We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of ...intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.
In this document we consider an exact sequence of group varieties e→N→G→Q→e over an algebraically closed field. We show that for l≠char(k) a prime there exists an isomorphism of graded Ql-algebras ...Hét∗(G,Ql)≅Hét∗(N,Ql)⊗QlHét∗(Q,Ql) that is compatible with pullback homomorphisms φ∗ of endomorphisms φ:G→G that stabilize N.
It is known that projective minimal models satisfy the celebrated Miyaoka-Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective and non-uniruled ...varieties.
We give an algebraic-geometric proof of the fact that for a smooth fibration $\pi: X \longrightarrow Y$ of projective varieties, the direct image $\pi_*(L\otimes K_{X/Y})$ of the adjoint line bundle ...of an ample (respectively, nef and $\pi$-strongly big) line bundle $L$ is ample (respectively, nef and big).
Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most ...recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.
We prove the existence and uniqueness of Kähler–Einstein metrics on
-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study ...analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.
A
bstract
The search for a theory of the S-Matrix over the past five decades has revealed surprising geometric structures underlying scattering amplitudes ranging from the string worldsheet to the ...amplituhedron, but these are all geometries in auxiliary spaces as opposed to the kinematical space where amplitudes actually live. Motivated by recent advances providing a reformulation of the amplituhedron and planar
N
= 4 SYM amplitudes directly in kinematic space, we propose a novel geometric understanding of amplitudes in more general theories. The key idea is to think of amplitudes not as functions, but rather as differential forms on kinematic space. We explore the resulting picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint
ϕ
3
scalar theory, we establish a direct connection between its “scattering form” and a classic polytope — the associahedron — known to mathematicians since the 1960’s. We find an associahedron living naturally in kinematic space, and the tree level amplitude is simply the “canonical form” associated with this “positive geometry”. Fundamental physical properties such as locality and unitarity, as well as novel “soft” limits, are fully determined by the combinatorial geometry of this polytope. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between the interior of this old “worldsheet associahedron” and the new “kinematic associahedron”, providing a geometric interpretation and simple conceptual derivation of the bi-adjoint CHY formula. We also find “scattering forms” on kinematic space for Yang-Mills theory and the Non-linear Sigma Model, which are dual to the fully color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact—“Color is Kinematics”— whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, all our scattering forms are well-defined on the projectivized kinematic space, a property which can be seen to provide a geometric origin for color-kinematics duality.
Abstract
This paper is the first in a series in which we offer a new framework for hermitian
$${\text {K}}$$
K
-theory in the realm of stable
$$\infty $$
∞
-categories. Our perspective yields ...solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie’s notion of a Poincaré
$$\infty $$
∞
-category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki’s algebraic Thom construction. For derived
$$\infty $$
∞
-categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on
$$\infty $$
∞
-categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré
$$\infty $$
∞
-categories, showing in particular that they form a bicomplete, closed symmetric monoidal
$$\infty $$
∞
-category. We also study the process of tensoring and cotensoring a Poincaré
$$\infty $$
∞
-category over a finite simplicial complex, a construction featuring prominently in the definition of the
$${\text {L}}$$
L
- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré
$$\infty $$
∞
-category using generators and relations. We extract its basic properties, relating it in particular to the 0th
$${\text {L}}$$
L
- and algebraic
$${\text {K}}$$
K
-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.
This monograph is concerned with counting rational points of bounded height on projective algebraic varieties. This is a relatively young topic, whose exploration has already uncovered a rich seam of ...mathematics situated at the interface of analytic number theory and Diophantine geometry. The goal of the book is to give a systematic account of the field with an emphasis on the role played by analytic number theory in its development. Among the themes discussed in detail are the Manin conjecture for del Pezzo surfaces, Heath-Brown's dimension growth conjecture, and the Hardy-Littlewood circle method. Readers of this monograph will be rapidly brought into contact with a spectrum of problems and conjectures that are central to this fertile subject area.
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