Math 277 - Section 3 - Topics in Differential Geometry - Fall 2009
D. Auroux -
Tue. & Thu., 9:30-11am, Room 3 Evans
"Why can't I see my reflection in the mirror on a television?"
-- Anonymous, on Yahoo! Answers
Homework:
Lecture notes:
- Thu Aug 27: the origins of mirror symmetry; overview of
the course (PDF)
- Tue Sep 1: a first statement of mirror symmetry;
complex structures, deformations of complex structures (PDF)
- Thu Sep 3: deformations of complex structures continued;
Kähler Hodge theory (PDF)
- Tue Sep 8: the Kodaira-Spencer map for Calabi-Yau manifolds;
J-holomorphic curves: basic definitions (PDF)
- Thu Sep 10: J-holomorphic curves: transversality; Gromov compactness
(PDF)
- Tue Sep 15: Gromov-Witten invariants; perturbations and
virtual fundamental class; the case of Calabi-Yau 3-folds
(PDF)
- Thu Sep 17: quantum cohomology and Yukawa coupling on
H^{1,1}; Kähler moduli space
(PDF)
- Tue Sep 22: the quintic 3-fold and its mirror
(PDF)
- Thu Sep 24: degenerations, monodromy; example: elliptic
curves
(PDF)
- Tue Sep 29: degenerations: weight filtration, large
complex structure limit, canonical coordinates
(PDF)
- Thu Oct 1: statement of mirror symmetry; the mirror quintic
and its holomorphic volume form
(PDF)
- Tue Oct 6: the mirror quintic: periods and Picard-Fuchs equation
(PDF)
- Thu Oct 8: the mirror quintic: canonical coordinates and
(2,1) Yukawa coupling
(PDF)
- Tue Oct 13: mirror symmetry for the quintic;
introduction to homological mirror symmetry
(PDF)
- Thu Oct 15: Lagrangian Floer homology: holomorphic strips,
Maslov index
(PDF)
- Tue Oct 20: Lagrangian Floer homology: compactness, d^2=0,
Hamiltonian isotopy invariance
(PDF)
- Thu Oct 22: Lagrangian Floer homology: grading, relation to
Morse homology, Oh spectral sequence
(PDF)
- Tue Oct 27: Lagrangian Floer homology: products, higher
products, Fukaya category
(PDF)
- Thu Oct 29: Fukaya categories continued: twisted
coefficients; CF(L,L); m_{0} and obstructions
(PDF)
- Tue Nov 10: coherent sheaves; derived functors, Ext groups
(PDF)
- Thu Nov 12: derived categories, mapping cones, exact
triangles (PDF)
- Tue Nov 17: Ext and Hom in the derived category;
twisted complexes and the derived category of an A∞ category
(PDF)
- Thu Nov 19: no class (Fukaya's talk at MSRI)
- Tue Nov 24: Lagrangian connected sums and mapping cones
in the Fukaya category; split-closure; statement of HMS for the
elliptic curve; theta functions
(PDF)
- Tue Dec 1: Homological mirror symmetry for the
elliptic curve
(PDF)
- Thu Dec 3: Introduction to the Strominger-Yau-Zaslow
conjecture, special Lagrangian fibrations and affine structures
(PDF)
Course outline
This course will focus on various aspects of mirror symmetry. It is
aimed at students who already have some basic knowledge in symplectic
and/or complex geometry (Math 242 helpful but not required).
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,
periods; pseudoholomorphic curves, Gromov-Witten invariants,
quantum cohomology; large complex structure limits;
mirror symmetry at the level of periods 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);
challenges: instanton corrections, ...
4. Beyond the Calabi-Yau case: Landau-Ginzburg models and
mirror symmetry for Fanos (if time permits)
Matrix factorizations; admissible Lagrangians; examples
(CP^{1}, CP^{2});
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:
- 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. A. Cox, S. Katz, Mirror symmetry and algebraic geometry,
Mathematical Surveys and Monographs 68, AMS, 1999.
- 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