Projective Variety Research Articles

Duality for projective morphisms finite in codimension one and the positivity of double point divisors of general inner projections

The purpose of this paper is to prove the positivity of double point divisors of general inner projections for projective S 2-varieties with invertible dualizing sheaves, without assuming the smoothness or the normality of varieties. As applications, we characterize two types of projective varieties: varieties with trivial double point divisors and varieties with d − e − 1 ≤ n for dimension n, degree d and codimension e.

A second main theorem of finite ramified coverings

In this paper, we study a second main theorem for holomorphic curves from finite ramified coverings of the complex line to complex projective varieties intersecting hypersurfaces in subgeneral position.

Abelian surfaces and the non-Archimedean Hodge-$\mathcal{D}$-conjecture – The semi-stable case

If X is a smooth projective variety over \mathbb{R} , the Hodge \mathcal{D} -conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general, but is true in some special cases like Abelian surfaces and K3 -surfaces – and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here, the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess (1999) in the case of products of elliptic curves and by the author in general (Sreekantan, 2014). The case of bad reduction was also studied by the author in Sreekantan (2008).

CODIMENSION ONE FOLIATIONS IN POSITIVE CHARACTERISTIC

AbstractWe investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.

Cohomology of a real toric variety and shellability of posets arising from a graph

AbstractGiven a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.

Categories of abelian varieties over finite fields II: Abelian varieties over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{F}_q}$$\\end{document} and Morita equivalence

The category of abelian varieties over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{F}_q}$$\\end{document} is shown to be anti-equivalent to a category of ℤ-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{F}_q}$$\\end{document}. On full subcategories cut out by a finite set w of conjugacy classes of Weil q-numbers, the anti-equivalence is represented by what we call w-locally projective abelian varieties.

Arithmetic Okounkov bodies and positivity of adelic Cartier divisors

In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic R {\mathbb {R}} -Cartier divisor D ¯ \overline {D} are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized R {\mathbb {R}} -divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of D ¯ \overline {D} coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.

Ample bodies and Terracini loci of projective varieties

AbstractWe introduce the notion of ample body of a projective variety and use it to prove emptiness results for Terracini loci and specific identifiability results for toric and homogeneous varieties.

A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields

We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27 -dimensional \mathrm{E_6} module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.

L^p-Cohomology, Heat Semigroup and Stratified Spaces

Let (M, g) be an incomplete Riemannian manifold of finite volume and let 2le p<infty . In the first part of this paper we prove that under certain assumptions the inclusion of the space of L^p-differential forms into that of L^2-differential forms gives rise to an injective/surjective map between the corresponding L^p and L^2 cohomology groups. Then in the second part we provide various applications of these results to the curvature and the intersection cohomology of compact Thom–Mather stratified pseudomanifolds and complex projective varieties with only isolated singularities.

Semiorthogonal decomposition via categorical action

We show that the categorical action of the shifted q=0 affine algebra can be used to construct semiorthogonal decomposition on the weight categories. In particular, this construction recovers Kapranov's exceptional collection when the weight categories are the derived categories of coherent sheaves on Grassmannians and n-step partial flag varieties. Finally, as an application, we use this result to construct a semiorthogonal decomposition on the derived categories of coherent sheaves on Grassmannians of a coherent sheaf with homological dimension ≤1 over a smooth projective variety X.

Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position

Let A be an annular end of a complete minimal surface S in Rm and let V be a k-dimension projective subvariety of Pn(C)(n=m−1). Let g be the generalized Gauss map of S into V⊂Pn(C). In this paper, we establish a modified defect relation of g on the annular end A for q hypersurfaces {Qi}i=1q of Pn(C) in N-subgeneral position with respect to V. Our result implies that the image g(A) cannot omit all q hypersurfaces Q1,…,Qq if g is nondegenerate over Id(V) and q>(2N−k+1)(M+1)(M+2d)2d(k+1), where M=HV(d)−1 and d is the least of common multiple of deg⁡Q1,…,deg⁡Qq. As our best knowledge, it is the first time the value distribution of the Gauss map on an annular end of a minimal surfaces with hypersurface targets is studied, in particular the product into sum inequality for holomorphic curves on Riemann surfaces with hypersurfaces targets is presented. This our result has been used to study the unicity of the gauss maps in the recent work of C. Lu and X. Chen [14].

노후 산업단지 유형화와 유형별 특성 분석

The government has implemented a variety of regeneration projects for old industrial complexes and declined regions. Due to the promotion of policies and projects focused on each target place, however, there is a lack of policy linkages among them. This study intends to categorize old industrial complexes nationwide and analyze the characteristics of each type while considering industrial complexes and surrounding regions. Ten indicators are selected to implement the principal component analysis using data from the national industrial complex status, industrial location information system, and statistical geographic information system. Based on the factor score produced by the principal component analysis, the cluster analysis indicates that there are four types of old industrial complexes: large sustaining and developing complex type, small growing complex type, old declining complex type and region-linked complex type. The old declining industrial complex continues to decline due to weak economy of both the complexes and surrounding areas, and the residential environment. Industrial restructuring and industrial complex regeneration projects are needed to strengthen the competitiveness of the complexes. Regeneration strategies for each type of industrial complex is required in consideration of the physical and economic characteristics of both the complexes and the surrounding areas.

Biochemical Society support in action: experiences from our community

Throughout the year, the Biochemical Society offers a programme of grants for all career stages supporting research, attendance at scientific conferences and the sponsorship of events. So far in 2023, we have awarded over £262,000 towards international conference participation, summer laboratory placements and the organization of independent lectures and diversity-related projects. In this article, we are delighted to share a few insights from our community that highlight the variety of projects and opportunities made possible by our funding. For more information about our funding opportunities and the application process, please visit our website.

Positivity of Higher Exterior Powers of the Tangent Bundle

Abstract We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth, or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under the assumption that $\rho (X)\ne 1$.

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf F \mathcal {F} endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition F ≃ U ⊕ A \mathcal {F}\simeq \mathcal {U}\oplus \mathcal {A} where U \mathcal {U} is a hermitian flat bundle and A \mathcal {A} is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles f ∗ ω X / Y ⊗ m f_* \omega _{X/Y}^{\otimes m} under a surjective morphism f : X → Y f\colon X \to Y of smooth projective varieties with m ≥ 2 m\geq 2 . This extends previous results of Fujita, Catanese–Kawamata, and Iwai.

Morelli-Włodarczyk cobordism and examples of rooftop flips

We introduce the notion of rooftop flip, namely a small modification among normal projective varieties which is modeled by a smooth projective variety of Picard number 2 admitting two projective bundle structures. Examples include the Atiyah flop and the Mukai flop, modeled respectively by P1×P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^1\ imes \\mathbb {P}^1$$\\end{document} and by PTP2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}\\left( T_{\\mathbb {P}^2}\\right) $$\\end{document}. Using the Morelli-Włodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb C}^*$$\\end{document}-action with only two fixed point components, induces a rooftop flip.

Terracini Loci for Maps

Let X be a smooth projective variety and f:X→Pr a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case dimX=1. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set B⊊X or solutions containing a prescribed p∈X). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools.

The unique tangent cone property for weakly holomorphic maps into projective algebraic varieties

The unique tangent cone property for weakly holomorphic maps into projective algebraic varieties

The number of irreducible components of the locus of nonbirational projection centers

We work over an algebraically closed field of characteristic zero. For a nondegenerate projective variety X⊆PN, the locus of points from which X is projected nonbirationally onto its image, is called the Segre locus of X. The purpose here is to give an upper bound of the number of the irreducible components of the Segre locus of a projective variety in terms of its invariants.