scholarly journals Generalized Equivariant Cohomology and Stratifications

2016 ◽  
Vol 59 (3) ◽  
pp. 483-496
Author(s):  
Peter Crooks ◽  
Tyler Holden
Keyword(s):  
Equivariant Cohomology ◽  
Direct Limit ◽  
Semisimple Group ◽  
Projective Varieties ◽  
Affine Grassmannian ◽  
Smooth Complex ◽  
Compact Torus ◽  
Complex Semisimple ◽  
Complex Projective ◽  
Systematic Framework

AbstractFor T a compact torus and a generalized T-equivariant cohomology theory, we provide a systematic framework for computing in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute as an (pt)-module when X is a direct limit of smooth complex projective Tℂ-varieties. We perform this computation on the affine Grassmannian of a complex semisimple group.

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 H1–semistability for projective groups

2016 ◽  
Vol 162 (1) ◽  
pp. 89-100 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAHAN MJ
Keyword(s):  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Universal Covers ◽  
Complex Projective

AbstractWe initiate the study of the asymptotic topology of groups that can be realised as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the H1-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran [EKPR], this implies that linear projective groups satisfy the H1-semistability conjecture.

scholarly journals Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

2015 ◽  
Vol 159 (3) ◽  
pp. 517-527
Author(s):  
ANGELO FELICE LOPEZ
Keyword(s):  
Open Subset ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Complex ◽  
Normal Varieties ◽  
Restricted Volume ◽  
Complex Projective ◽  
Base Loci

AbstractLet X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on X. Given an expression (*) D$\sim_{\mathbb R}$t1H1 +. . .+ tsHs with ti ∈ ${\mathbb R}$ and Hi very ample, we define the (*)-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z$\not\subseteq$B+(D). Then, using some recent results of Birkar [Bir], we generalise to ${\mathbb R}$-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterisation of B+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X − B+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

scholarly journals AN EFFECTIVE VERSION OF A THEOREM OF KAWAMATA ON THE ALBANESE MAP

2011 ◽  
Vol 13 (03) ◽  
pp. 509-532 ◽  
Author(s):  
ZHI JIANG
Keyword(s):  
Fiber Space ◽  
Projective Varieties ◽  
Effective Version ◽  
Smooth Complex ◽  
Albanese Map ◽  
Complex Projective

We study the Albanese map of smooth complex projective varieties with small plurigenera. We provide criteria for the Albanese map to be surjective and to be an algebraic fiber space. These criteria are optimal in some sense.

scholarly journals NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR ARITHMETIC REASONS

2019 ◽  
Vol 7 ◽  
Author(s):  
JEFFREY D. ACHTER ◽  
SEBASTIAN CASALAINA-MARTIN ◽  
CHARLES VIAL
Keyword(s):  
Projective Varieties ◽  
Normal Functions ◽  
Smooth Complex ◽  
Field Of Definition ◽  
The Family ◽  
Definition Of ◽  
Complex Projective ◽  
Cycle Classes

For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.

scholarly journals Varieties with vanishing holomorphic Euler characteristic

2014 ◽  
Vol 2014 (691) ◽  
Author(s):  
Jungkai Alfred Chen ◽  
Olivier Debarre ◽  
Zhi Jiang
Keyword(s):  
Euler Characteristic ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective

Abstract.We study smooth complex projective varieties

scholarly journals COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

2012 ◽  
Vol 23 (04) ◽  
pp. 1250037 ◽  
Author(s):  
MICHELE BOLOGNESI ◽  
SONIA BRIVIO
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Low Rank ◽  
Coherent Systems ◽  
Projective Varieties ◽  
Complex Curve ◽  
Smooth Complex ◽  
Moduli Theory ◽  
Modular Varieties ◽  
Small Genus

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

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