Page 385 -

360 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

be of type (1, 0) forms, and hence dh i, j = dh(s i , s j ) = ω i,k h k, j + ω j,k h i,k
k k
decomposes into forms of type (1, 0) and (0, 1) as follows

d h i, j = ω i,k h k, j and d h i, j = ω j,k h i,k .
k k
t

In matrix notation these relations reduce simply to d h = ωh and d h = h ω. Then

ω = (d h)h −1 is the unique solution to both equations, and since ω is well deter-
mined by compatibility under change of frame, it is globally defined. This proves the
theorem.
(4.9) Remark. If the local frame of a Hermitian vector bundle E in the previous
theorem is unitary, that is, if h(s i , s j ) = δ i, j ,thenwehave0 = dh(s i , s j ) = ω i, j +
ω j,i , and thus the matrix associated with a unitary frame is skew-Hermitian. In terms
of covariant differentiation parallel transport takes the following form.

(4.10) Parallel Transport. Let M be a smooth manifold with a connection ∇ on a
vector bundle q : E → M. A vector field v along a curve c :[a, b] → M is a lifting
v :[a, b] → E with qv = c. Let c :[a, b] → T (M) denote the tangent vector

lifting to c. The vector field v is a parallel transport (with respect to ∇) provided
∇ c (t) v = 0 for all t ∈ [a, b].

In local coordinates, this is a first order differential equation, and, as such, it has a
unique solution for given initial data. This initial data is a vector v a ∈ E c(a) , and the
solution is a vector v(t) ∈ E c(t) depending smoothly on t ∈ [a, b].
The parallel transport defined by these curves c is the linear transformation T c :
E c(a) → E c(b) assigning to a vector w ∈ E c(a) firstly the solution v(t) to the parallel
transport equation with v(a) = w and then the value v(b) = T c (v(a)) = T c (w) ∈
E c(b) .
We must remark that the theory of connections and vector fields was developed
only globally, and here we have used it for tangent vectors along a curve. The point-
wise properties are left to the reader to work out. For the product of two parame-
terized curves c.d,thatis, c followed by d, we have also the transitivity property
T c.d = T d T c : E (c.d)(a) → E (c.d)(b) so that the product of curves gives composition
of parallel transport. Also, the parallel transport of the inverse path is the inverse
linear map.

(4.11) HolonomyGroups. Let M be a smooth manifold with a connection ∇ on
a vector bundle q : E → M. The holonomy group for (E, ∇) at x ∈ M is the
subgroup of all parallel transport elements T c ∈ GL(E x ) for loops c at x. From the
composition of parallel transport property we know that T c is an automorphism and
the set of these elements is closed under multiplication of linear automorphisms of
E x . This is a closed subgroup of the linear group.

(4.12) Complex HolonomyGroups. Let X be a complex analytic manifold with
a holomorphic connection ∇ on a complex analytic vector bundle q : E → X