Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität, Bonn, 1986. / Includes bibliographical references (p. 139-142).

Inaug.-Diss.--Rheinische Friedrich-Wilhelms-Universität, 1992. / Includes bibliographical references (p. 91-92).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2005. / Includes bibliographical references: p. 116-119.

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

Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this dissertation is to define a class of hypersurfaces that exhibits such classically unexpected properties, and to offer a perspective with which to conceptualize such phenomena. Specifically, this dissertation proposes an analogy between the eponymous q-bic hypersurfaces—special hypersurfaces of degree q+1, with q any power of the ground field characteristic, a familiar example given by the corresponding Fermat hypersurface—and low degree hypersurfaces, especially quadrics and cubics. This analogy is substantiated by concrete results such as: q-bic hypersurfaces are moduli spaces of isotropic vectors for a bilinear form; the Fano schemes of linear spaces contained in a smooth q-bic hypersurface are smooth, irreducible, and carry structures similar to orthogonal Grassmannian; and the intermediate Jacobian of a q-bic threefold is purely inseparably isogenous to the Albanese variety of its smooth Fano surface of lines.

Xiao, Chang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 37-[38]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Algebraic geometry in positive characteristic has a quite different flavour than in characteristic zero. Many of the pathologies disappear when a variety admits a lift to characteristic zero. It is known since the sixties that such a lift does not always exist. However, for applications it is sometimes enough to lift a variety dominating the given variety, and it is natural to ask when this is possible. The main result of this dissertation is the construction of a smooth projective variety over any algebraically closed field of positive characteristic that cannot be dominated by another smooth projective variety admitting a lift to characteristic zero. We also discuss some cases in which a dominating liftable variety does exist.

We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K1, K2/F of non-Archimedean local fields F . Our formula works for all cases, K1 and K2 can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover Gross and Keating’s result on lifting endomorphism of formal modules.

No description available.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 113-119). Available also in a digital version from Dissertation Abstracts.