1998 年度実績報告書

代数幾何学とHodge理論

研究課題

研究課題/領域番号	08304002
研究機関	大阪大学
研究代表者	臼井三平大阪大学, 大学院・理学研究科, 教授 (90117002)
研究分担者	今野一宏大阪大学, 大学院・理学研究科, 助教授 (10186869) 森重文京都大学, 数理解析研究所, 教授 (00093328) 向井茂名古屋大学, 大学院・多元数理学研究科, 教授 (80115641) 加藤和也東京大学, 大学院・数理科学研究科, 教授 (90111450) 中村郁北海道大学, 大学院・理学研究科, 教授 (50022687)
キーワード	Hodge 理論 / Log 幾何学 / 代数多様体 / 多重標準形成 / K3曲面 / 標準曲線 / calabi-yau多様体 / Morse化問題
研究概要	10数年続けている研究集会を今年も開催した: 「Hodge理論・Log幾何学・退化」1998年10月12日〜16日,於山梨県北巨摩郡大泉村八ケ岳高原泉郷, 世話係:朝倉政典・荒川達也・臼井三平今年度はHodge増論・Log幾何学・退化というテーマについて以下の研究発表を行い、それらを巡る活発な討議が行われた.この研究集会の報告集を印刷し配付した. 昨年に続き小人数合宿形式による次のミニワークショップを行い密度の高い討議をした: 「Hodge理論と代数幾何学」1998年1月28日〜31日,於兵庫県多可郡八千代町エーデルささゆり, 世話係:臼井三平また昨年に続き,勤務時間外に自由参加として,中学生を主とした地元の人達との交流をした.今年度もこの試みは双方にとって大変好評であった. 個々の研究実績・発表として主なものは次の通りである:臼井三平と加藤和也は昨年度の共同研究の成果を発展させ,Hodge理論とLog幾何学を使ってHodge構造の分類空間Dの数論的群による商の部分コンパクト化についての最も望ましい結果を得た.これはDがHermite対象領域の時のMumford等によるトロイダルコンパクト化を含み,それらを一般化したものである.この成果について目下論文を作成中である.川又は標準特異点の変形および多重標準形式の拡張性を研究した.向井は偏極K3曲面の双対性についてEuroconferenceで講演した.1998年12月の奈良における谷口基金による最終国際学会で,森は代数多様体上の有理曲線について,加藤はBlochコンダクターとp進イプシロン元について講演した.1998年6月のカナダのBanffにおけるNATO高等研究集会において,臼井は対数的Hodge構造とその分類空間について,齋藤はCalabi-Yau多様体の湯川対のポテンシャルと鏡対称性について,朝倉政典は数論的Hodge構造について講演した.今野は,標準曲線の定義式系に関するReid氏の1-2-3予想を完全に解決した.足利正と荒川達也は共同研究により退化超楕円曲線束のMorse化問題を解決した.遊佐毅は幾何殻という概念を導入しそれの基本的性質そ調べた. 本科研費の分担者・協力者によるその他の成果については発表論文表参照.

研究成果

(47件)

すべてその他

すべて文献書誌 (47件)

[文献書誌] Usui, S.: "Recovery of vanishing cycles by log geometry : case of several variables" Proc. "Commutative algebra and Algebraic geometry" , Hanoi 1996, Lect. Note Math., Springer-Verlag. (to appear).
[文献書誌] Usui, S.: "Recovery of vanishing cycles by log geometry" Tohoku Math. J.(submitted).
[文献書誌] Kato, K. Usui, S.: "Logarithmic Hodge structures and classifying spaces (Summary)" Proc. NATO Advanced Study Institute / CRM summer School 1998 : "The Arithmetic ang Geometry of Algebraic Cycles" , Banff. (submitted).
[文献書誌] Nakamura, I.: "Global deformations of P^2-bundles over P^1-bundles" J. Math. Kyoto Univ.38. 29-54 (1998)
[文献書誌] Ito, Y. and Nakamura, I.: "Hilbert schemes and simple singularities" Proc. "New trends in Algebraic geometry" , Warwick 1996, cambridge Univ. Press. 151-233 (1999)
[文献書誌] Alexeev, V. and Nakamura, I.: "On Mumford's construction of degenerating abelian varieties" Tohoku Math. J.(to appear).
[文献書誌] Nakamura, I.: "Compactification of the moduli obabelian varieties over Z [ S_N, 1/N]" Comptes Rendus acad. scient. Paris. 327. 875-880 (1998)
[文献書誌] Nakamura, I.: "Stability of degenerate abelian varieties" Invent. Math.(to appear).
[文献書誌] Nakamura, I.: "Hilbert schemes of abelian group orbits" Max Plank Preprint series. 101. (1998)
[文献書誌] Ishida, M. -N. and Kato, F.: "The strong rigidity theorem for non-Archimedean uniformization" Tohoku Math. J.50. 537-555 (1998)
[文献書誌] Kawamata, Y.: "Index 1 covers of logterminal singularities" J. Alg. Geom.(to appear).
[文献書誌] Kawamata, Y.: "Subadjunction of log canonical divisors II" Amer. J. Math.120. 893-899 (1998)
[文献書誌] Kawamata, Y.: "Deformations of canonical singularities" J. Amer. Math. Soc.(to appear).
[文献書誌] Fujita, T.: "An appendix to Kawachi-Macek's paper on global generation of adjoint bundles on normal surfaces" J. Alg. Geom.7. 251-252 (1998)
[文献書誌] Mukai, S.: "Lattice theoretic construction of symplectic action on K3 surfaces" Duke Math. J.92. 593-603 (1998)
[文献書誌] Mukai, S.: "Duality of polarized K3 surfaces" Proc. Euroconference on Algebraic Geometry, cambridge University Press. 107-122 (1998)
[文献書誌] Mukai, S.: "Abelian variety and spin representations" Warwick preprint. 13. (1998)
[文献書誌] Mukai, S.: "Eguations defining a space curve" Warwick preprint. 14. (1998)
[文献書誌] Mukai, S.: "Simple Lie algebra and Legendre subvariety" Warwick preprint. 15. (1998)
[文献書誌] Miyanishi, M. and Yang, J. -G.: "Surfaces of general type whose canonical map is composed of a pencil of senus 3 with small invariants" J. Math. Kyoto Univ.38. 123-149 (1998)
[文献書誌] Miyanishi, M. and Kinugawa, K.: "Reduction modulo P of cyclic group actions of order pr" J. Pure Applied Algebra. (to appear).
[文献書誌] Miyanishi, M. and Masuda, K.: "Invariant subvarieties of low codimension in the affine space" Tohoku Math. J.(submitted).
[文献書誌] Miyanishi, M. and Aoki, M.: "Geometric construction of certain etale endmorphisms" J. Algebra. (submitted).
[文献書誌] Miyanishi, M. and Gurlar, R. V.: "On a generalization of the Jacobian conjecture" J. reine angew. Math.(submitted).
[文献書誌] Miyanishi, M. and Masuda, K.: "Etale endmorphisms of algebraic surfaces with Gm-actions" Math. Ann.(submitted).
[文献書誌] Konno, K.: "Clifford index and the slope of fibered surfaces" J. Alg. Geom.(to appear).
[文献書誌] Konno, K.: "Certain algebraic surfaces with non-reduced moduli" Portugal. Math.(to appear).
[文献書誌] Saito, M. -H., Hosono, S., Stienstpa, J.: "On the mirror symmetry conjecture for Schoen's Calabi-Yau 3-folds" proc. Taniguchi symp. 97 "Integrable Systems and Algebraic Geometry" , Kobe/Kyoto, World Scientific Co.194-235 (1998)
[文献書誌] Saito, M. -H.: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry" proc. NATO Aduanced Study Institute/CRM Summer School 1998 : "The Arithmetic and Geometry of Algebraic Cycles" , Banff. (Submitted).
[文献書誌] Sumihiro, H.: "Determinantal varieties associated to rank two vector bundles on complex projective spaces" Hiroshima Math. J.(Submitted).
[文献書誌] Nakayama, N.: "Invariance of the plurigenera of algebraic varieties" Research Institut of Mathematical Science, preprint. 1191. (1998)
[文献書誌] Nakayama, N.: "Normalized tautological divisors of semi-stable vector bundles" Research Institute of Mathematical Science, preprint. 1214. (1998)
[文献書誌] Arakawa, T.: "Bounds for the order of automorphism groups of hyperelliptic fibrations" Tohoku Math. J.50. 317-323 (1998)
[文献書誌] Arakawa, T.: "A note on normally generated line bundles on compact Riemann surfaces" Tsukuba J. Math.22-3. (1999)
[文献書誌] Arakawa, T.: "Partially negative cyales and projective embeddings of surfaces of genelal type" Osaka J. Math.(to appear).
[文献書誌] Arakawa, T.: "A note on normally generated line bundles on compact Riemann surfaces, II" Tsukuba J. Math.(submitted).
[文献書誌] Sato, E. and Kachi, Y.: "Polarized varieties whose points are joined by national curves of small degree" Illinois J. Math.(to appear).
[文献書誌] Sato, E.: "Behavior of national curves with the minimal degree in projective space bundle" (submitted).
[文献書誌] Sato, E.: "Smooth 4-folds which contain a P^1-bundle as an ample divisor" (submitted).
[文献書誌] Ashikaga, T.: "Surface singularities on cyclic coverings and an ineguality for the signature" J. Math. Soc. Japan. (to appear).
[文献書誌] Arakawa, T. and Ashikaga, T.: "Local splitting families of hpere lliptic pencils, I" Max Plank Institute preprint series. 33. (1998)
[文献書誌] Usa, T.: "Problems on geometric struotures of projective embeddings" Report Fac. Sci. Himeji Institute of Technology. 9. 12-29 (1998)
[文献書誌] 川又雄次郎: "代数多様体論" 共立出版, 223 (1997)
[文献書誌] 加藤和也、他二名: "現代数学の基礎 : 数論2" 岩波書店, 244 (1998)
[文献書誌] 向井茂: "現在数学の展望 : モジュライ理論1" 岩波書店, 176 (1998)
[文献書誌] 森重文, Kollar, J.: "現在数学の展望 : 双有理幾何学" 岩波書店, 328 (1998)
[文献書誌] Mori, S. and Kollar, J.: "Birational geometry of algebraic varieties" Cambridge University Press, (1998)

1998 年度 実績報告書

代数幾何学とHodge理論

研究代表者

臼井 三平 大阪大学, 大学院・理学研究科, 教授 (90117002)

研究成果

[文献書誌] Usui, S.: "Recovery of vanishing cycles by log geometry : case of several variables" Proc. "Commutative algebra and Algebraic geometry" , Hanoi 1996, Lect. Note Math., Springer-Verlag. (to appear).

[文献書誌] Usui, S.: "Recovery of vanishing cycles by log geometry" Tohoku Math. J.(submitted).

[文献書誌] Kato, K. Usui, S.: "Logarithmic Hodge structures and classifying spaces (Summary)" Proc. NATO Advanced Study Institute / CRM summer School 1998 : "The Arithmetic ang Geometry of Algebraic Cycles" , Banff. (submitted).

[文献書誌] Nakamura, I.: "Global deformations of P^2-bundles over P^1-bundles" J. Math. Kyoto Univ.38. 29-54 (1998)

[文献書誌] Ito, Y. and Nakamura, I.: "Hilbert schemes and simple singularities" Proc. "New trends in Algebraic geometry" , Warwick 1996, cambridge Univ. Press. 151-233 (1999)

[文献書誌] Alexeev, V. and Nakamura, I.: "On Mumford's construction of degenerating abelian varieties" Tohoku Math. J.(to appear).

[文献書誌] Nakamura, I.: "Compactification of the moduli obabelian varieties over Z [ S_N, 1/N]" Comptes Rendus acad. scient. Paris. 327. 875-880 (1998)

[文献書誌] Nakamura, I.: "Stability of degenerate abelian varieties" Invent. Math.(to appear).

[文献書誌] Nakamura, I.: "Hilbert schemes of abelian group orbits" Max Plank Preprint series. 101. (1998)

[文献書誌] Ishida, M. -N. and Kato, F.: "The strong rigidity theorem for non-Archimedean uniformization" Tohoku Math. J.50. 537-555 (1998)

[文献書誌] Kawamata, Y.: "Index 1 covers of logterminal singularities" J. Alg. Geom.(to appear).

[文献書誌] Kawamata, Y.: "Subadjunction of log canonical divisors II" Amer. J. Math.120. 893-899 (1998)

[文献書誌] Kawamata, Y.: "Deformations of canonical singularities" J. Amer. Math. Soc.(to appear).

[文献書誌] Fujita, T.: "An appendix to Kawachi-Macek's paper on global generation of adjoint bundles on normal surfaces" J. Alg. Geom.7. 251-252 (1998)

[文献書誌] Mukai, S.: "Lattice theoretic construction of symplectic action on K3 surfaces" Duke Math. J.92. 593-603 (1998)

[文献書誌] Mukai, S.: "Duality of polarized K3 surfaces" Proc. Euroconference on Algebraic Geometry, cambridge University Press. 107-122 (1998)

[文献書誌] Mukai, S.: "Abelian variety and spin representations" Warwick preprint. 13. (1998)

[文献書誌] Mukai, S.: "Eguations defining a space curve" Warwick preprint. 14. (1998)

[文献書誌] Mukai, S.: "Simple Lie algebra and Legendre subvariety" Warwick preprint. 15. (1998)

[文献書誌] Miyanishi, M. and Yang, J. -G.: "Surfaces of general type whose canonical map is composed of a pencil of senus 3 with small invariants" J. Math. Kyoto Univ.38. 123-149 (1998)

[文献書誌] Miyanishi, M. and Kinugawa, K.: "Reduction modulo P of cyclic group actions of order pr" J. Pure Applied Algebra. (to appear).

[文献書誌] Miyanishi, M. and Masuda, K.: "Invariant subvarieties of low codimension in the affine space" Tohoku Math. J.(submitted).

[文献書誌] Miyanishi, M. and Aoki, M.: "Geometric construction of certain etale endmorphisms" J. Algebra. (submitted).

[文献書誌] Miyanishi, M. and Gurlar, R. V.: "On a generalization of the Jacobian conjecture" J. reine angew. Math.(submitted).

[文献書誌] Miyanishi, M. and Masuda, K.: "Etale endmorphisms of algebraic surfaces with Gm-actions" Math. Ann.(submitted).

[文献書誌] Konno, K.: "Clifford index and the slope of fibered surfaces" J. Alg. Geom.(to appear).

[文献書誌] Konno, K.: "Certain algebraic surfaces with non-reduced moduli" Portugal. Math.(to appear).

[文献書誌] Saito, M. -H., Hosono, S., Stienstpa, J.: "On the mirror symmetry conjecture for Schoen's Calabi-Yau 3-folds" proc. Taniguchi symp. 97 "Integrable Systems and Algebraic Geometry" , Kobe/Kyoto, World Scientific Co.194-235 (1998)

[文献書誌] Saito, M. -H.: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and mirror symmetry" proc. NATO Aduanced Study Institute/CRM Summer School 1998 : "The Arithmetic and Geometry of Algebraic Cycles" , Banff. (Submitted).

[文献書誌] Sumihiro, H.: "Determinantal varieties associated to rank two vector bundles on complex projective spaces" Hiroshima Math. J.(Submitted).

[文献書誌] Nakayama, N.: "Invariance of the plurigenera of algebraic varieties" Research Institut of Mathematical Science, preprint. 1191. (1998)

[文献書誌] Nakayama, N.: "Normalized tautological divisors of semi-stable vector bundles" Research Institute of Mathematical Science, preprint. 1214. (1998)

[文献書誌] Arakawa, T.: "Bounds for the order of automorphism groups of hyperelliptic fibrations" Tohoku Math. J.50. 317-323 (1998)

[文献書誌] Arakawa, T.: "A note on normally generated line bundles on compact Riemann surfaces" Tsukuba J. Math.22-3. (1999)

[文献書誌] Arakawa, T.: "Partially negative cyales and projective embeddings of surfaces of genelal type" Osaka J. Math.(to appear).

[文献書誌] Arakawa, T.: "A note on normally generated line bundles on compact Riemann surfaces, II" Tsukuba J. Math.(submitted).

[文献書誌] Sato, E. and Kachi, Y.: "Polarized varieties whose points are joined by national curves of small degree" Illinois J. Math.(to appear).

[文献書誌] Sato, E.: "Behavior of national curves with the minimal degree in projective space bundle" (submitted).

[文献書誌] Sato, E.: "Smooth 4-folds which contain a P^1-bundle as an ample divisor" (submitted).

[文献書誌] Ashikaga, T.: "Surface singularities on cyclic coverings and an ineguality for the signature" J. Math. Soc. Japan. (to appear).

[文献書誌] Arakawa, T. and Ashikaga, T.: "Local splitting families of hpere lliptic pencils, I" Max Plank Institute preprint series. 33. (1998)

[文献書誌] Usa, T.: "Problems on geometric struotures of projective embeddings" Report Fac. Sci. Himeji Institute of Technology. 9. 12-29 (1998)

[文献書誌] 川又雄次郎: "代数多様体論" 共立出版, 223 (1997)

[文献書誌] 加藤和也、 他二名: "現代数学の基礎 : 数論2" 岩波書店, 244 (1998)

[文献書誌] 向井 茂: "現在数学の展望 : モジュライ理論1" 岩波書店, 176 (1998)

[文献書誌] 森 重文, Kollar, J.: "現在数学の展望 : 双有理幾何学" 岩波書店, 328 (1998)

[文献書誌] Mori, S. and Kollar, J.: "Birational geometry of algebraic varieties" Cambridge University Press, (1998)

1998 年度実績報告書

臼井三平大阪大学, 大学院・理学研究科, 教授 (90117002)

[文献書誌] 加藤和也、他二名: "現代数学の基礎 : 数論2" 岩波書店, 244 (1998)

[文献書誌] 向井茂: "現在数学の展望 : モジュライ理論1" 岩波書店, 176 (1998)

[文献書誌] 森重文, Kollar, J.: "現在数学の展望 : 双有理幾何学" 岩波書店, 328 (1998)