## MAT 614: Enumerative Geometry
## Stony Brook Fall 2007 |
---|

Here are Notes on Mirror Symmetry (updated 12/12). They are in preparation; only Sections 1 and 3 are currently ready; I may add more in the future. Please let me know if you have any comments or corrections (however minor).

Here is general information about the course (updated 8/28), including a fairly detailed syllabus.

*Name:*
Aleksey Zinger
*E-mail:* azinger@math
*Phone:* 432-8618

*Office:* Math Tower 3-117
*Office Hours:* Wednesday 9-12 (9-10 may be in P-143)

Date |
Topic | Read |

9/5, W | Course Overview; Review of Chern Classes, etc. | [K]: Ch 1,2,4-6 [MS]: Ch 10,11,13,14 [Z1]: Sec 1,2.1,2.2,A.1,A.3,A.4 |

9/10, M | Counting Lines in Projective Spaces Schubert Calculus |
[K]: Ch 7 [GH]: Ch 1, Sec 1 [MS]: Ch 14 |

9/12, W | Counting Lines in Projective Hypersurfaces | |

9/17, M | no class | |

9/19, W | Pseudocycles and Integral Homology | [Z2] |

9/24, M | ||

9/26, W | Counting Low-Degree Curves in P^{2}:
Simple Cases |
[K]: Ch 2 [Z1]: Sec 2, Subs 3.2,3.5.1-3.5.3 |

10/1, M | ||

10/3, W | Counting Low-Degree Curves in P^{n}:
Simple Cases |
[GH]: pp176,177 |

10/8, M | Degenerate Contributions: Overview and Computation in Simple Cases |
[K]: Ch 8 [Z3]: Sec 3 |

10/10, W | ||

10/15, M | Local Excess Intersection Approach: Fairly General Case |
[Z4]: Sec 2 |

10/17, W | ||

10/22, M | Local Excess Intersection Approach: Singular Spaces | [Z4]: Sec 2 |

10/24, W | Counting Rational Plane Quartics | [Z1]: Subs 3.4,3.5.6 |

10/29, M | Recursion for Counts of Rational Curves in P^{2} |
[Z1]: Sec 4 |

10/31, W | Gromov-Witten Invariants: Simple Cases | [K]: Ch 3 [RT]: Sec 1,2,10 |

11/5, M | ||

11/7, W | An Example of Obstruction Bundle in GW Theory | |

11/12, M | GW-Invariants of General Symplectic Manifolds | [FO],[LT] |

11/14, W | Calabi-Yau 3-Folds and GW-Invariants | [P1] |

11/19, M | Mirror Symmetry for Genus 0 GWs of a Quintic On Genus-0 GW-Invariants of Hypersurfaces |
[P2]: Sec 3; [K]: Ch 9 [MirSym]: Sec 26.1,29.1,29.2 |

11/21, W | no class (Friday schedule 9-5)
| |

11/26, M | Equivariant Cohomology | Notes: Sec 1 [AB]: Sec 1-3 |

11/28, W | Proof of Atiyah-Bott Localization Theorem | |

12/3, M | Localization Theorem and Stable Maps | [MirSym]: Sec 27.0-27.5 |

12/5, W | no class | |

12/10, M | Proof of Genus-0 Mirror Symmetry | Notes: Subs 3.1-3.5, [MirSym]: Sec 29.1-29.3,30.3,30.4 |

12/12, W | Notes: Subs 3.6-3.8, [MirSym]: Sec 29.4,30.1,30.2 | |

- [AB]: M. Atiyah and R. Bott,
*The Moment Map and Equivariant Cohomology*- Section 2 is an overview of equivariant cohomology.
- Section 3 contains a beautiful proof of their localization theorem.

- [GH] P. Griffiths and J. Harris,
*Principles of Algebraic Geometry*- Chapter 1, Section 5 consists of a more in-depth discussion of Grassmannians than Chapter 7 in [K].
- pp176,177 describe holomorphic maps to P
^{n}intrisically from the point of view of the domain.

- [FO] K. Fukaya and K. Ono,
*Arnold Conjecture and Gromov–Witten Invariant*

Like [LT], this paper defines GW-invariants for arbitrary symplectic manifolds. Chapter 2 concerns the topology of orbifolds and orbi-bundles. - [K] S. Katz,
*Enumerative Geometry and String Theory*

This book is intended for undergraduates; you will likely find it quite pleasant to read. However, please try to read it thoughtfully. The exercises in the book are fairly easy; please make sure you can do them, but do not hand in your solutions for them (except possibly for the exercises mentioned explcitly below).- Chapters 1,2 give a flavor of classical enumerative geometry
- Chapters 4-6 should be very familiar to you.
Among the less familiar aspects might be the Grassmannian G(2,4)
first introduced in Chapter 4 and
the homology intersection product introduced in Chapter 5.
Chapter 6 should be read very thoroughly as it discusses aspects
of P
^{n}that will be used throughout the course. - Chapter 7 relates line counting in P
^{n}to Grassmannians of 2-planes in C^{n+1}. Exercises 9,10,11 are of particular interest here. - Chapter 8 introduces excess intersection theory from
a more algebro-geometric point of view.
In particular, the statement in the 2nd paragraph on p115 that there may not exist
a section
*s'*linearly independent of*s*refers to holomorphic sections*s'*. Such a smooth section certainly exists, and this is one advantage of the more topological point of view presented in class. Exercise 4 is of a particular interest here;*k*is an example of the kinds of numbers we would like to compute, but in more interesting situations. - Chapter 3 describes in detail the moduli space of stable genus-0 degree-2 maps
to P
^{2}. This is a smooth compactification of the space of smooth conics, with more structure than the obvious compactification P^{9}. - Chapter 9 relates genus 0 Gromov-Witten invariants of a quintic threefold
to the euler class of a vector bundle on the moduli space of maps to P
^{4}and discuss both of the latter in detail.

- [LT] J. Li and G. Tian,
*Algebraic and symplectic geometry of Gromov-Witten invariants*

Like [FO], this paper defines GW-invariants for arbitrary symplectic manifolds. Section 3 describes the topology of the infinite-dimensional spaces involved. - [MirSym], Mirror Symmetry

The primary focus of this 900-page expository book is the mirror symmetry principle, especially for a quintic threefold, from both mathematical and physical perspectives.- Section 26.1 expresses Gromov-Witten invariants in terms of euler classes of vector bundle in some special cases.
- Sections 29.1 and 29.2 contain the statement of the mirror symmetry
prediction for hypersurfaces and an overview of the proof.
In the very last expression on p564, F(
**t**) should in fact be F(**T**), reflecting the mirror transformation. Similarly in equation (29.2), the J_{i}'s are meant to be functions of t, i.e. two lines above I_{i}(**T**)/I_{0}(**T**) should be I_{i}(**t**)/I_{0}(**t**).

- [MS] J. Milnor and J. Stasheff,
*Characteristic Classes*

- Chapters 10,11,13,14 contain most of the general statements concerning the topology of smooth manifolds that are essential for this course. Exercise 14-D is of a particular interest as it expresses the chern classes of tautological vector bundles in terms of Schubert classes.

- [P1] R. Pandharipande,
*Hodge Integrals and Degenerate Contributions*

This paper relates counts of curves in an ideal Calabi-Yau threefold, motivating the Gopakumar-Vafa integrality conjecture. - [P2] R. Pandharipande,
*Three Questions in Gromov-Witten Theory*

Section 3 of these notes for his ICM'02 talk concerns the Gopakumar-Vafa conjecture. - [RT] Y. Ruan and G. Tian,
*A Mathematical Theory of Quantum Cohomology*

- Sections 1 and 2 define Gromov-Witten invariants, with a fixed complex structure on the domain, for semi-positive symplectic manifolds.
- Section 10 proves a recursive formula for counts of rational curves
in P
^{n}.

- [Z1] A. Zinger,
*Counting Plane Rational Curves: Old and New Approaches*- Sections 1,2.1,2.2,A.1,A.3,A.4 contain a few basic statements concerning P^n
- Section 2 and Subsections 3.2,3.5.1-3.5.3 describe
examples of counting curves in P
^{2}with up to one singular point of a specific type. - Subsections 3.3,3.4,3.5.4-3.5.6 contain examples of counting curves
with in P
^{2}with two or three singular points via the Local Excess Intersection Approach. - Section 4 derives a recursive formula for counts of rational curves
in P
^{2}.

- [Z2] A. Zinger,
*Pseudocycles and Integral Homology*- With the exception of Subsection 3.3, this should be fairly easy to read.

- [Z3] A. Zinger,
*Enumeration of Genus-Two Curves with a Fixed Complex Structure in P*^{2}and P^{3}- Section 3 introduces the Local Excess Intersection Approach, in a variery of cases. As this section has no relation to Sections 1 and 2 in the paper, it is sufficient to print out just this section.

- [Z4] A. Zinger,
*Counting Rational Curves of Arbitrary Shape in Projective Spaces*- Section 2 describes the Local Excess Intersection Approach in the general case. As this section is self-contained, it is sufficient to print out just this section.

This page is maintained by Aleksey Zinger.