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2003 Fiscal Year Final Research Report Summary

Classification problems in Higher Dimensional Birational Geometry

Research Project

Project/Area Number 12440005
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

MORI Shigefumi  MORI,Shigefumi, 数理解析研究所, 教授 (00093328)

Co-Investigator(Kenkyū-buntansha) NAMIKAWA Yoshinori  KYOTO UNIVERSITY, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (80228080)
NAKAYAMA Noboru  Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (10189079)
MUKAI Shigeru  Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (80115641)
OGUISO Keiji  The University of Tokyo, Graduate School of Mathematical Science, Associate Professor, 大学院・数理科学研究科, 助教授 (40224133)
SAITO Masa-hiko  Kobe University, Faculty of Science, Professor, 理学部, 教授 (80183044)
Project Period (FY) 2000 – 2003
Keywordsflip / flop / Fano 3-fold / canonical divisor / symplectic variety / extremal neighborhood / elliptic structure / terminal singularity
Research Abstract

Mori and Fujino have generalized Kodaira's canonical bundle formula and proved, as an application, that algebraic varieties with Kodaira dimension at most three have finitely generated canonical rings. Mori, Miyaoka and Takagi, together with Kollar, have proved the boundedness of Fano 3-folds with only canonical singularities. Mori also gave an explicit description of every irreducible semistable extremal neighborhood with two non-Gorenstein points, in terms of coordinates with equations and patching.
Mukai proved that every canonical curve of genus 9 with maximal Clifford index, is a linear space section of the symplectic Grassmannian variety of dimension six.
Masahiko Saito have introduced the Okamoto-Painleve pair of an algebraic surface and its anti-canonical divisor which algebro-geometrically characterizes the initial value space of the Painleve equation. He also reconstructed the Painleve equation from the pair using its deformation theory.
Nakayama described certain elliptic fiber structures over a given analytic space upto bimeromorphic equivalence using the a-etale cohomology group.
Namikawa have studied birational maps between complex symplectic varieties and proved that the analogue of "Reid's dream" does not hold in the category of complex symplectic varieties. He also constructed a counterexample to birational Torelli problem for complex symplectic varieties.
Oguiso has generalized the characterization of the Klein curve to the case of K3 surfaces, which states that the Klein-Mukai surface is the only K3 surface which admits a faithful action of the quartic extension of the simple group of order 168.
Takagi has generalized Takeuchi's method, obtained a classification list of Q-Fano 3-folds with index two and confirmed the existence in several cases.
Fujino has proved that every sequence of log flips terminates for 4 dimensional canonical pairs.

  • Research Products

    (81 results)

All Other

All Publications (81 results)

  • [Publications] 森重文, V.Alexeev: "Bounding singular surfaces of general type""Algebra, Arithmetic and Geometry with Applications" ed. by Christensen et al., Springer-Verlag. 143-174 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森重文, 齋藤夏雄: "Fano threefolds with wild conic bundle structures"Proc. Japan Acad., Ser.A. 79. 111-113 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森重文: "On semistable extremal neighborhoods"Advanced Studies in pure math.. 35. 157-184 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森重文: "On a Hyperplane Section Theorem of Gurjar"Math.Ann.. 319. 533-537 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森重文, J.Kollar, 宮岡洋一, 高木寛通: "Boundedness of canonical Q-Fano 3-folds"Proc.Japan Acad., Ser.A.. 76. 73-77 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 森重文, 藤野修: "A canonical bundle formula"J.Diff.Geom.. 56. 167-188 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 向井茂: "Geometric realization of T-shaped root systems and counterexamples to Hilbert's fourteenth problem"in "Algebraic Transformation Groups and Algebraic Varieties" (ed) V.L.Popv, Springeer-Verlag. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 向井茂: "Plane quartics and Fano threefolds of genus twelve""Proceedings of the Fano Conference, Torino 2002", (eds) A.Conte et al.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 向井茂: "Vector boundles on a K3 surface"Proc. ICM Beijin 2002, Higher Education Press, Beijing. 2. 495-502 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 中山昇, Yuji Ishii: "Classification of normal quartic surfaces with irrational singularities"J.Math.Soc. Japan. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 中山昇: "Global structure of an elliptic fibration"Publ.RIMS, Kyoto Univ.. 38. 451-649 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 中山昇: "Ruled surfaces with non-trivial surjective endomorphisms"Kushu J.Math.. 56. 433-446 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Mukai flops and derived categories"J. Reine Angew.Math.. 560. 65-76 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Counter-example to global Torelli problem for irreducible symplectic manifolds"Math.Ann.. 324. 841-845 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Stratified local moduli of Calabi-Yau threefolds"Topology. 41. 1219-1237 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Projectivity criterion of Moishezon spaces and density of projective symplectic varieties"International J.Math.. 13. 125-135 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Global Smoothing of Calabi-Yau Threefolds II"Compositio Mathematica. 125. 55-68 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Deformation theory of singular symplectic n-folds"Math.Ann.. 319. 597-623 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 並河良典: "Extension of 2-forms and symplectic varieties"J.reine angew.Math.. 539. 123-147 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michiaki Inaba, 岩崎克則, 齋藤政彦: "Backlund transformations of the Sixth Painleve Equations in Terms of Riemann-Hilbert Correspondences"Internat.Math.Res.Notices. 1. 1-30 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 齋藤政彦, 武部尚志, 寺島ひとみ: "Deformation of Okamoto-Painleve Pairs and Painleve equations"J. of Algebraic Geometry. 11. 311-362 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 細野忍, 齋藤 政彦, 高橋篤史: "Relative Lefschetz action and BPS state counting"Internat.Math.Res.Notices. 15. 783-816 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 齋藤 政彦, 由井典子: "The modularity conjecture for rigid Calabi-Yau threefolds over Q"J.Math.Kyoto Univ.. 41-2. 403-419 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小木曽啓示, 細野忍, S-T.Yau: "Fourier-Mukai Partners of a K3 Surface of Picard Number One"Contemporary Mathematics. 322. 43-55 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小木曽啓示: "Local Families of K3 surfaces and Applications"J.Algebraic Geometry. 12. 405-433 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小木曽啓示, D.Q.Zhang: "The simple group of order 168 and K3 surfaces"Complex Geometry, Collection of papers dedicated to Hans Grauert ; Bauer, Catanese, Kawamata, Peternell, Siu(ed), Springer. 165-184 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小木曽啓示: "K3 surfaces via almost primes"Math.Res.Lett.. 9. 47-63 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小古曽啓示: "Seshadri constants in a family of surfaces"Math.Annalen. 323. 625-631 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 小木曽啓示, E.Viehweg: "On the isotiviality of families of elliptic surfaces"J.Algebraic Geom.. 10. 569-598 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 斎藤盛彦, A.Dimca: "Monodromy at infinity and the weights of cohomology"Compos.Math.. 138. 55-71 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] L.Barbieri-Viale, A.Rosenschon, 齋藤盛彦: "Deligne's conjecture on 1-motives"Ann.Math.. 158. 593-633 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 高木寛通: "On classification of Q-Fano 3-folds of Gorenstein index 2.I"Nagoya Math.J.. 167. 117-155 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 高木寛通: "On classification of Q-Fano 3-folds of Gorenstein index 2.II"Nagoya Math.J.. 167. 157-216 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 趙康治, 宮岡洋一, N.I.Shepherd-Barron: "Characterizations of projective space and applications to complex symplectic manifolds"Advanced Stud. in Pure Math.. 35. 1-88 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 藤野修: "Termination of 4-fold canonical flips"Publ.Res.Inst.Math.Sci.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 藤野修: "Notes on toric varieties from Mori theoretic viewpoint"Tohoku Math.J.. 55. 551-564 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 藤野修: "A canonical bundle formula for certain algebraic fiber spaces and its applications"Nagoya Math.J.. 172. 129-171 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 藤野修: "The indices of log canonical singularities"American J.Math.. 123. 229-253 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 藤野修: "Abundance theorem for semi log canonical threefolds"Duke Math.J.. 102. 513-532 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 早川貴之: "Flips in dimension three via crepant descent method"Proc. Japan Acad., Ser.A. 79. 46-51 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 早川貴之: "Blowing ups of 3-dimensional terminal singularities, II"Publ.RIMS, Kyoto Univ.. 36. 423-456 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shigefumi Mori: "V. Alexeev, Bounding singular surfaces of generay type"in "Algebra, and Geometry with Applications". ed. by Christensen et al. Springer-Verlag. 143-174 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigefumi Mori: "Natsuo Saito, Fano threefolds with wild conic bundle structures"Proc. Japan Acad., A. 79. 111-113 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigefumi Mori: "On semistable extremal neghborhoods"Advanced Studies in pure math. 35. 157-184 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigefumi Mori: "On a Hyperplane Section Theorem of Gurjar"Math. Ann.. 319. 533-537 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigefuji Mori: "J. Kollar, Yoichi Miyaoka, Hiromichi Takagi, Boundedness of canonical Q-Fano 3-folds"Proc. Japan Acad., Ser. A. 76. 73-77 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigefumi Mori, Osamu Fujino: "A canonical bundle formula"J. Diff. Geom.. 56. 167-188 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeru Mukai: "Geometric realization of T-shaped root systems and counterexamples to Hilbert's fourteenth problem"Algebraic Transformation Groups and Algebraic Varieties. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeru Mukai: "Plane quartics and Fano threefolds of genus twelve"Proceedings of the Fano Conference. (To appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigeru Mukai: "Vector bundles on a K3 surface"Proc. ICM Beijin. 2. 495-502 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] oboru Nakayama, Yuji Ishii: "Classification of normal quatric sufaces with irrational singularities"J. Math. Soc. Japan. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Noboru Nakayama: "Ruled surfaces with non-trivial surjective endomorphisms"Kushu J. Math.. 38. 451-649 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinori Namikawa: "Mukai flops and derived categories"J. Reine Angew. Math.. 56. 433-446 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] oshinori Namikawa: "Counter-example to global Torelli problem for irreducible symplectic manifolds"Math. Ann.. 560. 65-76 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinori Namikawa: "Stratified local mouduli of Calabi-Yau three-folds"Topology. 41. 1219-1237 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinori Namikawa: "Projectivity criterion of Moishezon spaces and density of projective symplectic varieties"International J. Math.. 13. 125-135 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] oshinori Namikawa: "Gloval Smoothing of Calabi-Yau Threefolds II"Compositio Mathematica. 125. 55-69 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinori Namikawa: "Deformation theory of singular symplectic n-folds"Math. Ann.. 319. 597-623 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshinori Namikawa: "Extension of 2-forms and symplectic varieties"J. reine angew. Math.. 539. 123-147 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiaki Inaba, Katsunori Iwasaki, Masa-Hiko Saito: "Backlund transformations of the Sixth Painleve Equations in Terms of Riemann-Hilbert correspondences"Internat. Math. Res. Notices. 1. 1-30 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masa-Hiko Saito, Takashi Takebe, Hitomi Terajima: "Deformation of Okamoto-Painleve Pairs and Painleve equations"J. of Algebraic Geometry. 11. 311-362 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shinobu Hosono, Masa-Hiko Saito, Atsushi Takahashi: "Relative Lefschetz action and BPS state counting"Internat. Math. Res. Notices. 15. 783-816 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masa-Hiko Saito, Noriko Yui: "The modularity conjecture for rigid Calabi-Yau threefolds over Q"J. Math. Kyoto Univ.. 41-2. 403-419 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso, Shinobu Hosono, S-T. Yau, Fourier-Mukai: "Partners of a K3 Surface of Picard Number One"Contemporary Mathematics. 322. 43-55 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso: "Local Families of K3 Surfaces and Applications"J. Algebraic Geometry. 12. 405-433 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso, D. Q. Zhang: "The simple group of order 168 and K3 surfaces"In: Complex Geometry, Collection of papers dedicated to Hans Grauert; Bauer, Catanese, Kawamata, Peternell, Siu (ed), Springer. 165-184 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso: "K3 surfaces via almost primes"Math. Res. Lett.. 9. 47-63 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso: "Seshadri constants in a family of surfaces"Math. Annalen. 323. 625-631 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiji Oguiso, E. Viehweg: "On the isotiviality of families of elliptic surfaces"J. Algebraic Geom.. 10. 569-598 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Morihiko Saito, A. Dimca: "Monodromy at infinity and the weights of cohomology"Compos. Math.. 138. 55-71 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] L.Barbieri-Viale, A.Rosenschon, Morihiko Saito: "Deligne's conjecture on 1-motives"Ann. Math.. 158. 593-633 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiromichi Takagi: "On classification of Q-Fano 3-folds of Gorenstein index 2. I"Nagoya Math. J.. 167. 117-155 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hiromichi Takagi: "On classification of. Q-Fano 3-folds of Gorenstein index 2. II"Nagoya Math. J.. 167. 157-216 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Koji Cho, Yoichi Miyaoka, N.I: "Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds"Advanced Stud. in Pure Math.. 35. 1-88 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Osamu Fujino: "Termination of 4-fold canonical flips"Publ. Res. Inst. Math. Sci.. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Osamu Fujino: "Notes on toric varieties from Mori theoretic view-point"Tohoku Math. J.. 55. 551-564 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Osamu Fujino: "A canonical bundle formula for certain algebraic fiber spaces and its applications"Nagoya Math. J.. 172. 129-171 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Osamu Fujino: "the indices of log canonical singularities"American J. Math.. 123. 229-253 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Osamu Fujino: "Abundance theorem for semi log canonical three-folds"Duke Math. J.. 102. 513-532 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takayuki Hayakawa: "Flips in dimension three via crepant descent method"Proc. Japan Acad., Ser. A. 79. 46-51 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takayuki Hayakawa: "Blowing ups of 3-dimensional terminal singularities, II"Publ. RIMS, Kyoto Univ.. 36. 423-456 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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