Research on Periods of Algebraic Varieties and Hypergeometric Functions
Research on Periods of Algebraic Varieties and Hypergeometric Functions
批准号:
09440015
负责人:
SAITO Masa-hiko
金额:
$8.26万
依托单位:
依托单位国家:
日本
项目类别:
Grant-in-Aid for Scientific Research (B)
财政年份:
1997
资助国家:
日本
项目状态:
已结题
起止时间:
1997 至 1999
中文摘要
During the period of the project,we have investigated the following subjects and obtained the following results。(I)Mirror Symmetry Conjecture for Calabi-Yau manifolds,(Ii)Counting Curves of higher genus in Rational Elliptic Surfaces,(Iii)Lie theoretic aspects on Painleve equations,Algebro-geometric aspects on Painleve equations,GKZ hypergeometric systems and their Grobner deformations。As for(I),we have been studying Gromov-Witteninvariants for certain Calabi-Yau3-folds and computed a part of A-model prepotentials by means of the theta function of the E I D28锡D2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series.As a result,we have checked MSC mathematically for those cases。Developing further,in(Ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces.We propose the holomorphic anomaly@equation(HAE),which the prepotential should satisfy。By using the Jacobi‘s triple product formula and the relative Lefschetz decomposition,we have checked the prepotential satisfies the HAE.As for the studies of Painleve equations((Iii)),our group have been developing Lie theoretic approach and algeblo-geometric approach,which clarify the relations among Painleve equations,the symmetry of Affine Weyl groups and the geometry of rational surfaces.
英文摘要
During the period of the project, we have investigated the following subjects and obtained the following results. (i)Mirror Symmetry Conjecture for Calabi-Yau manifolds, (ii)Counting Curves of higher genus in Rational Elliptic Surfaces, (iii)Lie theoretic aspects on Painleve equations, Algebro-geometric aspects on Painleve equations, GKZ hypergeometric systems and their Grobner deformations. As for (i), we have been studying Gromov-Witteninvariants for certain Calabi-Yau 3-folds and computed a part of A-model prepotentials by means of the theta function of the EィイD28ィエD2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series. As a result, we have checked MSC mathematically for those cases. Developing further, in (ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces. We propose the holomorphic anomaly equation(HAE), which the prepotential should satisfy. By using the Jacobi's triple product formula and the relative Lefschetz decomposition, we have checked the prepotential satisfies the HAE.As for the studies of Painleve equations ((iii)), our group have been developing Lie theoretic approach and algeblo-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.
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M.Noumi and Y.Yamada: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Commun. Math. Phys.. 199. 281-295 (1998)
M.Noumi 和 Y.Yamada:“仿射 Weyl 群、离散动力系统和 Painleve 方程”Commun。
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Takano,K.: "Defining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations, linear and nonlinear. 261-269 (2000)
Takano, K.:“定义 Painleve 方程的流形”对线性和非线性微分方程进行精确的 WKB 分析。
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S.Hosono: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv.Theor.Math.Phys.. 3. 177-208 (1999)
S.Hosono:“有理椭圆面的全纯异常方程和 BPS 状态计数”Adv.Theor.Math.Phys.. 3. 177-208 (1999)
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共 39 条
New developments and interaction between Algebraic Geometry and Integrable Systems
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批准号:19104002
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项目类别:Grant-in-Aid for Scientific Research (S)
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资助金额:$63.48万
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财政年份:2007
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负责人:SAITO Masa-hiko
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依托单位:
Research of new developments in moduli spaces and integrable systems
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批准号:16340009
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项目类别:Grant-in-Aid for Scientific Research (B)
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资助金额:$10.37万
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财政年份:2004
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负责人:SAITO Masa-hiko
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依托单位:
Geometry on String Theory and Moduli spaces
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批准号:12440008
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项目类别:Grant-in-Aid for Scientific Research (B)
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资助金额:$10.3万
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财政年份:2000
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负责人:SAITO Masa-hiko
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