Hodge conjecture and mixed motives. I

1991 ◽  
pp. 283-303 ◽  
Author(s):  
Morihiko Saito
Keyword(s):  
Hodge Conjecture ◽  
Mixed Motives

The Crowdfunding Paradigm

2019 ◽  
pp. 328-351
Author(s):  
Sharon F. Matusik ◽  
Jessica Jones
Keyword(s):  
Early Stage ◽  
Two Dimensions ◽  
Entrepreneurial Financing ◽  
Lock In ◽  
Mixed Motives ◽  
Theoretical Dynamics ◽  
Major Consideration

Crowdfunding has become a major consideration for individuals looking to fund their ideas, endeavors, and businesses. This phenomenon raises interesting questions for management scholars, such as what theories help to explain the nuance of crowdfunding as a form of entrepreneurial financing. With regard to what leads to crowdfunding campaign success, this chapter argues that there are mixed motives associated with contributing to these campaigns, and theoretical dynamics vary according to these different motives. The chapter also notes two fundamental differences of crowdfunding from more traditional means of funding early-stage ventures: the nature of engagement and preference toward product or person. Drawing on theory related to capabilities, the chapter identifies conditions under which crowdfunding is likely to be more and less advantageous based on these two dimensions. In summary, it provides a model that explains important sources of heterogeneity (i.e., motives) and homogeneity (i.e., diffused engagement and product lock-in) within the crowdfunding phenomenon that add nuance to theory in the entrepreneurial financing literature.

scholarly journals On the Hodge conjecture for quasi-smooth intersections in toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William Montoya
Keyword(s):  
Complete Intersection ◽  
Toric Variety ◽  
Toric Varieties ◽  
Picard Group ◽  
Hodge Conjecture ◽  
Smooth Hypersurface ◽  
Suitable Notion

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

On lambda operations on mixed motives

2013 ◽  
Vol 12 (2) ◽  
pp. 381-404 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Ring Structure ◽  
Triangulated Category ◽  
Grothendieck Ring ◽  
Basic Properties ◽  
Natural Notion ◽  
Mixed Motives

AbstractWe study the natural λ-ring structure on the Grothendieck ring of the triangulated category of mixed motives. Basic properties of a natural notion of characteristic-like series are developed in the context of equivariant objects.

scholarly journals Beilinson's Hodge Conjecture for Smooth Varieties

2013 ◽  
Vol 11 (2) ◽  
pp. 243-282 ◽  
Author(s):  
Rob de Jeu ◽  
James D. Lewis
Keyword(s):  
Projective Variety ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Generic Point ◽  
Higher Chow Group ◽  
Cycle Class

AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

scholarly journals Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alena Pirutka
Keyword(s):  
Function Field ◽  
Cohomology Group ◽  
Chow Group ◽  
Hodge Conjecture ◽  
Complex Curve ◽  
The Third ◽  
Generic Fibre ◽  
Nous Montrons ◽  
Unramified Cohomology ◽  
Cubic Threefold

En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de Hodge enti\`ere pour les classes de degr\'e 4 vaut pour les vari\'et\'es projectives et lisses de dimension 4 fibr\'ees en solides cubiques au-dessus d'une courbe, sans restriction sur les fibres singuli\`eres. --------------- We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension $2$ Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree $4$ classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres. Comment: in French

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