Fourier–Mukai and autoduality for compactified Jacobians, II

Melo, Margarida¹ ; Rapagnetta, Antonio² ; Viviani, Filippo³

¹ CMUC, Departamento de Matemática, Universidade de Coimbra, Coimbra, Portugal, Dipartimento di Matematica, Università Roma Tre, Roma, Italy
² Dipartimento di Matematica, Università di Roma II - Tor Vergata, Rome, Italy
³ Dipartimento di Matematica, Università Roma Tre, Roma, Italy

Geometry & topology, Tome 23 (2019) no. 5, pp. 2335-2395.

Résumé

To every reduced (projective) curve $X$ with planar singularities one can associate, following E Esteves, many fine compactified Jacobians, depending on the choice of a polarization on $X$, which are birational (possibly nonisomorphic) Calabi–Yau projective varieties with locally complete intersection singularities. We define a Poincaré sheaf on the product of any two (possibly equal) fine compactified Jacobians of $X$ and show that the integral transform with kernel the Poincaré sheaf is an equivalence of their derived categories, hence it defines a Fourier–Mukai transform. As a corollary of this result, we prove that there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-$1$ torsion-free sheaves on $X$ into the connected component of the algebraic space parametrizing rank-$1$ torsion-free sheaves on a given fine compactified Jacobian of $X$.

The main result can be interpreted in two ways. First of all, when the two fine compactified Jacobians are equal, the above Fourier–Mukai transform provides a natural autoequivalence of the derived category of any fine compactified Jacobian of $X$, which generalizes the classical result of S Mukai for Jacobians of smooth curves and the more recent result of D Arinkin for compactified Jacobians of integral curves with planar singularities. This provides further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R Donagi and T Pantev). Second, when the two fine compactified Jacobians are different (and indeed possibly nonisomorphic), the above Fourier–Mukai transform provides a natural equivalence of their derived categories, thus it implies that any two fine compactified Jacobians of $X$ are derived equivalent. This is in line with Kawamata’s conjecture that birational Calabi–Yau (smooth) varieties should be derived equivalent and it seems to suggest an extension of this conjecture to (mildly) singular Calabi–Yau varieties.

Détail

DOI : 10.2140/gt.2019.23.2335

Classification : 14B07, 14D20, 14F05, 14H20, 14H40
Keywords: compactified Jacobians, Fourier–Mukai transforms, Poincaré bundle, autoduality

Bibliographie

[1] A B Altman, A Iarrobino, S L Kleiman, Irreducibility of the compactified Jacobian, from: "Real and complex singularities, Oslo 1976" (editor P Holm), Sijthoff Noordhoff (1977) 1

[2] A B Altman, S L Kleiman, Compactifying the Picard scheme, II, Amer. J. Math. 101 (1979) 10 | DOI

[3] A B Altman, S L Kleiman, Compactifying the Picard scheme, Adv. Math. 35 (1980) 50 | DOI

[4] D Arinkin, Cohomology of line bundles on compactified Jacobians, Math. Res. Lett. 18 (2011) 1215 | DOI

[5] D Arinkin, Autoduality of compactified Jacobians for curves with plane singularities, J. Algebraic Geom. 22 (2013) 363 | DOI

[6] C Bartocci, U Bruzzo, D Hernández Ruipérez, Fourier–Mukai and Nahm transforms in geometry and mathematical physics, 276, Birkhäuser (2009) | DOI

[7] V V Batyrev, Birational Calabi–Yau n–folds have equal Betti numbers, from: "New trends in algebraic geometry" (editors K Hulek, F Catanese, C Peters, M Reid), London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press (1999) 1 | DOI

[8] J Briançon, Description de HilbnC{x,y}, Invent. Math. 41 (1977) 45 | DOI

[9] J Briançon, M Granger, J P Speder, Sur le schéma de Hilbert d’une courbe plane, Ann. Sci. École Norm. Sup. 14 (1981) 1 | DOI

[10] W Bruns, J Herzog, Cohen–Macaulay rings, 39, Cambridge Univ. Press (1993) | DOI

[11] R Donagi, T Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012) 653 | DOI

[12] E Esteves, Compactifying the relative Jacobian over families of reduced curves, Trans. Amer. Math. Soc. 353 (2001) 3045 | DOI

[13] E Esteves, M Gagné, S Kleiman, Autoduality of the compactified Jacobian, J. London Math. Soc. 65 (2002) 591 | DOI

[14] E Esteves, S Kleiman, The compactified Picard scheme of the compactified Jacobian, Adv. Math. 198 (2005) 484 | DOI

[15] B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, Fundamental algebraic geometry : Grothendieck’s FGA explained, 123, Amer. Math. Soc. (2005)

[16] M Granger, Géométrie des schémas de Hilbert ponctuels, 8, Soc. Mat. de France (1983) 84

[17] M Groechenig, Moduli of flat connections in positive characteristic, Math. Res. Lett. 23 (2016) 989 | DOI

[18] A Grothendieck, Éléments de géométrie algébrique, II : Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961) 5

[19] A Grothendieck, Éléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, I, Inst. Hautes Études Sci. Publ. Math. 20 (1964) 5

[20] A Grothendieck, Éléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, II, Inst. Hautes Études Sci. Publ. Math. 24 (1965) 5

[21] A Grothendieck, Éléments de géométrie algébrique, IV : Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28 (1966) 5

[22] M Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001) 941 | DOI

[23] R Hartshorne, Residues and duality, 20, Springer (1966) | DOI

[24] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[25] R Hartshorne, Generalized divisors on Gorenstein schemes, –Theory 8 (1994) 287 | DOI

[26] D Hernández Ruipérez, A C López Martín, F Sancho De Salas, Fourier–Mukai transforms for Gorenstein schemes, Adv. Math. 211 (2007) 594 | DOI

[27] D Huybrechts, Fourier–Mukai transforms in algebraic geometry, , Oxford Univ. Press (2006) | DOI

[28] A Iarrobino, Punctual Hilbert schemes, Bull. Amer. Math. Soc. 78 (1972) 819 | DOI

[29] A A Iarrobino, Punctual Hilbert schemes, 188, Amer. Math. Soc. (1977) | DOI

[30] Y Kawamata, D–equivalence and K–equivalence, J. Differential Geom. 61 (2002) 147 | DOI

[31] S L Kleiman, A B Altman, Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 (1979) 775 | DOI

[32] H Matsumura, Commutative ring theory, 8, Cambridge Univ. Press (1989) | DOI

[33] M Melo, A Rapagnetta, F Viviani, Fine compactified Jacobians of reduced curves, Trans. Amer. Math. Soc. 369 (2017) 5341 | DOI

[34] M Melo, A Rapagnetta, F Viviani, Fourier–Mukai and autoduality for compactified Jacobians, I, J. Reine Angew. Math. 755 (2019) 1 | DOI

[35] M Melo, F Viviani, Fine compactified Jacobians, Math. Nachr. 285 (2012) 997 | DOI

[36] L Migliorini, V Shende, F Viviani, A support theorem for Hilbert schemes of planar curves, II, preprint (2018)

[37] S Mukai, Duality between D(X) and D(X) with its application to Picard sheaves, Nagoya Math. J. 81 (1981) 153 | DOI

[38] S Mukai, On the moduli space of bundles on K3 surfaces, I, from: "Vector bundles on algebraic varieties", Tata Inst. Fund. Res. Stud. Math. 11, Oxford Univ. Press (1987) 341

[39] D Mumford, Abelian varieties, 5, Oxford Univ. Press (1970)

[40] E Sernesi, Deformations of algebraic schemes, 334, Springer (2006) | DOI