The work of this thesis is to motivate the following:
Statement: The Hodge conjecture holds for products of varieties
Z = XxC where (i) X is smooth, projective of dimension 2m-l, (ii) C is a smooth curve.
The basic setting of this thesis is depicted by the following
where (i) k⁻¹ (t) = Zt = Xt xC, {Xt } a
Lefschetz pencil of hyperplane sections of X (ii) £ is the singular set of k, i.e.,
k = k
is smooth and proper.
Corresponding to this diagram are the extended Hodge bundle
U H (Z . C) with integrable connection V , and the family of tt?1 t
intermediate Jacobians. U JCZ ) with corresponding normal functions
Now V induces an operator (also denoted by V)
on the normal functions, and those normal functions v satisfying the differential equation Vv = 0 are labeled horizontal, which includes those normal functions arising from the primitive algebraic cocycles
in H²m (Z). Now the known generalization of Lefschetz's techniques
state that every primitive integral class of type (m,m) in H²m (Z)
comes from a horizontal normal function in some natural way, so that
what's needed to prove the above statement is some way of converting a
normal function to an algebraic cocycle. We motivate this statement
by proving some results about the group of normal functions, in particular
our main result:
Theorem: The group of normal functions are horizontal.
To prove this theorem, we exhibit Vv as a global section of some holomorphic vector bundle over p¹, and then show that there
are no non-zero global sections of this vector bundle. The main idea is to compare the quasi-canonical extensions of certain holomorphic vector bundles with integrable connection with those extensions arising from algebra (hypercohomology), by calculating certain periods of growth. Once this comparison is made precise, we apply a vanishing theorem statement about the global sections of the algebraic extensions to our geometric extensions, thus concluding the proof of the theorem. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/23075 |
| Date | January 1981 |
| Creators | Lewis, James Dominic |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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