Lecture 10 Notes: Picard Fuchs Equation
1. The Quintic (contd.)
Recall that we had a quintic mirror family X with LCSL degeneration as
5
z = (5) 0. We had the PicardFuchs equation for periods of , and found 2
solutions given by
0
(5n)!
(z) =
(n!)
n=0
z
Lecture 1 Notes: Intro to Course
Mirror symmetry comes from statements in supersymmetric string theory. Ba sic idea of string theory:
replace particles with vibrating strings, which propogate through space and form surfaces. We thus get 2d
quantum field
Lecture 5 Notes: Associative Algebra
1. GromovWitten Invariants
Recall that if (X, ) is a symplectic manifold, J an almostcomplex structure, H2(X, Z), Mg,k(X, J, ) is the set of (possibly nodal) Jholomorphic maps to X of genus g representing
class with
Lecture 3 Notes: Hodge Theory
Last time, we say that a deformation of (X, J) is given by
1
0,1
cfw_s
(1)
(X, TX)s + 2[s, s] = 0/Di(X)
1
To first order, these are determined by Def1(X, J) = H (X, TX), but extending these to higher order is
2
obstructed b
Lecture 2 Notes: Deformations
Reference for today: M. Gross, D. Huybrechts, D. Joyce, CalabiYau Mani folds and Related Geometries, Chapter 14.
1. Deformations of Complex Structures
An (almost) complex structure (X, J) splits the complexified tangent and
Lecture 6 Notes: Distinguished Characteristics
The simplest CalabiYaus are hypersurfaces in toric varieties, especially smooth hypersurfaces X in CP
n+1
defined by a polynomial of degree d = n + 2, i.e. a
n+1
section of OPn+1 (d). Smoothness implies that
Lecture 8 Notes: Mirror Symmetry
Last time: 18.06 Linear Algebra.
Today: 18.02 Multivariable Calculus. / 18.04 Complex Variables Thursday: 18.03 Dierential Equations
1. Mirror Symmetry Conjecture
Last time, we said that if we have a large complex structur
Lecture 9 Notes: Preimages
To recall where we were, we had
4
4
X = cfw_(x0 : : x4) P  f =
(1)
with
cfw_
0
Z
4

Z
5
xi 5x0x1x2x3x4 = 0
0
cfw_
a
i
i
5
(2)
G=
(a
,.,a
)
(
)
/5
a
acting by diagonal multiplication xixi
5
Z Z
2i/5
(a, a, a, a,
= 0 /a)
,=e
=
Lecture 7 Notes: Monodromy
X
t
2
Last time, we considered families
D
X (with
=
where for t = 0, X
varying J) and for t = 0, X0 is typically singular. We saw that monodromy around t = 0
n
induces Aut(H (Xt0, Z).
are roots of unity:
Theorem 1. All eigenvalu
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