scholarly journals On the geometric André–Oort conjecture for variations of Hodge structures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiaming Chen
Keyword(s):  
Projective Variety ◽  
Hodge Structure ◽  
Finite Union ◽  
Shimura Variety ◽  
Hodge Structures ◽  
Smooth Complex ◽  
Classical Setting ◽  
Dominant Period

Abstract Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

scholarly journals On the boundary of moduli spaces of log Hodge structures, II: Nontrivial torsors

2014 ◽  
Vol 213 ◽  
pp. 1-20
Author(s):  
Tatsuki Hayama
Keyword(s):  
Moduli Spaces ◽  
Hodge Structure ◽  
Hodge Structures

AbstractIn this paper, we determine when a natural torsor arising in the work of Kato and Usui on partial compactification of period domains of pure Hodge structure is trivial, and we give an application to cycle spaces.

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
Kiumars Kaveh
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Projective Variety ◽  
Cohomology Ring ◽  
Holomorphic Vector Field ◽  
Zero Set ◽  
Graded Algebras ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

scholarly journals Hodge correlators

2019 ◽  
Vol 2019 (748) ◽  
pp. 1-138
Author(s):  
Alexander B. Goncharov
Keyword(s):  
Fundamental Group ◽  
Hodge Structure ◽  
Euler System ◽  
Feynman Integral ◽  
Mixed Hodge Structure ◽  
Modular Curves ◽  
Hodge Structures ◽  
Motivic Cohomology ◽  
Linear Differential ◽  
Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

scholarly journals MIXED HODGE STRUCTURES WITH MODULUS

2020 ◽  
pp. 1-35
Author(s):  
Florian Ivorra ◽  
Takao Yamazaki
Keyword(s):  
Hodge Structure ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
Classical Notion ◽  
Mixed Hodge Structures

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

scholarly journals Purity for graded potentials and quantum cluster positivity

2015 ◽  
Vol 151 (10) ◽  
pp. 1913-1944 ◽  
Author(s):  
Ben Davison ◽  
Davesh Maulik ◽  
Jörg Schürmann ◽  
Balázs Szendrői
Keyword(s):  
Projective Variety ◽  
Hodge Structure ◽  
Regular Function ◽  
Mixed Hodge Structure ◽  
Positive Weight ◽  
Quantum Cluster ◽  
Vanishing Cycle ◽  
Marked Points ◽  
Quivers With Potential ◽  
Hard Lefschetz Theorem

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

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