Vector Bundles in Algebraic Geometry

One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kahler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differential-geometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space. Thus the partitioning of the classification problem according to topological type provides a coarse, and more calculable, alternative to the partition by connected components. Furthermore, the topology of a variety strongly influences its geometric properties. A sociological observation is that the quest for understanding the topology of algebraic varieties has led to a rich set of techniques which found applications elsewhere, even in physics. Perhaps a word about the choice of ground field is appropriate. One could also look at the shapes of the real points of real algebraic varieties. However, by Weierstrass approximation, pretty much anything can arise if you let the degree get big enough. Thus the classical question in this case is “which shapes can occur for a given degree?” This is so much more difficult

An exceptional bundle is basically a bundle whose analytical or algebraic structure is uniquely determined by its topological one. This phenomenon was discovered in the mid eighties while studying vector bundles over complex algebraic surfaces or real 4-manifolds. Since then there were several attempts to investigate exceptional vector bundles and to use them in vector bundle theory and in the classification of 4-manifolds. Main idea of this article is to combine results and conjectures for bundles on a Del Pezzo surface. The author feels strongly that the context of Del Pezzo surfaces has several advantages and understanding here could develop really rapidly. Our aim is to influence such a development.

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

Matrices of linear forms of constant rank from vector bundles on projective spaces

According to Horrocks (Quart. J. Math. Oxford (2) 17 (1966), 14–18), a vector bundle E on the projective n-space extends stably to the projective N-space, N > n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N > n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coandă and Trautmann (Commun. Algebra 34 (2006), 2485–2488). Combining this result with a theorem of Mohan Kumar et al. (Manuscripta Math. 112 (2003), 183–189), we get a new effective version of the Babylonian tower theorem for vector bundles on projective spaces.

These notes record, in a slightly expanded way, the lectures given by the first two authors at the College on Moduli Spaces of Vector Bundles that took place at CIMAT in Guanajuato, Mexico, from November 27th to December 8th, 2006. The college, together with the ensuing conference on the same topic, was held in occasion of Peter Newstead's 65th anniversary. It has been a great pleasure and a privilege to contribute to celebrate Peter's outstanding achievements in algebraic geometry and his lifelong dedication to the progress of mathematical knowledge. We warmly thank the organizers of the college and conference for inviting us, thus allowing us to participate in Peter's celebration.

Part 1 Algebraic geometry: the Hasse-Witt matrix of an algebraic curve rational points of algebraic curves over functional fields correspondences, motive and monoidal transforms new directions in geometry arrangements of hyperplanes, higher braid groups and higher Bruhat orders. Part 2 Modular forms and diophantine equations: the p-torsion of elliptic curves is uniformly bounded parabolic points and zeta-functions of modular curves periods of p-adic Schottky groups rational points of bounded height of Fano varieties points of bounded height on del Pezzo surfaces. Part 3 Differential equations and mathematical physics: long wave equation with free boundary - 1 conservation laws and solutions long wave equations with a free surface - 2 Hamiltonian structure and higher equations a supersymmetric extension of the Kadomtsev-Petviashvili hierarchy the Momford form and the Polyakov measure in string theory Gromov-Witten classes, quantum cohomology and enumerative geometry and other papers.

Hodge theory is pivotal in studying algebraic varieties' intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi-Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory Zachary Alexander Maddock This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.

Theory of computing has given rise to some fundamental mathematical problems, notably the P ≠ NP conjecture, and the related lower bound problems concerning formula or circuit size. We develop an approach to these problems through geometric invariant theory. The goal of this approach is to reduce the hard nonexistence problems under consideration to tractable existence problems.Accordingly, we reduce the arithmetic (characteristic 0) version of the P ≠ NP conjecture, and other related lower bound problems to proving existence of obstructions. These are representations in the homogeneous coordinate rings of orbit-closures in geometric invariant theory [MFK], of a class of points which are partially stable and whose stabilizers have special representation-theoretic properties. However, the Luna-Vust complexity [LV] of these orbit closures is quite high, in contrast with the well-understood homogeneous or almost-homogeneous-spaces, such as G/P [LLM], toric varieties [F3], and spherical embed-dings [BLV], whose Luna-Vust complexity is zero.We take a step towards explicit construction of obstructions by proving two results regarding these orbit closures. The first is a generalization of the Borel-Weil theorem for G/P to these orbit-closures. Second, we conjecture a nice representation-theoretic set of generators for their ideals, and prove a weaker version of the conjecture. Such a set of generators had earlier been given for the ideal of G/P by Lakshmibai, Seshadri, Littelmann [LS, Li3, LLM] and Kostant (cf. [PK]).Finally, using these results, we reduce, in essence, the arithmetic non-existence problems under consideration to fundmental existence and construction problems in representation theory and algebraic geometry that are conjectured to be in the complexity class P. 2000 Mathematics Subject Classification68Q1514L2414L3520G0514M17Key words and phrasesComputational ComplexityGeometric Invariant TheoryRepresentation TheoryClassical GroupsPlethysmHomogeneous Spaces

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ( H 1 {H^1} in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.

This dissertation discusses Fano vector bundles on projective space and the quantum cohomology of the associated projective bundle. A bundle is said to be Fano if the associated projective bundle is a Fano variety. We prove a theorem that reconstructs the small quantum cohomology ring for the projective bundle from a subset of the quantum multiplication-table. Equivalently, in the spirit of Kontsevich’s first reconstruction theorem, the theorem reconstructs all genus-0 Gromov-Witten invariants from a small set of 2-point invariants. The theorem improves on Kontsevich’s reconstruction theorem, where they both apply, as the required subset of invariants is smaller. Qin and Ruan note that, unlike in the classical case, the quantum cohomology of a projectivised bundle need not be a module over the quantum cohomology of the base. The reconstruction theorem described in this thesis demonstrates that this ‘non-modularity’ contains all the essential quantum information. As an example we discuss the Fano bundle Ω(2)P4 , the second wedge of the cotangent bundle on P4, and calculate the quantum cohomology of the Fano 9-fold given by its projectivisation using the reconstruction theorem. We apply Givental formalism to this result to obtain the quantum periods for some Fano subvarieties embedded in P(Ω(2)P4). These novel calculations are of interest to the Fanosearch programme. In particular, we describe two 4-dimensional Fano subvarieties that have quantum periods suggestive of 4-folds which are not of toric complete-intersection type; they are not contained in a conjecturally complete list of such objects. We observe the presence of the Apery numbers in J-function of the Fano 9-fold P(Ω(2)P4) and comment, after Golyshev, that this may indicate hidden modularity in the J-function.

An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center by H. Baumgartel and F. Lledo Modular invariants and subfactors by J. Bockenhauer and D. E. Evans On the PCT-theorem in the theory of local observables by H. J. Borchers and J. Yngvason Transplantation of local nets and geometric modular action on Robertson-Walker space-times by D. Buchholz, J. Mund, and S. J. Summers Classification of subsystems, local symmetry generators and intrinsic definition of local observables by S. Carpi and R. Conti From local perturbation theory to Hopf- and Lie-algebras of Feynman graphs by A. Connes and D. Kreimer The flat tube theorem for vector valued functions by C. D'Antoni and L. Zsido Point interactions by G. Dell'Antonio Perturbative algebraic field theory, and deformation quantization by M. Dutsch and K. Fredenhagen Sum rules for the free energy in the mean field spin glass model by F. Guerra Fractals in noncommutative geometry by D. Guido and T. Isola What I woud like to understand by R. Haag The Rohlin property for automorphisms of $C^*$-algebras by M. Izumi Environment induced localization and superselection rules in a gas of pyramidal molecules by G. Jana-Lasinio, C. Presilla, and C. Toninelli Connes-Moscovici-Kreimer Hopf algebras by D. Kastler The structure of the automorhpism group of an approximately finite dimensional factor by Y. Katayama and M. Takesaki Braiding and extensions of endomrophisms of subfactors by Y. Kawahigashi Bicategories of operator algebras and Poisson manifolds by N. P. Landsman Notes for a quantum index theorem introduction by R. Longo Conformal field theory and Doplicher-Roberts reconstruction by M. Muger On the distance between MASA's in type II$_1$ factors by S. Popa Recent results concerning E$_o$-semigroups of $\mathfrak{B}(\mathfrak{H})$ by R. T. Powers Locality and modular invariance in 2D conformal QFT by K.-H. Rehren Tensor products of Banach spaces and the Stone-Weierstrass problem of $C^*$-algebras by S. Sakai Perron-Frobenius theory for positive maps on trace ideals by R. Schrader Space- and time-like superselection rules in conformal quantum field theory by B. Schroer Finite quantum groupoids and inclusions of finite type by K. Szlachanyi On generalizations of the spectrum condition by R. Verch Algebraic orbifold conformal field theories by F. Xu.

(II) fedN(X) = ^ e ± i ^ i J k is a tempered distributional function of t. The classical trace formula says that the singular support of (II) is contained in the length spectrum of X. Moreover, under suitable hypotheses on geodesic flow, the trace formula gives considerable information about the singularities in (II). (See [DG and C].) There is a fairly straightforward (and not terribly interesting) generalization of the trace formula to vector bundles. (See, for instance, the introduction to [DG].) We will be concerned in this article with a much more subtle generalization inspired by recent articles of Hogreve, Potthoff, and Schrader [HPS], and Schrader and Taylor [ST] in Communications in Mathematical Physics. Let G be a compact Lie group and ir: P —> X a principle G-bundle with connection. Given a finite-dimensional unitary representation, p, of G we will denote by Ep the vector bundle over X associated with p and by Dp the associated connection. Now consider a ladder {pe, e = 1,2,...} of irreducible representations of G. (This means that the maximal weight of pe is e times the maximal weight of p\.) For given e let Afc,e? # = 1, 2, O, . . . be the spectrum of the Laplace operator on C(Epe): Ae = D*peDpe + e .