Mirror symmetry of abelian varieties and multi-theta functions
Author:
Kenji Fukaya
Journal:
J. Algebraic Geom. 11 (2002), 393-512
DOI:
https://doi.org/10.1090/S1056-3911-02-00329-6
Published electronically:
February 27, 2002
MathSciNet review:
1894935
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We study homological mirror symmetry conjecture of symplectic and complex torus. We will associate a mirror torus $(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$ to each symplectic torus $(T^{2n},\omega )$ together with a closed 2 form $B$ which we call a $B$-field. We will associate a coherent sheaf ${\mathcal E}(L,{\mathcal L})$ on $(T^{2n},\omega +\sqrt {-1}B)^{\wedge }$ to each pair $(L,{\mathcal L})$ of affine Lagrangian submanifolds $L$ and a flat complex line bundle ${\mathcal L}$ on $L$. In the case of affine Lagrangian submanifolds, we show that the Floer homology of Langrangian submanifolds is isomorphic to the extension of the mirror sheaf ${\mathcal E}(L,{\mathcal L})$. We construct a canonical isomorphism in the case when a certain transversality condition is satisfied. Our isomorphism then is functorial.
ASKZ M. Alexandrov, A. Schwarz, M. Kontsevich, and O. Zaboronsky. The geometry of the master equation and topological quantum field theory. Intern. J. Modern Phys. A, 12:1405–1429, 1997.
BaK S. Baranikov and M. Kontsevich. Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices, 4:201 – 215, 1998.
BBS K. Becker, M. Becker, and A. Strominger. Fivebranes Membranes and Non- perturbative String theory. hep-th/9507158, 1995.
BF K. Behrend and B. Fantechi. The instrinsic normal cone. Invent. Math, 128:45 – 88, 1997.
BDPP G. Bini, C. De Concini, M. Polito, and C. Procesi. On the work of Givental relative to mirror symmetry. math.AG/9805097.
BoK A. Bondal and M. Kapranov. Framed triangulated categories. Math. USSR Sbornik, 181:93 – 107, 1991.
Bo R. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math, 132:491–562, 1998.
COGP P. Candelas, X. de.la Ossa, P. Green, and L. Parks. A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory. Nuclear Physics, B.359:21+, 1991.
Ch Y. Chekanov. Lagrangian tori in a symplectic vector space and global symplectomorphisms. Math. Z., 223:547–559, 1996.
Do1 S. Donaldson. Irrationality and h-cobordism conjecture. J. Diff. Geom., 26:275 – 297, 1986.
Do2 S. Donaldson. A lecture at University of Warwick, 1992.
Fl A. Floer. Morse theory for Lagrangian intersectoins. J. Diff. Geom., 28:513 – 547, 1988.
Fu1 K. Fukaya. Morse homotopy, ${A}^{\infty }$-categories, and Floer homologies. In H. J. Kim, editor, Proc. of the 1993 Garc Workshop on Geometry and Topology, volume 18 of Lecture Notes series, pages 1 – 102. Seoul Nat. Univ., 1993. http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya/ fukaya.html.
Fu3 K. Fukaya. Floer homology of connected sum of homology 3-spheres. Topology, 35:89 – 136, 1996.
Fu4 K. Fukaya. Floer homology for 3-manifolds with boundary I. preprint, never to appear, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya fukaya.html, 1997.
Fu2 K. Fukaya. Morse homotopy and its quantization. In W. Kazez, editor, Geometry and Topology, AMS/IP Studies in Advanced Mathematics, pages 409 – 440. International Press, 1997.
Fu5 K. Fukaya. Floer homology of Lagrangian foliations and noncommutative mirror symmetry. preprint, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya fukaya.html, 1998.
Fu6 K. Fukaya. Floer homology and mirror symmetry I, to appear in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, AMS/IP Studies in Advanced Math. 23 C. Vafa, S.T. Yau ed.
Fu7 K. Fukaya. Floer homology and mirror symmetry II. to appear in Advanced Studies in Pure Math. Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 2001.
Fu8 K. Fukaya. Floer homology for families - report of a project in progress -. to appear in Contemporary Math., 2001.
FOh K. Fukaya and Y. G. Oh. Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math., 1:99 – 180, 1997.
FOOO K. Fukaya, Y. G. Oh, H. Ohta, and K. Ono. Langrangian intersection Floer theory -anomaly and obstruction-. preprint, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya/fukaya.html, 2000.
FOn2 K. Fukaya and K. Ono. Arnold conjecture and Gromov-Witten invariants. Topology, 38, 1999.
FOn1 K. Fukaya and K. Ono. Arnold conjecture and Gromov-Witten invariants for general symplectic manifolds. Fields Institute Communications, 24:173 – 190, 1999.
GJ E. Getzler and J. Jones. ${A}_{\infty }$ algebra and cyclic bar complex. Illinois J. Math., 34, 1990.
Gi A. Givental. Equivariant Gromov-Witten invariants. International Mathematics Research Notices, 131:616 – 663, 1996.
GZ L. Göttsche and D. Zagier. Jacobi forms and the structure of Donaldson invariants for 4-manifolds with $b^+=1$. Selecta Mathematica, 4:69 – 115, 1998.
Gr M. Gromov. Pseudoholomorphic curves in symplectic geometry. Invent. Math., 82:307 – 347, 1985.
Ha R. Hartshone. Residues and Duality, volume 20 of Lecture notes in Mathematics. Springer-Verlag, 1966.
HZ H. Hofer and P. Zehnder. Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, 1994.
Ho L. Hörmander. Fourier integral operator 1. Acta Math., 127:79 – 183, 1971.
KS M. Kashiwara and P. Shapira. Sheaves on manifolds, volume 292 of Grund. der Math. Springer-Verlag, 1990.
K3 M. Kontsevich. Deformation of quantization of poisson manifolds I. preprint, q-alg/9709040.
K1 M. Kontsevich. ${A}_{\infty }$ algebras in mirror symmetry. MPI Arbeitstagung talk, 1993.
K2 M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, Zürich, volume I, pages 120 – 139. Birkhäuser, 1995.
LB S. Lang and Ch. Birkenhake. Complex Abelian Varieties, volume 302 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1980.
LiT J. Li and G. Tian. Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In Topics in symplectic $4$-manifolds (Irvine, CA, 1996), volume 1 of First Int. Press Lect. Ser., pages 47 – 83. International Press, 1997.
LLY B. Lian, K. Liu, and S. Yau. Mirror principle I. Asian. J. Math., 1:729 – 763, 1997.
LuT G. Liu and G. Tian. Floer homology and Arnold conjecture. J. Diff. Geom., 49:1 – 74, 1998.
Ma Y. Manin. Three constructions of Frobenius manifolds. Asian. J. Math., 3:179 – 220, 1999.
MS1 D. McDuff and D. Salamon. J-holomorphic curves and quantum cohomology, volume 6 of University Lecture Seires. Amer. Math. Soc., 1994.
MS2 D. McDuff and D. Salamon. Introduction to Symplectic topology. Oxford Science Publ., 1995.
LMP F. Lalonde, D. McDuff and L. Polterovich. On the flux conjectures. dg-ga/9706015.
Mu1 S. Mukai. Semi-homogeneous vector bundles on an abelian variety. J. Math. Kyoto Univ., 18:239 – 272, 1978.
Mu2 S. Mukai. Duality between ${D}({X})$ and ${D}({X} \hat {} )$ with its application to Picard sheaves. Nagoya Math. J., 81:153 – 175, 1981.
Mu3 S. Mukai. Abelian variety and spin representation. preprint, 1998.
Mum D. Mumford. Abelian variety. Oxford Univ. Press, 1970.
N S. Novikov. Multivalued functions and functional - an analogue of the Morse theory. Sov. Math. Dokl., 24:222 – 225, 1981.
Oh Y. G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I,II. Comm. Pure and Appl. Math., 46:949 – 994, 995 – 1012, 1993.
PSS S. Piunikhin, D. Salamon, and M. Schwarz. Symplectic Floer-Donaldson theory and quantum cohomology. In C. Thomas, editor, Contact and Symplectic Geometry, pages 151 – 170. Cambridge Univ. Press, 1996.
Po A. Polchinski. Tasi lectures on D-branes. In C.Efthimiou and B.Green, editors, Fields, String and Duality. World Scientific, 1997.
Pl A. Polishchuk. Massey and Fukaya products on elliptic curves. alg-geo/9803017.
PZ A. Polishchuk and E. Zaslow. Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys., 2:443 – 470, 1998.
Poz M. Pozniak. Floer homology, Novikov rings and clean intersections. PhD thesis, Univ. of Warwick, 1994.
Ru Y. Ruan. Virtual neighborhood and pseudoholomorphic curves. Turkish J. Math., 23:161 – 231, 1999.
RT Y. Ruan and G. Tian. Bott-type symplectic Floer cohomology and its multiplicative structure. Math. Res. Lett., 2:203 – 219, 1995.
Sch V. Schechtman. Remarks on formal deformation and Batalin-Vilkovsky algebras. mathAG/9802006.
Sei P. Seidel. Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128:103 – 149, 2000.
Sie B. Siebert. Gromov-Witten invariants for general symplectic manifolds. dg-ga /9608005.
Sil V. D. Silva. Products on symplectic Floer homology. PhD thesis, Oxford Univ., 1997.
St1 J. Stasheff. Homotopy associativity of H-spaces I, II. Trans. Amer. Math. Soc., pages 275 – 312, 1966.
St2 J. Stasheff. Deformation theory and the Batalin-Vilkovsky master equation. In Deformation theory and Symplectic Geometry, volume 20 of Mathematical Physics Studies, pages 271 – 284. Kluwer Academic Publications, 1996.
SYZ A. Strominger, S. Yau, and E. Zaslow. Mirror symmetry is T-duality. Nucl. Phys. B, 479:243 – 259, 1996.
Va C. Vafa. Extending mirror conjecture to Calabi-Yau with bundles. Commun. Contemp. Math., 1:65 – 70, 1999.
ASKZ M. Alexandrov, A. Schwarz, M. Kontsevich, and O. Zaboronsky. The geometry of the master equation and topological quantum field theory. Intern. J. Modern Phys. A, 12:1405–1429, 1997.
BaK S. Baranikov and M. Kontsevich. Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices, 4:201 – 215, 1998.
BBS K. Becker, M. Becker, and A. Strominger. Fivebranes Membranes and Non- perturbative String theory. hep-th/9507158, 1995.
BF K. Behrend and B. Fantechi. The instrinsic normal cone. Invent. Math, 128:45 – 88, 1997.
BDPP G. Bini, C. De Concini, M. Polito, and C. Procesi. On the work of Givental relative to mirror symmetry. math.AG/9805097.
BoK A. Bondal and M. Kapranov. Framed triangulated categories. Math. USSR Sbornik, 181:93 – 107, 1991.
Bo R. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math, 132:491–562, 1998.
COGP P. Candelas, X. de.la Ossa, P. Green, and L. Parks. A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory. Nuclear Physics, B.359:21+, 1991.
Ch Y. Chekanov. Lagrangian tori in a symplectic vector space and global symplectomorphisms. Math. Z., 223:547–559, 1996.
Do1 S. Donaldson. Irrationality and h-cobordism conjecture. J. Diff. Geom., 26:275 – 297, 1986.
Do2 S. Donaldson. A lecture at University of Warwick, 1992.
Fl A. Floer. Morse theory for Lagrangian intersectoins. J. Diff. Geom., 28:513 – 547, 1988.
Fu1 K. Fukaya. Morse homotopy, ${A}^{\infty }$-categories, and Floer homologies. In H. J. Kim, editor, Proc. of the 1993 Garc Workshop on Geometry and Topology, volume 18 of Lecture Notes series, pages 1 – 102. Seoul Nat. Univ., 1993. http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya/ fukaya.html.
Fu3 K. Fukaya. Floer homology of connected sum of homology 3-spheres. Topology, 35:89 – 136, 1996.
Fu4 K. Fukaya. Floer homology for 3-manifolds with boundary I. preprint, never to appear, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya fukaya.html, 1997.
Fu2 K. Fukaya. Morse homotopy and its quantization. In W. Kazez, editor, Geometry and Topology, AMS/IP Studies in Advanced Mathematics, pages 409 – 440. International Press, 1997.
Fu5 K. Fukaya. Floer homology of Lagrangian foliations and noncommutative mirror symmetry. preprint, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya fukaya.html, 1998.
Fu6 K. Fukaya. Floer homology and mirror symmetry I, to appear in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, AMS/IP Studies in Advanced Math. 23 C. Vafa, S.T. Yau ed.
Fu7 K. Fukaya. Floer homology and mirror symmetry II. to appear in Advanced Studies in Pure Math. Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 2001.
Fu8 K. Fukaya. Floer homology for families - report of a project in progress -. to appear in Contemporary Math., 2001.
FOh K. Fukaya and Y. G. Oh. Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math., 1:99 – 180, 1997.
FOOO K. Fukaya, Y. G. Oh, H. Ohta, and K. Ono. Langrangian intersection Floer theory -anomaly and obstruction-. preprint, http://www.kusm.kyoto-u.ac.jp/$\sim$fukaya/fukaya.html, 2000.
FOn2 K. Fukaya and K. Ono. Arnold conjecture and Gromov-Witten invariants. Topology, 38, 1999.
FOn1 K. Fukaya and K. Ono. Arnold conjecture and Gromov-Witten invariants for general symplectic manifolds. Fields Institute Communications, 24:173 – 190, 1999.
GJ E. Getzler and J. Jones. ${A}_{\infty }$ algebra and cyclic bar complex. Illinois J. Math., 34, 1990.
Gi A. Givental. Equivariant Gromov-Witten invariants. International Mathematics Research Notices, 131:616 – 663, 1996.
GZ L. Göttsche and D. Zagier. Jacobi forms and the structure of Donaldson invariants for 4-manifolds with $b^+=1$. Selecta Mathematica, 4:69 – 115, 1998.
Gr M. Gromov. Pseudoholomorphic curves in symplectic geometry. Invent. Math., 82:307 – 347, 1985.
Ha R. Hartshone. Residues and Duality, volume 20 of Lecture notes in Mathematics. Springer-Verlag, 1966.
HZ H. Hofer and P. Zehnder. Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser, 1994.
Ho L. Hörmander. Fourier integral operator 1. Acta Math., 127:79 – 183, 1971.
KS M. Kashiwara and P. Shapira. Sheaves on manifolds, volume 292 of Grund. der Math. Springer-Verlag, 1990.
K3 M. Kontsevich. Deformation of quantization of poisson manifolds I. preprint, q-alg/9709040.
K1 M. Kontsevich. ${A}_{\infty }$ algebras in mirror symmetry. MPI Arbeitstagung talk, 1993.
K2 M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, Zürich, volume I, pages 120 – 139. Birkhäuser, 1995.
LB S. Lang and Ch. Birkenhake. Complex Abelian Varieties, volume 302 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1980.
LiT J. Li and G. Tian. Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds. In Topics in symplectic $4$-manifolds (Irvine, CA, 1996), volume 1 of First Int. Press Lect. Ser., pages 47 – 83. International Press, 1997.
LLY B. Lian, K. Liu, and S. Yau. Mirror principle I. Asian. J. Math., 1:729 – 763, 1997.
LuT G. Liu and G. Tian. Floer homology and Arnold conjecture. J. Diff. Geom., 49:1 – 74, 1998.
Ma Y. Manin. Three constructions of Frobenius manifolds. Asian. J. Math., 3:179 – 220, 1999.
MS1 D. McDuff and D. Salamon. J-holomorphic curves and quantum cohomology, volume 6 of University Lecture Seires. Amer. Math. Soc., 1994.
MS2 D. McDuff and D. Salamon. Introduction to Symplectic topology. Oxford Science Publ., 1995.
LMP F. Lalonde, D. McDuff and L. Polterovich. On the flux conjectures. dg-ga/9706015.
Mu1 S. Mukai. Semi-homogeneous vector bundles on an abelian variety. J. Math. Kyoto Univ., 18:239 – 272, 1978.
Mu2 S. Mukai. Duality between ${D}({X})$ and ${D}({X} \hat {} )$ with its application to Picard sheaves. Nagoya Math. J., 81:153 – 175, 1981.
Mu3 S. Mukai. Abelian variety and spin representation. preprint, 1998.
Mum D. Mumford. Abelian variety. Oxford Univ. Press, 1970.
N S. Novikov. Multivalued functions and functional - an analogue of the Morse theory. Sov. Math. Dokl., 24:222 – 225, 1981.
Oh Y. G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I,II. Comm. Pure and Appl. Math., 46:949 – 994, 995 – 1012, 1993.
PSS S. Piunikhin, D. Salamon, and M. Schwarz. Symplectic Floer-Donaldson theory and quantum cohomology. In C. Thomas, editor, Contact and Symplectic Geometry, pages 151 – 170. Cambridge Univ. Press, 1996.
Po A. Polchinski. Tasi lectures on D-branes. In C.Efthimiou and B.Green, editors, Fields, String and Duality. World Scientific, 1997.
Pl A. Polishchuk. Massey and Fukaya products on elliptic curves. alg-geo/9803017.
PZ A. Polishchuk and E. Zaslow. Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys., 2:443 – 470, 1998.
Poz M. Pozniak. Floer homology, Novikov rings and clean intersections. PhD thesis, Univ. of Warwick, 1994.
Ru Y. Ruan. Virtual neighborhood and pseudoholomorphic curves. Turkish J. Math., 23:161 – 231, 1999.
RT Y. Ruan and G. Tian. Bott-type symplectic Floer cohomology and its multiplicative structure. Math. Res. Lett., 2:203 – 219, 1995.
Sch V. Schechtman. Remarks on formal deformation and Batalin-Vilkovsky algebras. mathAG/9802006.
Sei P. Seidel. Graded Lagrangian submanifolds. Bull. Soc. Math. France, 128:103 – 149, 2000.
Sie B. Siebert. Gromov-Witten invariants for general symplectic manifolds. dg-ga /9608005.
Sil V. D. Silva. Products on symplectic Floer homology. PhD thesis, Oxford Univ., 1997.
St1 J. Stasheff. Homotopy associativity of H-spaces I, II. Trans. Amer. Math. Soc., pages 275 – 312, 1966.
St2 J. Stasheff. Deformation theory and the Batalin-Vilkovsky master equation. In Deformation theory and Symplectic Geometry, volume 20 of Mathematical Physics Studies, pages 271 – 284. Kluwer Academic Publications, 1996.
SYZ A. Strominger, S. Yau, and E. Zaslow. Mirror symmetry is T-duality. Nucl. Phys. B, 479:243 – 259, 1996.
Va C. Vafa. Extending mirror conjecture to Calabi-Yau with bundles. Commun. Contemp. Math., 1:65 – 70, 1999.
Additional Information
Kenji Fukaya
Affiliation:
Department of Mathematics, Faculty of Sciences, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto, 602-8502 Japan
MR Author ID:
217346
Email:
fukaya@kusm.kyoto-u.ac.jp
Received by editor(s):
July 29, 1998
Published electronically:
February 27, 2002
Additional Notes:
Partially supported by Grant-in-Aid for Scientific Research 13852001
