Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\C^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

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This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaf 71). Issued in print and online. Available via ProQuest Digital Dissertations.

In this bachelor’s thesis an introduction to the fundamentals of algebraic geometry is given. Some concepts in algebraic geometry are introduced such as Spec of a ring and Proj of a graded ring and several results related to these are either proven or stated. Special focus is directed towards defining the so called ”Hilbert scheme” which is the main topic in a lot of modern algebraic geometry research.

This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.

Le premier chapitre montre par des méthodes toriques ($G-$graphes) que pour tout entier positif $n$, le quotient de l'espace affine à $n$ dimensions par le groupe cyclique $G_n$ d'ordre $2^n-1$ admet le $G_n$-schema de Hilbert comme résolution lisse crepante. Le deuxième chapitre contient des résultats sur les champs algébriques (construction du champ algébrique lisse associé à une log-paire). Le troisième chapitre montre l'équivalence entre la catégorie dérivée bornée des faisceaux cohérents $G_n-$équivariants sur l'espace affine et celle des faisceaux cohérents sur la résolution $G_n-$Hilb. Chapitre 4 donne une réalisation géométrique de la conjecture de Broué via la correspondance de McKay. L'annexe contient des résultats sur les groupes trihédraux, y compris un programme magma.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 44).

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T01:41:17Z (GMT). No. of bitstreams: 1 Santos_PatriciaBorgesdos_M.pdf: 505981 bytes, checksum: 9ca0829d764a16d8fca0b3e0ebcd0e9b (MD5) Previous issue date: 2010 / Resumo: Exibiremos uma bijeção entre o esquema de Hilbert de c pontos no espaço afim de dimensão n e uma variedade algébrica quase projetiva dada pela variedade das n matrizes c por c que comutam duas a duas e satisfazem uma condição de estabilidade módulo conjugação. Em seguida, estudamos o caso n = 2 mais cuidadosamente, mostrando que o esquema de Hilbert de c pontos é uma variedade quase projetiva, não-singular e de dimensão 2c. / Abstract: We exhibit a bijection between the Hilbert scheme of points in the n-dimensional affine space and a quasi-projective algebraic variety given by c x c matrices commuting two by two and satisfying a stability condition modulo conjugation. Next, we study the n = 2 case more closely, showing that the Hilbert scheme of points is a non-singular, irreducible, quasi-projective variety of dimension 2c. / Mestrado / Algebra / Mestre em Matemática

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