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.

• [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 Pⁿ 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 Pⁿ that will be used throughout the course.
  • Chapter 7 relates line counting in Pⁿ to Grassmannians of 2-planes in Cⁿ⁺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 P². This is a smooth compactification of the space of smooth conics, with more structure than the obvious compactification P⁹.
  • 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 P⁴ 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)/I₀(T) should be I_i(t)/I₀(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 Pⁿ.
• [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 P² 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 P² with two or three singular points via the Local Excess Intersection Approach.
  • Section 4 derives a recursive formula for counts of rational curves in P².
• [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 P² and P³
  • 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.