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When two varieties X, X′ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one point of projection, there are many very different shadows with very different degrees. We give the geometric properties of some of them. These shadows are birational to the variety in which they are a shadow. We compute the minimum degree of all such shadows. For most smooth varieties X⊂Pr, r≥3, it is the integer deg(X)−1.

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structuresAn analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base ringsA construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).

The main aim of this paper is to construct a complex analytic family of symmetric projective K3 surfaces through a compactifiable deformation family of complete quasi-projective varieties from$\\operatorname{CP}^2 \\#9\\overline{\\operatorname{CP}}^2$ . Firstly, for an elliptic curve$C_0$embedded in$\\operatorname{CP}^2$ , let$S \\cong \\operatorname{CP}^2 \\#9\\overline{\\operatorname{CP}}^2$be the blow up of$\\operatorname{CP}^2$at nine points on the image of$C_0$and$C$be the strict transform of the image. Then if the normal bundle satisfies the Diophantine condition, a tubular neighborhood of the elliptic curve$C$can be identified through a toroidal group. Fixing the Diophantine condition, a smooth compactifiable deformation of$S\\backslash C$over a 9-dimensional complex manifold is constructed. Moreover, with an ample line bundle fixed on$S$ , complete Kähler metrics can be constructed on the quasi-projective variety$S\\backslash C$ . So complete Kähler metrics are constructed on each quasi-projective variety fiber of the smooth compactifiable deformation family. Then a complex analytic family of symmetric projective K3 surfaces over a 10-dimensional complex manifold is constructed through the smooth compactifiable deformation family of complete quasi-projective varieties and an analogous deformation family.

For all integers n≥1 and k≥2, the Hadamard product v1★⋯★vk of k elements of Kn+1 (with K being the complex numbers or real numbers) is the element v∈Kn+1 which is the coordinate-wise product of v1,…,vk (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map h:Pn(K)k⤏Pn(K). When K=C, k=2 and Xi(C)⊂Pn(C), i=1,2 are irreducible, we prove four theorems for the case dimX2(C)=1, three of them with X2(C) as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over R. Under mild assumptions, we prove that by knowing X1(R)★⋯★Xk(R), we know X1(C)★⋯★Xk(C). The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen Pn(R), very often provides real ghosts, i.e., images that do not come from a point of X1(R)×⋯×Xk(R). We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen.

We prove that an affine cone admits a surjective morphism from an affine space if and only if is unirational.

We discuss in this note the algebra H^0(X, Sym*TX) for a smooth complex projective variety X . We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.

Our goal in this paper is to establish some difference analogue of second main theorems for holomorphic curves into projective varieties intersecting arbitrary families of c -periodical hypersurfaces (fixed or moving) with truncated counting functions in various cases. Our results generalize and improve the previous results in this topic.

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools.For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry.This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness.Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods.No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.

Let B be a commutative ℤ -graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that B ( f ) is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B . Also, given d ≥ 1, we give sufficient conditions that guarantee that every derivation of B ( d ) = ⊕ i ∈ ℤ B d i can be extended to a derivation of B . We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety ( Y , H ) to the existence of a nontrivial G a -action on the affine cone over ( Y , H ).