Research activity information

• aCM bundles on a general K3 surface of degree 2

  Kōta Yoshioka

  Elsevier BV, Nov. 2024, Journal of Algebra, 658, 415 - 449, English

  [Refereed]

  Scientific journal


• Wall crossing for moduli of stable sheaves on an elliptic surface

  Kōta Yoshioka

  Springer Science and Business Media LLC, Dec. 2023, Mathematische Zeitschrift, 306(1) (1), English

  [Refereed]

  Scientific journal


• The cohomology of the general stable sheaf on a K3 surface

  Izzet Coskun, Howard Nuer, Kōta Yoshioka

  Elsevier BV, Aug. 2023, Advances in Mathematics, 426, 109102 - 109102, English

  [Refereed]

  Scientific journal


• Some moduli spaces of 1-dimensional sheaves on an elliptic ruled surface

  Kōta Yoshioka

  Springer Science and Business Media LLC, May 2023, Geometriae Dedicata, 217(3) (3), English

  [Refereed]

  Scientific journal


• Birational automorphim groups of a generalized Kummer manifold for an abelian surface with Picard number 1

  Kōta Yoshioka

  Springer Science and Business Media LLC, Apr. 2023, manuscripta mathematica, 173(1-2) (1-2), 727 - 751, English

  [Refereed]

  Scientific journal


• Moduli of Stable Sheaves on a K3 Surface of Picard Number 1

  Akira MORI, Kōta YOSHIOKA

  Tokyo Journal of Mathematics, Dec. 2022, Tokyo Journal of Mathematics, 45(2) (2), English

  [Refereed]

  Scientific journal


• aCM bundles on a general abelian surface

  Kōta Yoshioka

  Springer Science and Business Media LLC, May 2021, Archiv der Mathematik, 116(5) (5), 529 - 539, English

  [Refereed]

  Scientific journal


• MMP via wall-crossing for moduli spaces of stable sheaves on an Enriques surface.

  Nuer, Howard, Yoshioka, Kota,

  Oct. 2020, Adv. Math., 372, 107283, 119pp, English

  [Refereed]

  Scientific journal


• Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces.

  Yoshioka, Kota

  Aug. 2020, J. Algebra, 556, 448 - 466, English

  [Refereed]

  Scientific journal


• Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational.

  Meachan, Ciaran, Mongardi, Giovanni, Yoshioka, Kota

  Apr. 2020, Math. Z., 294(3-4) (3-4), 871 - 880, English

  [Refereed]

  Scientific journal


• Moduli spaces of stable sheaves on Enriques surfaces

  Kōta Yoshioka

  Duke University Press, Dec. 2018, Kyoto Journal of Mathematics, 58(4) (4), 865914, English

  [Refereed]

  Scientific journal


• The wall-crossing behavior for Bridgeland's stability conditions on abelian and K3 surfaces

  H. Minamide, S. Yanagida, Kota YOSHIOKA

  Feb. 2018, J. reine angew. Math, 735, 1 - 107, English

  [Refereed]

  Scientific journal


• Fourier-Mukai duality for K3 surfaces via Bridgeland stability condition

  Kota YOSHIOKA

  Dec. 2017, J. Geometry and Physics, 122, 103 - 118, English

  [Refereed]

  Scientific journal


• A note on stable sheaves on Enriques surfaces

  Kota YOSHIOKA

  Sep. 2017, Tohoku Math. J, 69(3) (3), 369 - 382, English

  [Refereed]

  Scientific journal


• A note on stable sheaves on Enriques surfaces II

  Kota Yoshioka

  May 2017, MANUSCRIPTA MATHEMATICA, 153(1-2) (1-2), 147 - 158, English

  [Refereed]

  Scientific journal


• On the strange duality conjecture for abelian surfaces II.

  Barbara Bolognese, Alina Marian, Dragos Oprea, Kota Yoshioka

  2017, JOURNAL OF ALGEBRAIC GEOMETRY, 26(3) (3), 475 - 511, English

  [Refereed]

  Scientific journal


• Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface.

  Kota Yoshioka

  2016, DEVELOPMENT OF MODULI THEORY - KYOTO 2013, Adv. Stud. Pure Math., 69, 473 - 537, English

  [Refereed]

  International conference proceedings


• A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3 or generalized Kummer deformation type

  Markman Eyal, YOSHIOKA Kota

  Dec. 2015, Int. Math. Res. Not. IMRN 2015, 24, 13563 - -13574, English

  [Refereed]

  Scientific journal


• Perverse coherent sheaves and Fourier-Mukai transforms on surfaces, II

  Kota Yoshioka

  Jun. 2015, KYOTO JOURNAL OF MATHEMATICS, 55(2) (2), 365 - 459, English

  [Refereed]

  Scientific journal


• Some moduli spaces of Bridgeland's stability conditions

  YOSHIOKA Kota, MINAMIDE Hiroki, YANAGIDA Shintaro

  Nov. 2014, Int. Math. Res. Not. IMRN 2014, 19, 5264 - 5327, English

  [Refereed]

  Scientific journal


• Bridgeland's stabilities onabelian surfaces

  Shintarou Yanagida, YOSHIOKA kota

  Feb. 2014, Math.Z., 276, 571 - 610, English

  [Refereed]

  Scientific journal


• Semi-homogeneous sheaves, Fourier-Mukaitransforms and moduli of stablesheaves on Abelian surfaces

  Shintarou Yanagida, YOSHIOKA kota

  Nov. 2013, J. reine angew. Math., 684, 31 - 86, English

  [Refereed]

  Scientific journal


• Perverse coherent sheaves and Fourier-Mukai transformson surfaces I

  YOSHIOKA kota

  May 2013, Kyoto J. of Math., 53, 261 - 344, English

  [Refereed]

  Scientific journal


• Birational Maps of Moduli Spaces of Vector Bundles on $K3$ Surfaces

  Masanori KIMURA, Kōta YOSHIOKA

  Tokyo Journal of Mathematics, Dec. 2011, Tokyo Journal of Mathematics, 34(2) (2), 473 - 491, English

  [Refereed]

  Scientific journal


• Perverse coherent sheaves on blow-up. I. A quiver description.

  Nakajima, Hiraku, Yoshioka, Kota

  2011, Adv. Stud. Pure Math., 61, 349 - 386, English

  [Refereed]


• Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing

  Hiraku Nakajima, Kōta Yoshioka

  Duke University Press, Jan. 2011, Kyoto Journal of Mathematics, 51(2) (2), 263 - 335, English

  [Refereed]

  Scientific journal


• Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula

  Hiraku Nakajima, Kōta Yoshioka

  American Mathematical Society (AMS), Jan. 2011, Journal of Algebraic Geometry, 20(1) (1), 47 - 100, English

  [Refereed]

  Scientific journal


• Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting

  Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka

  Jan. 2010, Publ.Res.Inst.Math.Sci., 47(1) (1), 307 - 359, English

  [Refereed]

  Scientific journal


• Instanton counting and Donaldson invariants

  Nakajima, Hiraku, Yoshioka, Kota

  2010, Sugaku Expositions, 23(2) (2), 189 - 212, English

  [Refereed]

  Scientific journal


• An action of a Lie algebra on the homology groups of moduli spaces of stable sheaves

  Kota Yoshioka

  2010, SAPPORO 2007: ALGEBRAIC AND ARITHMETIC STRUCTURES OF MODULI SPACES, 58, 403 - 459, English

  [Refereed]

  International conference proceedings


• Fourier–Mukai transform on abelian surfaces

  Kōta Yoshioka

  Springer Science and Business Media LLC, Nov. 2009, Mathematische Annalen, 345(3) (3), 493 - 524, English

  [Refereed]

  Scientific journal


• K-theoretic Donaldson Invariants Via Instanton Counting

  Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka

  International Press of Boston, 2009, Pure and Applied Mathematics Quarterly, 5(3) (3), 1029 - 1112, English

  [Refereed]

  Scientific journal


• Stability and the Fourier–Mukai transform II

  Kota Yoshioka

  We consider the problem of preservation of stability under the Fourier–Mukai transform ℱ_ℰ:D(X)→D(Y) on an abelian surface and aK3 surface. IfYis the moduli space ofμ-stable sheaves onXwith respect to a polarizationH, we have a canonical polarization$\widehat {H}$onYand we have a correspondence between (X,H) and$(Y,\widehat {H})$. We show that the stability with respect to these polarizations is preserved under ℱ_ℰ, if the degree of stable sheaves onXis sufficiently large.

  Lead, Wiley, Jan. 2009, Compositio Mathematica, 145(1) (1), 112 - 142, English

  [Refereed]

  Scientific journal


• Holomorphic vector bundles on non-algebraic tori of dimension 2

  Koichi Kurihara, Kōta Yoshioka

  Springer Science and Business Media LLC, Jun. 2008, manuscripta mathematica, 126(2) (2), 143 - 166, English

  [Refereed]

  Scientific journal


• Instanton counting and Donaldson invariants

  Gottsche, Lothar, Nakajima, Hiraku, Yoshioka, Kota

  2008, J. Differential Geom., 80(3) (3), 343 - 390, English

  [Refereed]

  Scientific journal


• Moduli of vector bundles on algebraic surfaces

  Kota Yoshioka

  2007, Sugaku Expositions, 20(1) (1), 115 - 135, English

  [Refereed]

  Scientific journal


• インスタントンの数え上げとDonaldson不変量

  中島 啓, 吉岡 康太

  2007, 数学, 2, 131 - 153, Japanese


• Moduli spaces of twisted sheaves on a projective variety

  Kōta Yoshioka

  Mathematical Society of Japan, 2006, Moduli Spaces and Arithmetic Geometry (Kyoto, 2004), 45, 1 - 42, English

  [Refereed]

  International conference proceedings


• Instanton counting on blowup. I. 4-dimensional pure gauge theory

  Hiraku Nakajima, Kota Yoshioka

  Springer Science and Business Media LLC, Nov. 2005, Inventiones mathematicae, 162(2) (2), 313 - 355, English

  Scientific journal


• Instanton counting on blowup. II. K-theoretic partition function

  Hiraku Nakajima, Kota Yoshioka

  May 2005, Transformation Groups, 20(3-4) (3-4), 489 - 519, English

  [Refereed]


• 代数曲面上のベクトル束のモジュライ空間

  Kota Yoshioka

  The Mathematical Society of Japan, Jul. 2004, Sugaku, 56(3) (3), 225 - 247, Japanese


• Lectures on instanton counting

  Hiraku Nakajima, Kota Yoshioka

  2004, CRM Workshop on Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, 38, 31 - 101, English

  [Refereed]


• Stability and the Fourier-Mukai transform I

  Kota Yoshioka

  Springer Science and Business Media LLC, Dec. 2003, Mathematische Zeitschrift, 245(4) (4), 657 - 665, English

  [Refereed]

  Scientific journal


• Singularities on the 2-Dimensional Moduli Spaces of Stable Sheaves on K3 Surfaces

  Nobuaki Onishi, Kōta Yoshioka

  We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on k3 surfaces. We show that the moduli space is normal, in particular the siguralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

  World Scientific Pub Co Pte Lt, Oct. 2003, International Journal of Mathematics, 14(08) (08), 837 - 864, English

  [Refereed]

  Scientific journal


• Twisted stability and Fourier-Mukai transform II

  Kota Yoshioka

  Springer Science and Business Media LLC, Apr. 2003, manuscripta mathematica, 110(4) (4), 433 - 465, English

  [Refereed]

  Scientific journal


• A note on moduli of vector bundles on rational surfaces

  Kōta Yoshioka

  Duke University Press, Jan. 2003, Kyoto Journal of Mathematics, 43(1) (1), 139 - 163, English

  [Refereed]

  Scientific journal


• Twisted Stability and Fourier–Mukai Transform I

  Kōta Yoshioka

  Lead, Cambridge University Press (CUP), 2003, Compositio Mathematica, 138(3) (3), 261 - 288, English

  [Refereed]

  Scientific journal


• Moduli spaces of stable sheaves on abelian surfaces

  Kota Yoshioka

  Springer Science and Business Media LLC, Dec. 2001, Mathematische Annalen, 321(4) (4), 817 - 884, English

  [Refereed]

  Scientific journal


• Brill-Noether problem for sheaves on K3 surfaces (Proceedings of the Workshop "Algebraic Geometry and Integrable Systems related to String Theory")

  Yoshioka, Kota

  Jan. 2001, Sūrikaisekikenkyūsho Kōkyūroku, 1232, 109 - 124, English

  [Refereed]

  Scientific journal


• String partition functions and infinite products

  Toshiya Kawai, Kōta Yoshioka

  International Press of Boston, Feb. 2000, Advances in Theoretical and Mathematical Physics, 4(2) (2), 397 - 485, English

  [Refereed]

  Scientific journal


• Some examples of Mukai's reflections on K3 surfaces

  K. Yoshioka

  Walter de Gruyter GmbH, Oct. 1999, Journal für die reine und angewandte Mathematik (Crelles Journal), 1999(515) (515), 97 - 123, English

  [Refereed]

  Scientific journal


• Some notes on the moduli of stable sheaves on elliptic surfaces

  Kota Yoshioka

  Nagoya University, 1999, Nagoya Math.J., 154, 73 - 102, English

  [Refereed]

  Scientific journal


• Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces

  Kota Yoshioka

  1999, Communications in mathematical physics, 205, 501 - 517, English

  [Refereed]

  Scientific journal


• A note on the universal family of moduli of stable sheaves

  Kota Yoshioka

  Walter de Gruyter GmbH, Mar. 1998, Journal für die reine und angewandte Mathematik (Crelles Journal), 1998(496) (496), 149 - 161, English

  [Refereed]

  Scientific journal


• A NOTE ON A PAPER OF J.-M.DREZET ON THE LOCAL FACTORIALITY OF SOME MODULI SPACES

  KŌTA YOSHIOKA

  World Scientific Pub Co Pte Lt, Dec. 1996, International Journal of Mathematics, 07(06) (06), 843 - 858, English

  [Refereed]

  Scientific journal


• CHAMBER STRUCTURE OF POLARIZATIONS AND THE MODULI OF STABLE SHEAVES ON A RULED SURFACE

  KŌTA YOSHIOKA

  World Scientific Pub Co Pte Lt, Jun. 1996, International Journal of Mathematics, 07(03) (03), 411 - 431, English

  [Refereed]

  Scientific journal


• Numbers of Fq-rational points of the moduli of stable sheaves on elliptic surfaces.

  Kota Yoshioka

  1996, Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math., 179, 297 - 305, English

  [Refereed]


• The Picard group of the moduli space of stable sheaves on a ruled surface

  Kota Yoshioka

  Duke University Press, Jan. 1996, Kyoto Journal of Mathematics, 36(2) (2), 279 - 309, English

  [Refereed]

  Scientific journal


• The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface

  Kota Yoshioka

  Springer Science and Business Media LLC, May 1995, Mathematische Annalen, 302(1) (1), 519 - 540, English

  [Refereed]

  Scientific journal


• The Betti numbers of the moduli space of stable sheaves of rank2 on P2.

  Kota Yoshioka

  1994, Journal für die reine und angewandte Mathematik, 453, 193 - 220, English

  [Refereed]

  Scientific journal

• Vector bundles on algebraic surfaces

  YOSHIOKA Kota

  東京大学, Sep. 2014, 第59回代数学シンポジウム報告集, 54 - 66, Japanese

  Introduction other


• Examples of movable divisors on a generalized Kummer variety and an application

  YOSHIOKA Kota

  京都大学数理解析研究所, 2014, 2014 年度代数幾何学シンポジウム報告集, 2014, 3 - 11, English

  [Refereed]

  Introduction other


• PROBLEMS ON HOLOMORPHIC VECTOR BUNDLES ON COMPLEX MANIFOLDS (Open Problems in Complex Geometry)

  YOSHIOKA KOTA

  Kyoto University, Mar. 2011, RIMS Kokyuroku, 1731, 162 - 174, English


• The Betti numbers of the moduli space of stable sheaves of rank 2 on Pⁿ

  Yoshioka, Kota

  京都大学数理解析研究所, 1992, 代数幾何学シンポジューム記録, 1992, 40 - 54, Japanese

• Moduli of stable vector bundles on abelian surfaces

  Yoshioka Kota

  UNIBO Seminar, Oct. 2024, English

  [Invited]

  Invited oral presentation


• aCM bundles on a generic K3 surface of degree 2

  Yoshioka Kota

  One-day workshop "Bundles and lattices", Aug. 2024

  [Invited]

  Invited oral presentation


• アーベル曲面上の安定ベクトル束

  吉岡康太

  第3回仙台保型形式小集会 「保型形式, 代数幾何, (保型)微分作用素、頂点作用素代数」, Feb. 2024, Japanese

  [Invited]

  Invited oral presentation


• Moduli of stable sheaves on an elliptic surface

  Kōta YOSHIOKA

  Gauge Theory, Moduli Spaces and Representation Theory, Kyoto 2023 In honor of the 60th birthday of Hiraku Nakajima, Feb. 2023, English

  Invited oral presentation


• Moduli of stable sheaves on an abelian surface

  Yoshioka, Kota

  Japanese-European Symposium on symplectic varieties and moduli spaces, Mar. 2022, English

  [Invited]

  Invited oral presentation


• 楕円曲面上の安定層のモジュライ空間

  吉岡 康太

  第17回代数・解析・幾何学セミナー, Feb. 2022, Japanese

  [Invited]

  Invited oral presentation


• Moduli of stable sheaves on an elliptic surface

  Yoshioka, Kota

  Workshop in honour of Lothar Gottsche's 60th birthday,, Dec. 2021, English

  [Invited]

  Invited oral presentation


• Weak Brill-Noether for the moduli of stable sheaves on a generic K3 surface

  Kota Yoshioka

  Categorical and Analytic Invariants in Algebraic Geometry VII, Nov. 2019, English, 大阪大学理学部

  [Invited]

  Invited oral presentation


• Moduli of stable sheaves on Enriques surfaces

  Kota Yoshioka

  School and Workshop on Gauge Theories and Differential Invariants ICTP Italy, Oct. 2019, English

  [Invited]

  Invited oral presentation


• Weak Brill-Noether for the moduli of stable sheaves on a generic K3 surface

  Kota Yoshioka

  fourth Japanese-European Symposium on Symplectic Varieties and Moduli Spaces, Oct. 2019, English

  [Invited]

  Invited oral presentation


• Weak Brill-Noether for the moduli of stable sheaves on a generic K3 surface

  Kota YOSHIOKA

  Algebraic Geometry and Moduli Theory, Mar. 2019, English, 京都大学数理解析研究所, International conference

  [Invited]

  Invited oral presentation


• Moduli of stable sheaves on Enriques surfaces

  Kota Yoshioka

  Japanese-Europian Symposium on Symplectic Varieties and Moduli Space, English, Levico-Terme (Italy), Italy

  [Invited]

  Invited oral presentation


• Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces

  Kota Yoshioka

  Derived category and birational geometry, Feb. 2017, 大阪大学, Japan

  [Invited]

  Invited oral presentation


• Vector bundles on Enriques surfaces

  Kota Yoshioka

  代数的層のモジュライの研究とその周辺, Feb. 2017, 京都大学数理解析研究所, Japan

  [Invited]


• Moduli spaces of stable sheaves on Enriques surfaces

  Kota Yoshioka

  Derived categories and Chow groups of hyperkaehler and Calabi-Yau varieties, Sep. 2016, English, Simons center for geometry and physics, Stony Brook, New York (U.S.A.), United States, International conference

  [Invited]

  Invited oral presentation


• Stable sheaves on Enriques surfaces

  Kota Yoshioka

  Higher dimensional algebraic geometry and around Kobe -Kyoto, 2016, Feb. 2016, English, Kobe University, Japan

  [Invited]


• Fourier-Mukai duality for K3 surfaces

  Kota Yoshioka

  Vector bundles on algebraic curves 2015, Jun. 2015, English, Warwick University, United Kingdom, International conference

  [Invited]

  Invited oral presentation

• Algebraic Geometry and Integrable Systems -- Moduli theory and Equations of Painleve type

  齋藤 政彦, 山田 泰彦, 岩木 耕平, 望月 拓郎, 吉岡 康太, Rossman W.F, 稲場 道明, 光明 新

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Kobe Gakuin University, 01 Apr. 2022 - 31 Mar. 2027


• 安定層のモジュライ空間の研究

  吉岡 康太

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2023 - 31 Mar. 2026

  代数多様体上の安定層のコホモロジー群は底空間の性質を反映し、また安定層のモジュライ空間の性質を理解する上でも重要な対象である。これまでの研究においてはしばしば曲面のピカール数を1に制限した。それにより代数多様体の変形によらない性質等が得られた。一方標準因子が数値的に自明な曲面の場合、この制限は大きな制約となる。そこでR5年度はピカール数が1でない場合の曲面として、楕円曲線の直積などのアーベル曲面の場合に安定層のモジュライ空間の性質を調べた。とくにnon-isogeneousである楕円曲線の直積について、モジュライ空間の一般元について、そのコホモロジー群がリーマンロッホの定理から期待されるふるまいをすること、すなわちweak Brill-Noether性が成り立つことを確かめることができた。モジュライ空間を調べるのに、ピカール数が2以上の場合、ample錐の形の複雑さが解析を困難にする。そこで安定性の偏極への依存性について、既存の結果等と比較しながら研究を進めた。これはIzzet Coskun (University of Illinois Chicago, USA)とHoward Nuer (Technion, Israel Institute of Technology)との共同研究である。
  楕円曲面上の層に対し、相対的フーリエ向井変換と相性が良い安定性を導入できるが、この安定性がある種の条件のもと相対的フーリエ向井変換で保たれることを確かめた。


• Symplectic algebraic geometry and moduli spaces

  並河 良典, 望月 拓郎, 吉川 謙一, 尾高 悠志, 吉岡 康太, 森脇 淳, 疋田 辰之, 松下 大介

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Kyoto University, 05 Apr. 2021 - 31 Mar. 2026


• 連接層と複体のモジュライの研究

  吉岡 康太

  科学研究費補助金／基盤研究(B), Apr. 2018 - Mar. 2023, Principal investigator

  Competitive research funding


• 代数幾何と可積分系の融合 - 理論の深化と数学・数理物理学における新展開 -

  齋藤 政彦

  科学研究費補助金／基盤研究(S), Apr. 2017 - Mar. 2022

  Competitive research funding


• 代数幾何と可積分系の融合 - 種々のモジュライ空間と数学・数理物理学の新展開 -

  齋藤 政彦

  科学研究費補助金／基盤研究(A), Apr. 2017 - Mar. 2022

  Competitive research funding


• 代数多様体上の複体とそのモジュライの研究

  吉岡 康太

  科学研究費補助金／基盤研究(B), Apr. 2014 - Mar. 2018, Principal investigator

  Competitive research funding


• Geometry of instanton moduli spaces and representation theory

  Nakajima Hiraku, YOSHIOKA Kota

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyoto University, 01 Apr. 2011 - 31 Mar. 2016

  A duality between gauge theories and W-algebras was found by physicists Alday-Gaiotto-Tachikawa. We understand it as a mathematically rigorous statement that the equivariant cohomology group of a framed moduli space of instantons has a structure of a representation of a W-algebra, and various cohomology classes are described as vertex operators of the W-algebra. We study several results in this direction. In particular, jointly with Braverman-Finkelberg, we prove that the equivariant intersection cohomology group of a framed moduli space of instantons (more precisely its Uhlenbeck partial compactification) has an expected structure of a representation.


• 代数幾何と可積分系の融合と進化

  齋藤 政彦

  科学研究費補助金／基盤研究(S), Jun. 2012 - Mar. 2016

  Competitive research funding


• モジュライ理論による代数幾何と可積分系の新たな展開

  齋藤 政彦

  科学研究費補助金／基盤研究(A), Apr. 2012 - Mar. 2016

  Competitive research funding


• Constructions and developments of geomerty on moduli spaces and arithmetic varieties

  MORIWAKI Atsushi, MUKAI Shigeru, NAKAJIMA Hiraku, NAMIKAWA Yoshinori, YOSHIKAWA Kenichi, MOCHIZUKI Takuro, YOSIOKA Kota, KAWAGUCHI Shu, FUJINO Osamu, ABE Takeshi, INABA Michiaki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Kyoto University, 01 Apr. 2010 - 31 Mar. 2015

  Our six teams made international and surprising progresses of constructions and developments on geometries on moduli spaces and arithmetic varieties. For example, Prof. Mochizuki, who is a member of this research project, was elected as a plenary speaker of Soul International congress of Mathematicians 2014. Moreover, we supported the intercity seminars between Paris, Barcelona and Kyoto by this project. We could produce a lot of significant international researches such as joint works of Moriwaki and H. Chen at Fourier Institute, Grenoble University.


• Geometry of quiver varieties and representation theory

  NAKAJIMA Hiraku, ISHII Akira, YOSHIOKA Kota

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyoto University, 2007 - 2010

  Consider the blowup of a complex algebraic surface at a point. Together with Yoshioka, I introduced an abelian category in the derived category of coherent sheaves, called the category of perverse coherent sheaves, and study its moduli spaces. As an application, further with Gottsche, I proved Witten's conjecture equating Donaldson invariants and Seiberg-Witten invariants for complex surfaces.


• 複体を用いたベクトル束の研究

  吉岡 康太

  科学研究費補助金／基盤研究(B), 2010, Principal investigator

  Competitive research funding


• Research on lattices, automorphic forms and moduli spaces

  KONDO Shigeyuki, SHOJI Toshiaki, KANNO Hiroaki, ITO Yukari, MUKAI Shigeru, FUJINO Osamu, SHIMADA Ichiro, OGUISO Keiji, YOSHIOKA Kota

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Nagoya University, 2006 - 2009

  The purpose of the research is to study symmetries of algebraic varieties defined as the set of zeros of several polynomials and to describe the moduli of them. To do this, we employ the theory of lattices (a generalization of integral points on the plane) and invariant functions (automorphic functions, forms), which is a key idea of this research. We have obtained several results on symmetries and moduli of interesting algebraic varieties called K3 surfaces and related varieties.


• Fano varieties and moduli spaces with emphasis on the Verlinde Formula and the 14^ problem of Hilbert

  MUKAI Shigeru, MORI Shigefumi, NAKAYAMA Noboru, ABE Takeshi, NAKAMURA Iku, KURANO Kazuhiko, YOSHIOKA Kota, TAKEUCHI Kiyohiko, TAKAGI Hiromichi, IDE Manabu, MORI Shigefumi, NAKAYAMA Noboru, ABE Takeshi, NAKAMURA Iku, KURANO Kazuhiko, YOSHIOKA Kota, TAKEUCHI Kiyohiko, TAKAGI Hiromichi, IDE Manabu

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyoto University, 2005 - 2008

  代数多様体の中には、曲線、K3曲面とファノ多様体という3つの良い族がある.これらは単独でも興味深いが、互いにモジュライという関係で繋がっていることを観察することによって、より深い理解に到達できると思う.今回の研究課題では、フェアリンデ型公式との関係や不変式環への応用から研究を始めて、エンリケス曲面の位数2のある種の自己同型や偏極K3曲面のモジュライの単有理性問題への応用を研究した.また、Mumfordのpathologyとして有名な現象をよく理解するために、3次元多様体内の曲線の変形に対する障害類が消えないための充分条件についても研究した.


• The global geometry of moduli spaoes

  NAKAMURA Iku, YOSHIOKA Kota, KONDO Shigeyuki, WENG Lin, KATO Fumiharu, MATSUMOTO Keiji

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), Hokkaido University, 2004 - 2007

  Nakamura proved a new theorem on McKay correspondence of a simple singularity C2/G, G being a finite group of SL(2,C). The theorem shows certain natural modules V and Vdagger have simple structures. Among others, the structure of Vdagger explains completely the known bijetive correspondence of the extended Dynkin diagram and all the irreducible representations of G.. The paper is now in print. He also proved an important vanishing theorem for degenerate quasi-abelian varieties. Weng is constructing a new important theory of geometric class field theory, which is modeled after Mumford's stability of GIT, Seshadri-Narashiman's theory of unitary vector bundles. In this program he defined nonabelian L-functions, and in some cases he proved a theorem analogous to Riemann hypothesis. Kato studied with Fujiwara the fundamental theory of p-adic geometry and rigid geometry. He proved also that Mumford fake projective plane and the other known fake projective planes are among Shimura varieties. Kondo constructed a uniformization by a 5-dimensional complex ball of the moduli of ordered 8 points of the projective line, by using Borcherds modular forms. Matsumoto gave a very precise description of a link and its complement in S3 by using real theta functions. Yoshioka proved a formula of counting the number of instantons on certain complex surfaces with Nakajima.


• 代数幾何と可積分系の融合と新しい展開

  齋藤 政彦

  科学研究費補助金／基盤研究(S), 2007

  Competitive research funding


• Quiver varieties, moduli spaces and representation theory

  NAKAJIMA Hiraku, ISHII Akira, YOSHIOKA Kota

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kyoto University, 2005 - 2006

  Together with Kota Yoshioka and Lothar Gottsche I have studied relation between Donaldson invariants and Nekrasov's partition function. Further more, Takuro Mochizuki have joined our group to continue the research. Donaldson invariants are defined as integration of natural cohomology classes over moduli spaces of instantons on 4-manifolds. When the underlying 4-manifold has b_+ =1, the invariants depends on the choice of a Riemannian metric. The wall-crossing formula gives the difference of Donaldson invariants with respect to two Riemannian metrics. We express the wall-crossing formua in terms of Nekrasov's partition function, when the rank of vector bundles is 2. This was proved via a study of the torus action on the moduli space when the underlying 4-manifold is a toric surface. This research is an expansion of one started in 2004, we give the proof that the same formula holds for arbitrary projective surfaces, not necessarily toric surfaces. We further study the K-theoretic version of instanton counting. We study the theta function associated with the Seiberg-Witten curve which is a mirror of K-theoretic instanton couting and show that the Seiberg-Witten curve can be reconstructed from the coordinate ring of moduli spaces. This result is the K-theoretic generalization of the Nekrasov's conjecture. We also prove the wall-crossing formula similar to above, but we do not understand how to define the invariants using the instanton moduli spaces, due to singularieties. So we restrict our attention to the case of projective surfaces, and define invariants as holomorphic Euler characteristic of natural line bundles over the algebra-geometric compactification of the moduli spaces. With helps of Takuro Mochizuki, we study higher rank (>2) cases, and we prove a recursive expression of the wall-crossing formula, and proved that it is again given by the Nekrasov's partition function.


• 安定層のモジュライ空間の研究

  吉岡 康太

  科学研究費補助金／基盤研究(B), 2006, Principal investigator

  Competitive research funding


• 超幾何・パンルヴェ系の総合的研究

  野海 正俊

  科学研究費補助金／基盤研究(A), 2005

  Competitive research funding


• 対称性が豊富な多様体上の連接層の研究

  吉岡 康太

  科学研究費補助金／若手研究(B), 2005, Principal investigator

  Competitive research funding


• モジュライ空間と可積分系の新しい展開

  齋藤 政彦

  科学研究費補助金／基盤研究(B), 2005

  Competitive research funding


• The study of vector bundles on an algebraic manifold with the trivial canonical bundle

  YOSHIOKA Kota, YAMADA Yasuhiko, SAITO Masahiko

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), KOBE UNIVERSITY, 2000 - 2002

  Yoshioka find the condition for the existence of stable sheaves on K3 and abelian surfaces and showed the connectedness of the moduli spaces. If the moduli space is compact, then he determined the albanese map and showed that the fiber is a hyperkaehler manifold. Moreover he showed that the deformation type of the manifold is determined by the dimension. By this result, the study of the topological type of the moduli space is reduced to the rank 1 case. In order to get these results, he used the Fourier-Mukai transform and the deformation of the underlying K3 surfaces. Hence he also studied the relation between the stability and the Fourier-Mukai transform. Moreover he studied the moduli of stable sheaves on an elliptic surface, surface components of the moduli of stable sheaves on a K3 surface and the Gromov-Witten invariants and got some results. Saito studied Gopakumar-Vafa conjecture on BPS states. He proposed a mathematical definition of the BPS invariants by using the moduli of purely 1-dimensional sheaves and checked the consistency for some cases. Yamada studied D-brane on a rational elliptic surface. Mathematical beautiful structures such as monodromy group, Mordell-Weil group, affine Weyl group are studied from the Physical point of view. Saito and yamada studied the Painleve equation in terms of the symmetry and the geometry. In particular, Saito showed that the Backlund transform is the flop in the birational geometry and the Painleve equation is derived from the deformation theory of a rational surface, which has an application of the classification of the Riccati solution.


• Geometry on String Theory and Moduli spaces

  SAITO Masa-hiko, HOSONO Shinobu, YAMADA Yasuhiko, NOUMI Masatoshi, YOSHIOKA Kota, MUKAI Shigeru

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 2000 - 2002

  During the period of the project, we have investigated the following subjects and obtained the following results. (i) Gromov-Witten invariants and BPS invariants for Calabi-Yau manifolds-Gopakumar-Vafa conjecture, (ii) Homological Mirror Symmetry and Geometry of derived categories, (iii) Algebraic Geometry of spaces of initial conditions of Painleve equations, (iv) Lie theoretic studies for Painleve equations and its generalizations, (v) Symmetries of the moduli spaces of vector bundles, (vi) New development of invariant theory. As for (i), we proposed a mathematical definition of BPS invariants for Calabi-Yau 3-fold and verified their compatibility for the known calculation of Gromov-Witten invariants assuming Gopakumar-Vafa conjecture. In (iii), we introduce the notion of Okamoto-Painleve pairs and proved that one can derive Painleve equations through deformation theory of Okamoto-Painleve pairs. As for the studies of Painleve equations (iv), our group have been developing Lie theoretic approach and algebro-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.


• Research on Periods of Algebraic Varieties and Hypergeometric Functions

  SAITO Masa-hiko, TAKANO Kyoichi, SASAKI Takeshi, NOUMI Masatoshi, YOSHIOKA Kota, TAKAYAMA Nobuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Kobe University, 1997 - 1999

  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.


• ベクトル束のモジュライ空間の研究

  吉岡 康太

  日本学術振興会, 科学研究費助成事業, 奨励研究(A), 神戸大学, 1997 - 1998

  最近楕円有理曲面上のベクトル束のモジュライ空間のオイラー数が数理物理の進展により計算された。 これはN=4位相的ヤンミルズ理論における双対性と新たに提出された正則アノマリーに関する漸化式予想にモジュライ空間の基本的性質を組合わせて得られた。 私は数理物理を使って計算されたオイラー数を数学的手法により階数が2の場合に計算し、その値が完全に一致することを確かめた。この事は双対性の存在の根拠を与え、また正則アノマリーに関する漸化式予想に対する根拠を与える。N=4位相的ヤンミルズ理論における双対性を確認するのは一般に大変難しく、知られている例は射影平面とK3曲面の場合だけであった。私の計算は新たな例を与えると同時に、K3曲面の場合より複雑で、しかもきれいな関数で記述できた点でも興味深いと思われる。さて私の利用した方法はwa11 crossing公式を適用するというもので、高い階数の場合には今のところ通用しないと思っている。楕円ファイバー構造を利用した計算方法を確立し、他の階数の場合も扱えるようにするのは今後の課題である。


• ベクトル束のモジュライの研究

  吉岡 康太

  日本学術振興会, 科学研究費助成事業, 奨励研究(A), 広島大学, 1996 - 1996

  最近の研究で楕円ファイバー構造を持つ曲面上のベクトル束のモジュライ空間はよい性質を持ち、またかなり有用である事が分かってきた。例えばアーベル曲面上の階数2のベルトル束がなすモジュライ空間の場合、アーベル曲面を楕円曲線の直積に変形することにより、ベッチ数を計算できることがわかっている。今年度は階数が一般の場合にモジュライ空間の既約性や双有理的構造を調べた。特にそのことからデルペソ曲面上の単純なトーションフリー層のモジュライ空間の既約性を示した。既約性は半安定層のモジュライ空間の局所分解性や普遍族の存在性とも関係があり、意味のある事と思われる。また第一チャーン類に関するある条件のもと、モジュライ空間の有理性を示すことができた。この事実を示すことが、楕円ファイバー構造を持つ場合を扱ったひとつの動機となっている。更にアーベル曲面上のモジュライ空間の場合には第一チャーン類に関するある条件のもとピカ-ル群やアルバネ-ゼ写像を決定できた。この結果自体は線織面の場合の結果から予想できるものであるが、逆に線織面の場合の定式化が一般性をもつことを示唆しており意味のある事である。さて今度はアーベル曲面上のベクトル束をさらに詳しく解析するつもりである。アーベル曲面やK3曲面の場合はフーリエ-向井変換という重要な操作があるので、それとの関連も調べたい。なお階数を一般にするのは、フーリエ-向井変換では階数はそんなに意味があるものではなく、向井ベクトルの長さのみが本質的な量だからである。


• 代数群の表現論の代数解析学的研究

  谷崎 俊之, 竹内 潔, 吉岡 康太, 都築 暢夫, 菅野 孝史, 隅廣 秀康

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 広島大学, 1996 - 1996

  主として,筆者の定義したゲルファント超幾何方程式の拡張と,それに関連して生ずる表現論の問題について考察し,結果を得た. 以前の研究で,この方程式がホロノミー系になるためのひとつの十分条件は得られていたが,これについて更にリー群論的立場からの考察を行い,巾零軌道との関係を明らかにした. また,解の積分表示の存在について更に研究を進めた.どのような場合に解の積分表示がある種のラドン変換で与えられるかを定め,その場合のラドン変換の一般的性質について考察した.柏原正樹との共同研究で扱ったラドン変換は,射影直線に関するものであったが,今回は半単純リー群のある種の一般型放物部分群に関するものを対象とした.特に,ラドン変換の像がどのような方程式を満たすかという問題をD加群論の立場から研究した.ラドン変換の像が常にある種の方程式系(ヴァーマ加群と関連して具体的に書ける)をみたすことは証明できた.予想は,ラドン変換がもとの関係空間をその方程式系を満たす関数の全体に1対1にうつすということだが,全射性は有る意味で(D加群論的定式化のもとで)一般に言えた.単射性は,現在のところ一般線形群の場合のみが言えている.なお,この結果は実リー群の場合の大島利雄・関口英子の結果の複素ヴァージョンである.ある種のコホモロジー群の消滅が言えば,我々の結果から大島・関口の結果は従うのであるが,それは今後の問題である. その他,我々の超幾何方程式を定義する際の前提になっている,ヴァーマ加群の最大真部分加群についての有る事実の量子群版についても研究を行った.これは,今後我々の方程式を差分化する際に必要になるであろう基本的結果である.


• VECTOR BUNDLES ON MANIFOLDS

  SUMIHIRO Hideyasu, YOSHIOKA Kouta, FURUSHIMA Mikio, MATSUMOTO Keishi, SUGANO Takashi, TANISAKI Toshiyuki

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), HIROSHIMA UNIVERSITY, 1996 - 1996

  In this project, we studied vector bundles on manifolds from the following various points of view : 1)algebraic geometric method, 2)algebraic analytic method, 3)number theoretic method. In 1), we obtained (1) a necessary and sufficient condition for a rank 2 bundle on projective space P^n (n*4) to split into line bundles which gives us a new approach to important conjectures concerning splitting of vector bundles on P^n, (2) we classified Non-projective Moishezon compactifications (X,Y) of affine C^3 space by clarifing numerically effectiveness of the boundary divisor Y and moreover, showed an equivalence between the Stein ess of a C^1- fiber space over a non-Stein manifold S and the triviality of certain rank 2 bundle on S., (3) we proved under some conditions on the first Chern class, the rationality of the moduli of bundles on surfaces with elliptic curves as fibers and have determined the Picard group and Albanese map of the moduli of bundles on elliptic surfaces. In 2), we studied (1) heighest weight modules of semi-simple Lie algebras, especially those corresponding to compact Hermitian symmetric spaces, (2) we have introduced the definition of Radon transformation on Flag manifolds of general type and founed usefulness of bundles to study the Radon transformation, (3) we got a duality concerning generalized hypergeometric functions by using the intersection theory on twisted cohomology group and the exterior products. In 3), (1) we gave an expression by integrals in terms of their Fourier coefficients of the L-functions which are liftings of cusp forms with haif integer weights to the modular forms on orthogonal groups and proved the meromorphic continuation and the functional equation under some technical conditions which can be viewed as a generalization of Kohnen-Skoruppa's result on quadratic Siegel cusp forms, (2) we showed an categorical equivalence between the category of p-adic Galoi representations over local fields with positive characteristic and the category of etale differential modules with Frobenius map and in particular, the ones whose representation of inertial group factors through finite representations correspond to overconvergent modules, (3) investigated sheaficatin of p-adic Hodge theory and their relativeness, which are analogue to Riemann-Hilbert correspondence in the case of complex manifolds.