令X 為個帶有凱勒形式(Kähler form ω) 以及全純四形式( holomorphic four- form Ω )的四維緊致卡拉比丘空間(Calabi-Yau manifolds) 。在一些假設條件下，通過研究Donaldson- Thomas方程所決定的模空間，我們定義了四維Donaldson-Thomas不變量。我們也對四維局部卡拉比丘空間(local Calabi-Yau four-folds) 定義了四維Donaldson-Thomas 不變量，並且將之聯繫到三維Fano空間的Donaldson- Thomas 不變量。在一些情況下，我們還研究了DT/GW不變量對應。最后，我們在模空間光滑時計算了一些四維Donaldson- Thomas不變量。 / Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kahler form ω and a holomorphic four-form Ω. Under certain assumptions, we de ne Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over X = KY with the Donaldson- Thomas invariants for the associated compact three-fold Y . In some specialcases, we prove the DT/GW correspondence for X. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth. / Detailed summary in vernacular field only. / Cao, Yalong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 100-105). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- The *4 operator --- p.18 / Chapter 2.1 --- The *4 operator for bundles --- p.18 / Chapter 2.2 --- The *4 operator for general coherent sheaves --- p.20 / Chapter 3 --- Local Kuranishi structure of DT₄ moduli spaces --- p.22 / Chapter 4 --- Compactification of DT₄ moduli spaces --- p.34 / Chapter 4.1 --- Stable bundles compactification of DT₄ moduli spaces --- p.34 / Chapter 4.2 --- Attempted general compactification of DT₄ moduli spaces --- p.36 / Chapter 5 --- Virtual cycle construction --- p.39 / Chapter 5.1 --- Virtual cycle construction for DT₄ moduli spaces --- p.40 / Chapter 5.2 --- Virtual cycle construction for generalized DT₄ moduli spaces --- p.48 / Chapter 6 --- DT4 invariants for compactly supported sheaves on local CY₄ --- p.52 / Chapter 6.1 --- The case of X = KY --- p.52 / Chapter 6.2 --- The case of X = T*S --- p.57 / Chapter 7 --- DT₄ invariants on toric CY₄ via localization --- p.66 / Chapter 8 --- Computational examples --- p.70 / Chapter 8.1 --- DT₄=GW correspondence in some special cases --- p.71 / Chapter 8.1.1 --- The case of Hol(X) = SU(4) --- p.72 / Chapter 8.1.2 --- The case of Hol(X) = Sp(2) --- p.77 / Chapter 8.2 --- Some remarks on cosection localizations for hyper-kähler four-folds --- p.79 / Chapter 8.3 --- Li-Qin's examples --- p.80 / Chapter 8.4 --- Moduli space of ideal sheaves of one point --- p.83 / Chapter 9 --- Appendix --- p.85 / Chapter 9.1 --- Local Kuranishi models of Mc° --- p.85 / Chapter 9.2 --- Some remarks on the orientability of the determinant line bundles on the (generalized) DT₄ moduli spaces --- p.87 / Chapter 9.3 --- Seidel-Thomas twists --- p.90 / Chapter 9.4 --- A quiver representation of Mc --- p.92

This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t₀(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·_X,s on X, and in [25], we construct a natural motive MF_X,s, in a certain quotient ring M^μ_X of the μ-equivariant motivic Grothendieck ring M^μ_X, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K_L,K_M have square roots K^1/2_L, K^1/2_M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·_L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DT^α(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields <b>K</b> of characteristic zero, rather than <b>K = C</b>, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.

This thesis contains two main results. The first is a comparison formula for the Donaldson-Thomas invariants of two (complex, smooth and projective) Calabi-Yau threefolds related by a flop; the second is a proof of the projective case of the Crepant Resolution Conjecture for Donaldson-Thomas invariants, as stated by Bryan, Cadman and Young. Both results rely on Bridgeland’s category of perverse coherent sheaves, which is the heart of a t-structure in the derived category of the given Calabi-Yau variety. The first formula is a consequence of various identities in an appropriate motivic Hall algebra followed by an implementation of the integration morphism (using the technology of Joyce and Song). Our proof of the crepant resolution conjecture is a quick and elegant application of the first formula in the context of the derived McKay correspondence of Bridgeland, King and Reid. The first chapter is introductory and is followed by two chapters of background material. The last two chapters are devoted to the proofs of the main results.

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 x P2.

We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and K-theoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the Donaldson-Thomas theory of threefolds fibered in those surfaces. We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasi-ruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of Eswara-Rao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras. Using Fourier-Mukai transforms and their relative action on families of dg-categories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the Laumon-Rothstein deformation of the Fourier-Mukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the Donaldson-Thomas theory of local curves.

We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed.