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Normal functions of product varieties Lewis, James Dominic
Abstract
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 2ml, (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 nonzero global sections of this vector bundle. The main idea is to compare the quasicanonical 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.
Item Metadata
Title 
Normal functions of product varieties

Creator  
Publisher 
University of British Columbia

Date Issued 
1981

Description 
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 2ml, (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 nonzero global sections of this vector bundle. The main idea is to compare the quasicanonical 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.

Genre  
Type  
Language 
eng

Date Available 
20100330

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080157

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.