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國立成功大學 應用數學研究所 碩士論文 格拉斯曼流形、五次三維流形以及相交理論 Grassmannians, quintic threefolds, and intersection theory 研 究 生：宋政蒲 指導教授：章源慶 中 華 民 國 一 ○ 二 年 六 月 摘要 在這篇論文裡，我們將會探討一些格拉斯曼流形與五次三維流形的 相交理論。我們也將介紹格羅莫夫-威騰不變量，以及利用大量子積的結 合性，推導康采維奇的著名公式：計算出通過(3d-1)個一般點的 d 次平面 有理曲線的數目。 關鍵字：格拉斯曼流形、五次三維流形、相交理論、穩定映射、 格羅莫夫-威騰不變量、康采維奇公式。 ABSTRACT . In this thesis, we will investigate the intersection theory of some Grassmannians and the quintic threefolds. We will also introduce the notion of Gromov-Witten invariants, and derive Kontsevich's celebrated formula for the number of plane rational curves of degree d passing through 3d 1 general points from the associativity of the big quantum product. − Keywords: Grassmannian, quintic threefold, intersection theory, stable map, Gromov-Witten invariant, Kontsevich's formula. 誌謝 首先要感謝家人，老爸不斷督促我，老媽也持續地給我鼓勵，讓我度過不少難 關，兩個哥哥雖然因不認同我的想法和做法，而給了我很多壓力，但我仍很感謝他們 讓我很有家的感覺，且因他們讓家裡有穩定的收入，我才能無後顧之憂地做研究。 還有興大保育社、成大野鳥社、墾丁解說志工認識的朋友，讓我在這三年可以到 處遊歷賞鳥，讓我在研究和學習遇到瓶頸時，能夠藉著大自然來抒發壓力和尋找靈感， 讓我的碩士生涯不是苦悶的三年，而是最充實的三年。特別感謝傻剛、婉瑄和國權， 陪我到處賞鳥到處遊玩，還常常討論許多我們關心的時事，也給了我很多觀點。 接下來是在應數所上認識的夥伴，416 的大家、學長們和學弟，常常舉辦賽局理 論研討會，讓我在學習和研究之間得以喘息。而他們也常常給我很多鼓勵和意見。特 別感謝孫維良，總是可以從他那邊聽到很多數學上有趣的事情，也是少數會和我談論 數學的人，同時在學術上，他也是我的楷模。另外還有黛玉，在我趕著論文，神經緊 繃而沮喪的時候，用簡短的一句話化解了我的緊張。以及呂庭維，在口試當天也幫我 跑東跑西的，還幫我煮了一壺讓夏杼教授讚譽有加的咖啡。 再來要感謝夏杼老師和陳若淳老師，同時也是我的口試委員。在口試前總是用玩 笑的口吻，跟我交談，減輕了我口試前的緊張。而在口試結束後也給予我許多鼓勵和 肯定，讓我能下更多的決心走向研究路線。 最後便是我的指導教授－章源慶老師，很感謝他在我一開始想請他當指導教授時， 沒多說甚麼就收了我當學生，也時常給予我指導，而不論我問甚麼樣的問題，總是會 告訴我線索或者相關書籍，讓我不至於沒有方向。在研究上也沒有給我太大的壓力， 且常常會鼓勵我或者是在我因瓶頸而沮喪時安慰我。非常感謝老師這兩年多的照顧！ 最後，要謝謝的人太多了，只能謝天又謝地了。 CONTENTS 1. The Grassmannians G 2; n 1 ::::::::::::::::::::::::::::::::::: 7 ( + ) 2. Quintic Threefold :::::::::::::::::::::::::::::::::::::::::::: 12 3. Lines On the Quintic Threefold ::::::::::::::::::::::::::::::::::: 13 4. Rational Curves of Degree d 2 On the Quintic Threefold ::::::::::::::::::: 16 ≥ 5. Gromov-Witten Invariant ::::::::::::::::::::::::::::::::::::::: 24 6. Kontsevich's Formula for Genus Zero Curves on P2 ::::::::::::::::::::::: 27 Bibliography ::::::::::::::::::::::::::::::::::::::::::::::::: 31 1. The Grassmannians G(2; n + 1) 7 1. THE GRASSMANNIANS G 2; n 1 ( + ) We would like to study the enumerative geometry of G 2; n 1 , the set of 2-dimensional subspaces Cn+1 Pn of , which is also the set of lines of the n dimensional( projection+ ) space . A typical chart of G 2; n 1 is ( + ) 1 0 a11 a1 n 1 ( − ) : 0 1 a a ⎛ 21 ⋯ 2(n−1) ⎞ ⎜ ⎟ So the dimension of G 2; n 1 is 2 ⎝n 1 . ⋯ ⎠ ( + ) ( − ) Cn+1 Now we fix a basis e1; : : : ; en+1 for , and let Vi be the linear space of e1; : : : ; ei . Then we have a collection{ of subspaces} { } −−⇀ Cn+1 0 V0 V1 Vn+1 ; which is called a flag V . { } = ⊂ ⊂ ⋯ ⊂ = Definition 1. For a flag V , we define the Schubert cycle σa;b n 1 a b 0 by P ( P− ≥ ≥ ≥ ) σa;b V ` G 2; n 1 ` Vn−a ; ` Vn+1−b : ( ) = { ∈ ( + ) T ∩ ( ) ≠ ∅ ⊂ ( )} When b is zero, we simply write σa;0 as σa. A chart of σa;b is given by 1 a11 a12 a1 n a 1 0 0 ( − − ) ; 0 1 a a 0 0 ⎛ 21 ⋯ 1(n+⋯1−b−2) ⋯ ⋯ ⎞ ⎜ ⎟ so the dimension of σ⎝a;b is n a 1⋯n ⋯1 b ⋯2 2n a b 2. ⋯ ⎠ ( − − ) + ( + − − ) = − − − 1. The Grassmannians G(2; n + 1) 8 The σa;b V is the closure of cellular decomposition , has codimension 2 n 1 2n a b 2 a b, 2 a b and σa;b V( ) H ( + ) G 2; n 1 . ( − )−( − − − ) = + The[ total( )] cohomology∈ ( (H∗ G+ 2)); n 1 is spanned by σa;b V n 1 a b 0 . Furthermore, if V ′ is another( ( flag,+ there)) exists a deforming{[ ( map)]TV− V≤ ′,≤ we≤ have} ′ → σa;b V σa;b V σa;b: [ ( )] = [ ( )] = Now we are ready to calculate σa;b σc;d. G(2;n+1) Because the dimension of G 2; nS 1 is 2 n ⋅ 1 , σa;b σc;d is not zero unless G(2;n+1) ( + ) ( − ) ⋅ c d 2 n 1 S a b : 1 ;+ if= c ( n− 1) −b( ; d+ )n 1 a: Theorem 1. σa;b σc;d G(2;n+1) ⎧ 0 ; otherwise: ⎪ = − − = − − S ⋅ = ⎨ Proof. ⎪ ⎩⎪ The number of σa;b σc;d is equal to G(2;n+1) S ⋅ ′ σa;b V σc;d V ; G(2;n+1) ′ S [ ( ) ∩ ′ ( )] where Vi intersect Vj transversely. (i.e. if i j n 2;Vi Vj .) Note that + < + ∩ = ∅ P P σa;b V ` G 2; n 1 ` Vn−a ; ` Vn+1−b ; σ (V ′) = {`∈ G(2; n+ 1) T `∩ P(V ′ ) ≠ ∅ ; ` ⊂ P(V ′ )}: c;d n−c n+1−d For a line L σ V ( σ) = {V ′∈ , L( must+ contain) T ∩ ( points) ≠p∅ P ⊂V ( and )}q P V ′ . a;b c;d n−a n−c Because L is in P V , the point q contained in P V ′ has to be in P V ∈ ( n)+∩1−b ( ) n−∈c ( ) ∈ n(+1−b ) Therefore, q P V ′ P V . Similarly, p P V P V ′ . (n−c ) n+1−b n(−a ) n+1−d ( ) Then P V ′ P V and P V P V ′ are not empty n−c∈ ( n)+1∩−b ( ) n−a n+1∈−d ( ) ∩ ( ) ( ) ∩ ( ) ( ) ∩ ( ) P V ′ P Vn 1 b n c n 1 b n 2 n 1 b c n−c + − : P V P V ′ n a n 1 d n 2 n 1 a d ( n−a) ∩ ( n+1−d) ≠ ∅ ⇒ − + + − ≥ + − − ≥ ⇒ ⇒ However, c d(must) ∩ equal( to 2 )n≠ ∅1 ⇒ a −b .+ Therefore,+ − ≥c +n 1 b and− −d ≥n 1 a. + ( − ) − ( + ) = − − = − − 1. The Grassmannians G(2; n + 1) 9 So we know that σa;b ; σc;d 0 unless c is equal to n 1 b and d is equal to n 1 a. How many lines are⟨ in σa;b ⟩V= σc;d V ′ ? The question− is− easy, there is a unique− line− containing both p and q. ( ) ∩ ( ) ∗ The dual basis to σa;b , the basis of H G 2; n 1 , is σn−b−1;n−a−1 . For a cohomology g H2d G 2; n 1 , g a σ , if we want to find a , we can calculate { } ( ( + ))i d;d−i{ } i d 0≤i≤ 2 [ ] ∈ ( ( + )) [ ] = Q g σn−1−(d−i);n−1−d. G(2;n+1) STheorem[ ] 2.⋅ Pieri's formula for G 2; n 1 , σa σb;c( + ) σd;e : {d+e=a+b+c;n−1≥d≥b≥e≥c} ⋅ = Q Proof. Because the codimension of σa σb;c is a b c, then σa σb;c can be written as ⋅ +m+d;e σd;e : ⋅ d+e=a+b+c Furthermore, Q md;e σa σb;c ; σn−1−d;n−1−b σa σb;c σn−1−e;n−1−d G(2;n+1) = ⟨ ⋅ ⟩ = ( ⋅ ) ⋅ ′ ∗ S σa V σb;c V σn−1−e;n−1−d V ; G(2;n+1) ′ ∗ = [ ( ) ∩ ( ) ∩ ( )] where Vi ;Vj ;Vk intersect transversely. S Suppose md;e is not zero, σ V ′ ` G 2; n 1 ` P V ′ ; ` P V ′ b;c n−b n+1−c σ ( ) =V{∗ ∈ (` G+2;) nT ∩1 (` P)V≠∗∅ ⊂ ; `( P V)}∗ n−1−e;n−1−d 1+e 2+d ` P V ′ ,L P V ∗ T n−b (2+d ) = { ∈ ( + ) ∩ ( ) ≠ ∅ ⊂ ( )} P V ′ P V ∗ n b 2 d n 2 d b (1). ∵ ∩ n(−b ) ≠ ∅2+d ⊂ ( ) ` P V ∗ ,L P V ′ ⇒ ( 1)+e∩ ( ) ≠ ∅ ⇒n+1−c− + + ≥ + ⇒ ≥ P V ∗ P V ′ 1 e n 1 c n 2 e c (2). ∵ ∩ 1(+e ) ≠ ∅n+1−⊂c ( ) Now we consider σ V , if line L is also contained to σ V , L must contain point s in P V . ⇒ ( ) ∩ ( a ) ≠ ∅ ⇒ + + + − ≥ + ⇒a ≥ n−a Here P V and P V ′ P V ∗ intersect transversely. n−a n(+1−)c 2+d ( ) ( ) If σ V σ V ′ σ V ∗ is nonempty set, then there exists a line L, L P V a( ) b;c ( n−1)−e;n∩ −1(−d ) n−a and L P V ′ P V ∗ . ( ) ∩n+1−c( ) ∩ 2+d ( ) ∩ ( ) ≠ ∅ ⊂ ( ) ∩ ( ) 1. The Grassmannians G(2; n + 1) 10 Therefore, the intersection P V and P V ′ P V ∗ , the dimension is n−a n+1−c 2+d n( 1 )c 2( d )n∩ 1( d) c 2; is nonempty. ( + − ) + ( + ) − ( + ) = − + So, d c 2 n a is greater than n 1. d c a 0 d c d e b c 0 b e (3). In conclusion,− + + from− (1)(2)(3) we can+ see⇒ that−b; c;− d;≥ e are⇒ related− − ( by+ − − ) ≥ ⇒ ≥ n 1 d b e c 0 : Thus we get a necessary condition of m−d;e ≥0,≥ now≥ we≥ calculate≥ md;e. ≠ Since P V ′ and P V ∗ intersect transversely, we can choose a local coordinate x ; x ; : : : ; x n+1−c 2+d 1 2 n+1 such that the points of P V ′ are represented by ( ) ( n+1)−c ( ) ( ) x1; x2;:::;:::;:::;xn+1−c; 0;:::; 0 and the points of P V ∗ are represented( by ) 2+d ( ) 0;:::; 0; yn−d;:::;:::;:::;yn; yn+1 : The points of P V ′ P (V ∗ can be represented by ) n+1−c 2+d ( ) ∩ ( ) 0;:::; 0; zn−d; : : : ; zn+1−c; 0;:::; 0 : A line L in σ V ′ σ ( V ∗ must contain a point p in) P V ′ and a point q in P V ∗ . b;c n−1−e;n−1−d n−b 1+e Therefore, the( line)∩L can be constructed( ) by ( ) ( ) a11 a1 n b 0 0 ::: 0 0 ( − ) : 0 0 ::: 0 0 a a ⎛ ⋯ 2(n+1−e) ⋯ 2(n+1) ⎞ ⎜ ⎟ But L is in P V ′ P V ∗ which is locally represented by n+1−c ⎝ 2+⋯d ⋯ ⎠ ( ) ∩ ( ) 0;:::; 0; zn−d; : : : ; zn+1−c; 0;:::; 0 : So the line L can be constructed( by ) 0 0 zn−d zn−b 0 0 0 0 0 0 0 0 z z 0 0 ⎛ ⋯ ⋯ ⋯ n⋯+1−e ⋯n ⋯+1−c ⋯ ⎞ ⎜ ⎟ ⎝ n⋯ d 1⋯zeros ⋯ ⋯ ⋯b e zeros ⋯ c⋯zeros⎠ ↑ ( − − ) ↑ ( − ) ↑ 1. The Grassmannians G(2; n + 1) 11 Look at the above matrix, the space it constructs has n d 1 b e c n a 1 zeros, and intersects P V only in one dimension n−a ( − − ) + ( − ) + = − − That means the point s contained in P V and the line L is unique.
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  • Arxiv:1703.06955V1 [Math.AG]

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    THE GENUS-ONE GLOBAL MIRROR THEOREM FOR THE QUINTIC THREEFOLD SHUAI GUO AND DUSTIN ROSS Abstract. We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau con- jecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit sym- plectomorphism determined by genus-zero invariants. This gives the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the “genus zero controls higher genus” principle, in the sense of Givental’s quantization formalism, for non-semisimple cohomological field theories. 1. Introduction Over the last twenty-five years, there have been a number of important developments that have advanced our understanding of Gromov-Witten theory. Among these results, the genus-zero mir- ror theorems have provided closed formulas for the genus-zero Gromov-Witten potentials of a large number of target geometries [Giv98, LLY97, Ber00, CFK14], and Teleman’s classification theorem for semisimple cohomological field theories [Tel12] has led to explicit formulas for all- genus partition functions in terms of Givental’s quantization formula [Giv01a, Giv01b]. One of the most important remaining open problems is to understand the all-genus partition functions of non-semisimple cohomological field theories, for which the Gromov-Witten theory of the quintic 5 5 P4 threefold X := V (W = x0 + + x4) is the prototypical example. The Landau-Ginzburg/Calabi-Yau· · · correspondence,⊆ which first arose in the study of string theory [GVW89, VW89, Mar89], suggests an equivalence between the Gromov-Witten theory of a Calabi- Yau hypersurface and the Landau-Ginzburg model of the defining equation of the hypersurface.
  • Rational Curves on Calabi-Yau Threefolds

    Rational Curves on Calabi-Yau Threefolds

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server RATIONAL CURVES ON CALABI-YAU THREEFOLDS SHELDON KATZ In the conformal field theory arising from the compactification of strings on a Calabi-Yau threefold X, there naturally arise fields corre- spond to harmonic forms of types (2,1) and (1,1) on X [4]. The un- corrected Yukawa couplings in H2;1 and H1;1 are cubic forms that can be constructed by techniques of algebraic geometry — [20] contains a nice survey of this in a general context in a language written for mathe- maticians. The cubic form on the space Hp;1 of harmonic (p; 1) forms is given by the intersection product (!i;!j;!k) RX !i !j !k for p =1, while for p = 2 there is a natural formulation7→ in terms∧ of∧ infinitesimal variation of Hodge structure [20, 6]. The Yukawa couplings on H2;1 are exact, while those on H1;1 receive instanton corrections. In this con- text, the instantons are non-constant holomorphic maps f : CP1 X. The image of such a map is a rational curve on X,whichmayormay→ not be smooth. If the rational curve C does not move inside X,then the contribution of the instantons which are generically 1-1 (i.e. bi- rational) maps with image C can be written down explicitly — this contributions only depends on the topological type of C,ormoreor less equivalently, the integrals RC f ∗J for (1files, a,1) forms J on X. If the conformal field theory could also be expressed in terms of a “mirror manifold” X0, then the uncorrected Yukawa couplings on 2;1 H (X0) would be the same as the corrected Yukawa couplings on H1;1(X).
  • Open Strings and Extended Mirror Symmetry

    Open Strings and Extended Mirror Symmetry

    arXiv:yymm.nnnn Open Strings and Extended Mirror Symmetry Johannes Walcher School of Natural Sciences, Institute for Advanced Study Princeton, New Jersey, USA Abstract The classical mirror theorems relate the Gromov-Witten theory of a Calabi- Yau manifold at genus 0 to the variation of Hodge structure of an associated mirror manifold. I review recent progress in extending these closed string re- sults to the open string sector. Specifically, the open Gromov-Witten theory of a particular Lagrangian submanifold of the quintic hypersurface is related to the Abel-Jacobi map for a particular object in the derived category of coherent sheaves of the mirror quintic. I explain the relevance for the homo- logical mirror symmetry program. Contribution to Proceedings of BIRS Workshop “Modular Forms and String Duality,” June 3–8, 2006 September 2007 2 1 INTRODUCTION Contents 1 Introduction 2 2 A-model 6 2.1 OpenGromov-Witteninvariants. 7 2.2 Schubertcomputationindegree1 . 9 2.3 Result ............................... 11 2.4 Proofbylocalization . 12 3 B-model 14 3.1 VariationofHodgestructure . 14 3.2 Normalfunctions ......................... 16 3.3 Extended Picard-Fuchs equation . 17 4 Mirror Symmetry 19 4.1 Matrixfactorizations . 19 4.2 A homological mirror symmetry conjecture . 21 4.3 Assembly ............................. 22 4.4 FloerHomologyoftherealquintic . 24 References 26 1 Introduction Mirror symmetry is one of the best-developed examples of the refreshed in- teraction between theoretical physics and certain areas of pure mathematics that has taken place over the past decades. From the string theory point of view, it is natural to expect that there exists a far reaching generalization of classical concepts of geometry that will incorporate the idea that the funda- mental building blocks are extended objects with some degree of non-locality.

  • Gromov-Witten Invariants of Hypersurfaces

    Gromov-Witten invariants of hypersurfaces Dem Fachbereich Mathematik der Technischen Universitat¨ Kaiserslautern zur Erlangung der venia legendi vorgelegte Habilitationsschrift von Andreas Gathmann geboren am 9.4.1970 in Hannover 2003 Contents Preface 1 Chapter 1. Gromov-Witten invariants of projective spaces 9 1.1. Stable curves and stable maps 12 1.2. Gromov-Witten invariants 18 1.3. Basic relations among Gromov-Witten invariants 21 1.4. The Virasoro relations 28 1.5. Proof of the Virasoro algorithm 32 1.6. Numerical examples and applications 39 Chapter 2. Relative Gromov-Witten invariants in genus zero 45 2.1. Moduli spaces of stable relative maps 47 2.2. Raising the multiplicities 56 N 2.3. Proof of the main theorem for hyperplanes in P 59 2.4. Proof of the main theorem for very ample hypersurfaces 64 2.5. Enumerative applications 69 Chapter 3. The mirror theorem 81 3.1. The mirror transformation 85 3.2. Examples 94 3.3. Proof of the main technical lemmas 97 Chapter 4. The number of plane conics 5-fold tangent to a given curve 101 4.1. The Gromov-Witten approach 103 4.2. The components of M¯ (2;2;2;2;2) 108 4.3. Computation of the virtual fundamental classes 111 4.4. Application to rational curves on K3 surfaces 117 Chapter 5. Relative Gromov-Witten invariants in higher genus 121 5.1. Stable relative and non-rigid maps 123 5.2. The virtual push-forward theorem 132 5.3. Push-forwards in codimension 0 145 5.4. Push-forwards in codimension 1 155 5.5.
  • Arxiv:Alg-Geom/9606016V1 25 Jun 1996

    Arxiv:Alg-Geom/9606016V1 25 Jun 1996

    RECENT DEVELOPMENTS IN TORIC GEOMETRY David A. Cox Abstract. This paper will survey some recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic geometry, combinatorics and mirror symmetry. Introduction A toric variety over C is a n-dimensional normal variety X containing (C∗)n as a Zariski open set in such a way that the natural action of (C∗)n on itself extends to an action of (C∗)n on X. This seemingly simple definition leads to a fascinating combinatorial structure and some surprisingly rich mathematics. In this article, we will discuss some recent developments in toric varieties, including novel applications and new foundations for the entire theory. Toric varieties were discovered in the early 1970’s independently by several groups of people. From the beginning, the theory of toric varieties has led to some notable applications, including the following: • The characterization of algebraic subgroups of maximal rank of Cremona groups, in Demazure’s 1970 paper [Dem]. • The stable reduction theorem, in the 1973 book [KKMS] by Knudsen, Kempf, Mumford and Saint-Donat. • The construction of nice (meaning “toroidal”) compactifications of discrete quotients of bounded symmetric domains, begun in the 1973 paper [Sat] by Satake and the 1975 book [AMRT] by Ash, Mumford, Rapaport and Tai. • The rich connections between Newton polytopes, toric varieties and singu- larities, first explored by Kushnirenko [Kus] in 1976 and Khovanskii [Kho] in 1977. • The use of Hard Lefschetz for simplicial toric varieties to prove McMullen’s arXiv:alg-geom/9606016v1 25 Jun 1996 conjectures for the number of vertices, edges, faces, etc.