Bounded sets of sheaves on relative analytic spaces
Annales Henri Lebesgue, Volume 4 (2021), pp. 1531-1563.

Metadata

Keywords bounded sets of coherent sheaves, relative analytic space, Douady space, Harder-Narasimhan filtration

Abstract

We extend previous results on boundedness of sets of coherent sheaves on a compact Kähler manifold to the relative and not necessarily smooth case. This enlarged context allows us to prove properness properties of the relative Douady space as well as results related to semistability of sheaves such as the existence of relative Harder–Narasimhan filtrations.


References

[AG06] Ancona, Vincenzo; Gaveau, Bernard Differential forms on singular varieties. De Rham and Hodge theory simplified., Pure and Applied Mathematics (Boca Raton), 273, Chapman & Hall/CRC, 2006 | Zbl

[And09] André, Yves Slope filtrations, Confluentes Math., Volume 1 (2009) no. 1, pp. 1-85 | DOI | MR | Zbl

[Bar78] Barlet, Daniel Convexité de l’espace des cycles, Bull. Soc. Math. Fr., Volume 106 (1978) no. 4, pp. 373-397 | DOI | Zbl

[BFM75] Baum, Paul; Fulton, William; MacPherson, Robert Riemann–Roch for singular varieties, Publ. Math., Inst. Hautes Étud. Sci., Volume 45 (1975), pp. 101-145 | DOI | Numdam | Zbl

[BFM79] Baum, Paul; Fulton, William; MacPherson, Robert Riemann–Roch and topological K theory for singular varieties, Acta Math., Volume 143 (1979) no. 3-4, pp. 155-192 | DOI | MR | Zbl

[BH69] Bloom, Thomas; Herrera, Miguel E. De Rham cohomology of an analytic space, Invent. Math., Volume 7 (1969), pp. 275-296 | DOI | MR | Zbl

[Bin83] Bingener, Jürgen On deformations of Kähler spaces. I, Math. Z., Volume 182 (1983) no. 4, pp. 505-535 | DOI | Zbl

[BM14] Barlet, Daniel; Magnússon, Jón Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés (Paris), 22, Société Mathématique de France, 2014 | Zbl

[Bre97] Bredon, Glen E. Topology and geometry, Graduate Texts in Mathematics, 139, Springer, 1997 (corrected third printing of the 1993 original) | Zbl

[BS77] Bănică, Constantin; Stănăşilă, Octavian Méthodes algébriques dans la théorie globale des espaces complexes, Varia Mathematica, 1-2, Gauthier-Villars, 1977 (avec une préface de Henri Cartan, troisième édition revue et augmentée)

[BTT17] Buchdahl, Nicholas; Teleman, Andrei; Toma, Matei A continuity theorem for families of sheaves on complex surfaces, J. Topol., Volume 10 (2017) no. 4, pp. 995-1028 | DOI | MR | Zbl

[CCP19] Campana, Frédéric; Cao, Junyan; Păun, Mihai Subharmonicity of direct images and applications (2019) (https://arxiv.org/abs/1906.11317v1)

[CP19] Campana, Frédéric; Păun, Mihai Foliations with positive slopes and birational stability of orbifold cotangent bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 129 (2019), pp. 1-49 | DOI | MR | Zbl

[Dem93] Demailly, Jean-Pierre A numerical criterion for very ample line bundles, J. Differ. Geom., Volume 37 (1993) no. 2, pp. 323-374 | MR | Zbl

[Dem07] Demailly, Jean-Pierre Complex analytic and differential geometry, 2007 (OpenContentBook, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.2767&rep=rep1&type=pdf)

[Dim92] Dimca, Alexandru Singularities and topology of hypersurfaces, Universitext, Springer, 1992 | DOI | Zbl

[Dim04] Dimca, Alexandru Sheaves in topology, Universitext, Springer, 2004 | DOI | Zbl

[DP75] Dolbeault, Pierre; Poly, Jean Differential forms with subanalytic singularities; integral cohomology; residues, Several complex variables (Proceedings of Symposia in Pure Mathematics. Part 1), Volume 30, American Mathematical Society, 1975, pp. 255-261 | Zbl

[DV76] Douady, Adrien; Verdier, Jean-Louis Séminaire de Géométrie Analytique, Astérisque, 36-37, Société Mathématique de France, 1976 | Numdam

[Fis76] Fischer, Gerd Complex analytic geometry, Lecture Notes in Mathematics, 538, Springer, 1976 | DOI | Zbl

[FL85] Fulton, William; Lang, Serge Riemann–Roch algebra, Grundlehren der Mathematischen Wissenschaften, 277, Springer, 1985 | DOI | Zbl

[Fle81] Flenner, Hubert Eine Bemerkung über relative Ext -Garben, Math. Ann., Volume 258 (1981) no. 2, pp. 175-182 | DOI | Zbl

[FM81] Fulton, William; MacPherson, Robert Categorical framework for the study of singular spaces, Memoirs of the American Mathematical Society, 31, American Mathematical Society, 1981 no. 243 | Zbl

[Fog69] Fogarty, John Truncated Hilbert functors, J. Reine Angew. Math., Volume 234 (1969), pp. 65-88 | MR | Zbl

[Fuj78] Fujiki, Akira Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci., Volume 14 (1978) no. 1, pp. 1-52 | DOI | Zbl

[Fuj82] Fujiki, Akira Projectivity of the space of divisors on a normal compact complex space, Publ. Res. Inst. Math. Sci., Volume 18 (1982) no. 3, pp. 1163-1173 | DOI | MR | Zbl

[Fuj84] Fujiki, Akira On the Douady space of a compact complex space in the category 𝒞. II, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 3, pp. 461-489 | DOI | MR | Zbl

[Ful95] Fulton, William Algebraic topology, Graduate Texts in Mathematics, 153, Springer, 1995 | DOI | MR | Zbl

[GH78] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | MR | Zbl

[GKP16] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Movable curves and semistable sheaves, Int. Math. Res. Not., Volume 2016 (2016) no. 2, pp. 536-570 | MR | Zbl

[Gor81] Goresky, R. Mark Whitney stratified chains and cochains, Trans. Am. Math. Soc., Volume 267 (1981) no. 1, pp. 175-196 | DOI | MR | Zbl

[Gro61] Grothendieck, Alexander Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki (1960/61) (Séminaire Bourbaki), Volume 13, Société Mathématique de France, 1961, pp. 249-276 | Zbl

[GRT16] Greb, Daniel; Ross, Julius; Toma, Matei Variation of Gieseker moduli spaces via quiver GIT, Geom. Topol., Volume 20 (2016) no. 3, pp. 1539-1610 | DOI | MR | Zbl

[GT17] Greb, Daniel; Toma, Matei Compact moduli spaces for slope-semistable sheaves, Algebr. Geom., Volume 4 (2017) no. 1, pp. 40-78 | DOI | MR | Zbl

[Har77] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI | Zbl

[Her66] Herrera, Miguel E. Integration on a semianalytic set, Bull. Soc. Math. Fr., Volume 94 (1966), pp. 141-180 | DOI | Numdam | MR | Zbl

[HL10] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010 | DOI | Zbl

[Ive86] Iversen, Birger Cohomology of sheaves, Universitext, Springer, 1986 | DOI | Zbl

[Kob87] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987 (also published by Iwanami Shoten Publishers, vol. 5 of “Kanô Memorial Lectures”, 1987) | DOI | Zbl

[Lev87] Levy, Roni N. The Riemann–Roch theorem for complex spaces, Acta Math., Volume 158 (1987) no. 3-4, pp. 149-188 | DOI | MR | Zbl

[Lev08] Levy, Roni N. Riemann–Roch theorem for higher bivariant K-functors, Ann. Inst. Fourier, Volume 58 (2008) no. 2, pp. 571-601 | DOI | Numdam | MR | Zbl

[LT95] Lübke, Martin; Teleman, Andrei The Kobayashi–Hitchin correspondence, World Scientific, 1995 | DOI | Zbl

[Mar96] Maruyama, Masaki Construction of moduli spaces of stable sheaves via Simpson’s idea, Moduli of vector bundles. Papers of the 35th Taniguchi symposium, Sanda, Japan, and a symposium held in Kyoto, Japan, 1994 (Lecture Notes in Pure and Applied Mathematics), Volume 179, Marcel Dekker, 1996, pp. 147-187 | MR | Zbl

[MM07] Ma, Xiaonan; Marinescu, George Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, Birkhäuser, 2007 | MR | Zbl

[Ryd08] Rydh, David Families of cycles and the Chow scheme, Ph. D. Thesis, KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Stockholm, Sweden (2008) (218 pages, electronic version available from the following URL: https://www.diva-portal.org/smash/get/diva2:14103/FULLTEXT01.pdf)

[Sto71] Stoll, Wilhelm Fiber integration and some applications, Symposium on Several Complex Variables (Park City, Utah, 1970) (Lecture Notes in Mathematics), Volume 184, Springer, 1971, pp. 109-120 | DOI | MR | Zbl

[Tom16] Toma, Matei Bounded sets of sheaves on Kähler manifolds, J. Reine Angew. Math., Volume 710 (2016), pp. 77-93 | Zbl

[Tom20] Toma, Matei Properness criteria for families of coherent analytic sheaves, Algebr. Geom., Volume 7 (2020) no. 4, pp. 486-502 | DOI | MR | Zbl

[Var89] Varouchas, Jean Kähler spaces and proper open morphisms, Math. Ann., Volume 283 (1989) no. 1, pp. 13-52 | DOI | MR | Zbl