18.969 - Topics in Geometry - Spring 2009
D. Auroux - Tuesdays &
Thursdays, 11-12:30 in 2-142.
Lecture notes:
There are two sets of notes: handwritten notes by the lecturer, and
typeset notes taken by Kartik Venkatram. Both are provided as is,
without any guarantee of readability or accuracy.
- Tue Feb 3: the origins of mirror symmetry; overview of
the course
(DA, KV)
- Thu Feb 5: deformations of complex structures
(DA, KV)
- Tue Feb 10: deformations continued, Hodge theory;
pseudoholomorphic curves, transversality
(DA, KV)
- Thu Feb 12: pseudoholomorphic curves, compactness,
Gromov-Witten invariants
(DA, KV)
- Thu Feb 19: quantum cohomology and Yukawa coupling on
H1,1; Kähler moduli space
(DA, KV)
- Tue Feb 24: the quintic 3-fold and its mirror;
complex degenerations and monodromy
(DA, KV)
- Thu Feb 26: monodromy weight filtration, large
complex structure limit, canonical coordinates
(DA, KV)
- Tue Mar 3: canonical coordinates and mirror symmetry;
the holomorphic volume form on the mirror quintic and its periods
(DA, KV)
- Thu Mar 5: Picard-Fuchs equation and canonical coordinates
for the quintic mirror family
(DA, KV)
- Tue Mar 10: Yukawa couplings and numbers of rational curves on the quintic;
introduction to homological mirror symmetry
(DA, KV)
- Thu Mar 12: Lagrangian Floer homology
(DA, KV)
- Tue Mar 17: Lagrangian Floer theory: Hamiltonian isotopy
invariance, grading, examples
(DA, KV)
- Thu Mar 19: Lagrangian Floer theory: product structures,
A_∞ equations
(DA, KV)
- Tue Mar 31: Fukaya categories: first version; Floer homology
twisted by flat bundles; defining CF(L,L)
(DA, KV)
- Thu Apr 2: Defining CF(L,L) continued; discs and obstruction.
Coherent sheaves, examples, introduction to Ext.
(DA, KV)
- Tue Apr 7: Ext groups; motivation for the derived category.
(DA, KV)
- Thu Apr 9: The derived category; exact triangles; Homs and Exts.
(DA, KV)
- Tue Apr 14: Twisted complexes and the derived Fukaya
category; Dehn twists, connected sums and exact triangles.
(DA, KV)
- Thu Apr 16: Homological mirror symmetry: the elliptic curve;
theta functions and Floer products
(DA, KV)
- Thu Apr 23: HMS for the elliptic curve: Massey products;
motivation for the SYZ conjecture
(DA, KV)
- Tue Apr 28: The SYZ conjecture; special Lagrangian
submanifolds and their deformations
(DA, KV)
- Thu Apr 30: The moduli space of special
Lagrangians: affine structures; mirror complex structure and
Kähler form
(DA, KV)
- Tue May 5: SYZ continued; examples: elliptic curves,
K3 surfaces
(DA, KV)
- Thu May 7: SYZ from toric degenerations (K3 case);
Landau-Ginzburg models, superpotentials; example: the mirror of CP^1.
(DA, KV)
- Tue May 12: homological mirror symmetry for CP^1: matrix
factorizations, admissible Lagrangians, etc.
(DA, KV)
Course description
This course will focus on various aspects of mirror symmetry. It is
aimed at students who already have some basic knowledge in symplectic
and complex geometry (18.966, or equivalent).
The geometric concepts needed to formulate various mathematical
versions of mirror symmetry will be introduced along
the way, in variable levels of detail and rigor. The main topics will be
as follows:
1. Hodge structures, quantum cohomology, and mirror symmetry
Calabi-Yau manifolds; deformations of complex structures, Hodge theory
and periods; pseudoholomorphic curves, Gromov-Witten invariants,
quantum cohomology; mirror symmetry at the level of Hodge numbers,
Hodge structures, and quantum cohomology.
2. A brief overview of homological mirror symmetry
Coherent sheaves, derived categories; Lagrangian Floer homology and
Fukaya categories (in a limited setting); homological mirror symmetry
conjecture; example: the elliptic curve.
3. Lagrangian fibrations and the SYZ conjecture
Special Lagrangian submanifolds and their deformations;
Lagrangian fibrations, affine geometry, and tropical geometry;
SYZ conjecture: motivation, statement, examples (torus, K3);
large complex limits; challenges: instanton corrections, ...
4. Beyond the Calabi-Yau case: Landau-Ginzburg models and
mirror symmetry for Fanos
Matrix factorizations; admissible Lagrangians; examples (An
singularities; CP1, CP2);
the superpotential as a Floer
theoretic obstruction; the case of toric varieties.
Bibliography
This very incomplete list tries to provide some of the more accessible references
on the material. There are many other excellent references, but those often
require a higher level of dedication.
Books:
- D. A. Cox, S. Katz, Mirror symmetry and algebraic geometry,
Mathematical Surveys and Monographs 68, AMS, 1999.
- K. Hori et al., Mirror symmetry,
Clay Mathematics Monographs 1, AMS-CMI, 2003.
- M. Gross, D. Huybrechts, D. Joyce, Calabi-Yau manifolds and related
geometries, Lectures from the Summer School held in Nordfjordeid, June
2001, Universitext, Springer, 2003.
- D. McDuff, D. Salamon, J-holomorphic curves and symplectic
topology, AMS Colloquium Publ. 52, AMS, 2004.
Papers:
- R. P. Thomas, The geometry of mirror symmetry,
Encyclopedia of Mathematical Physics, Elsevier, 2006,
pp. 439-448;
arXiv:math.AG/0512412
- A. Polishchuk, E. Zaslow, Categorical mirror symmetry: the elliptic
curve, Adv. Theor. Math. Phys. 2 (1998), 443-470;
arXiv:math.AG/9801119
- R. P. Thomas, Derived categories for the working mathematician,
Winter school on mirror symmetry (Cambridge MA, 1999), AMS/IP Stud. Adv. Math.
23, AMS, 2001, pp. 363-377;
arXiv:math.AG/0001045
- D. Auroux, Mirror symmetry and T-duality in the complement of an
anticanonical divisor,
J. Gökova Geom. Topol. 1 (2007), 51-91;
arXiv:math.SG/0706.3207