loop spaces, preprint, 1994.
  • [GK] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), 203-272. MR 96a:18004
  • [Gr] A. Grothendieck, Techniques de construction et théorèmes d'existence en géometrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki 221 (1960/61). MR 27:1339
  • [Hi1] V. Hinich, Dg-coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), 209-250. MR 2002f:14008
  • [Hi2] V. Hinich, Deformations of homotopy algebras, preprint math.AG/9904145.
  • [I] L. Illusie, Complexe Cotangent et Déformations (Lecture Notes in Math. 239), Springer-Verlag, 1972. MR 58:10886a
  • [Ka] M. Kapranov, Injective resolutions of BG and derived moduli spaces of local systems, J. Pure Appl. Algebra 155 (2001), 167-179. MR 2002b:18017
  • [Kol] J. Kollar, Rational Curves in Algebraic Varieties, Springer-Verlag, 1996. MR 98c:14001
  • [Kon] M. Kontsevich, Enumeration of rational curves via torus actions, in:`` Moduli Space of Curves" (R. Dijkgraaf, C. Faber, G. van der Geer Eds.) pp. 335-368, Birkhauser, Boston, 1995. MR 97d:14077
  • [KS] M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and Deligne's conjecture, preprint math.QA/0001151.
  • [LM] G. Laumon and L. Moret-Bailly, Champs Algébriques, Springer-Verlag, 2000. MR 2001f:14006
  • [LT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. of the AMS 11 (1998), 119-174. MR 99d:14011
  • [Lo] J.L. Loday, Cyclic Homology, Springer-Verlag, 1995. MR 94a:19004
  • [Man] M. Manetti, Extended deformation functors, I, Int. Math. Res. Not. 2002, no. 14, 719-756.
  • [Mar] M. Markl, Models for operads, Comm. Algebra 71 (1996), 1471-1500. MR 96m:18012
  • [May] J. P. May, Geometry of Iterated Loop Spaces (Lecture Notes in Math. 271), Springer-Verlag, 1972. MR 54:8623b
  • [Q] D. Quillen, On the (co)homology of commutative rings, Proc. Sympos. Pure Math. 17 (1970), 65-87. MR 41:1722
  • [Re] C. Rezk, Spaces of algebra structures and cohomology of operads, thesis, MIT (1996).
  • [Se] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197-278. MR 16:953c
  • [St] J. D. Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, Lecture Notes in Math. 1510 (1992), 120-137. MR 93j:17055
  • [Vi] E. Viehweg, Quasi-projective Moduli for Polarized Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 30, Springer-Verlag, Berlin, 1995. MR 97j:14001

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14M30, 18G50

Retrieve articles in all journals with MSC (2000): 14M30, 18G50

Additional Information

Ionut Ciocan-Fontanine
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Mikhail M. Kapranov
Affiliation: Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada M5S 3G3

Received by editor(s): August 14, 2000
Published electronically: June 21, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society