scholarly journals Parity sheaves and Smith theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Spencer Leslie ◽  
Gus Lonergan
Keyword(s):  
Fixed Point ◽  
Prime Number ◽  
Algebraic Variety ◽  
Reductive Group ◽  
Derived Category ◽  
Geometric Construction ◽  
Affine Grassmannian ◽  
Complex Algebraic Variety ◽  
Smith Theory

Abstract Let p be a prime number and let X be a complex algebraic variety with an action of ℤ / p ⁢ ℤ {\mathbb{Z}/p\mathbb{Z}} . We develop the theory of parity complexes in a certain 2-periodic localization of the equivariant constructible derived category D ℤ / p ⁢ ℤ b ⁢ ( X , ℤ p ) {D^{b}_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_{p})} . Under certain assumptions, we use this to define a functor from the category of parity sheaves on X to the category of parity sheaves on the fixed-point locus X ℤ / p ⁢ ℤ {X^{\mathbb{Z}/p\mathbb{Z}}} . This may be thought of as a categorification of Smith theory. When X is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

scholarly journals OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

2019 ◽  
Vol 7 ◽  
Author(s):  
A. ASOK ◽  
J. FASEL ◽  
M. J. HOPKINS
Keyword(s):  
Vector Bundle ◽  
Complex Manifold ◽  
Algebraic Variety ◽  
Vector Bundles ◽  
Necessary Condition ◽  
Chern Classes ◽  
Smooth Complex ◽  
Steenrod Operations ◽  
Complex Algebraic Variety ◽  
Cycle Class

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

scholarly journals Base change for semiorthogonal decompositions

2011 ◽  
Vol 147 (3) ◽  
pp. 852-876 ◽  
Author(s):  
Alexander Kuznetsov
Keyword(s):  
Algebraic Variety ◽  
Base Change ◽  
Derived Category ◽  
Intermediate Step ◽  
Coherent Sheaves ◽  
Base Scheme ◽  
Compatible System ◽  
Perfect Complexes

AbstractLet X be an algebraic variety over a base scheme S and ϕ:T→S a base change. Given an admissible subcategory 𝒜 in 𝒟b(X), the bounded derived category of coherent sheaves on X, we construct under some technical conditions an admissible subcategory 𝒜T in 𝒟b(X×ST), called the base change of 𝒜, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of 𝒟b (X) is given, then the base changes of its components form a semiorthogonal decomposition of 𝒟b (X×ST) . As an intermediate step, we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on X and of the category of perfect complexes on X. As an application, we prove that the projection functors of a semiorthogonal decomposition are kernel functors.

Locally free (ℙn)-modules

1992 ◽  
Vol 112 (2) ◽  
pp. 233-245 ◽  
Author(s):  
S. C. Coutinho ◽  
M. P. Holland
Keyword(s):  
Projective Space ◽  
Algebraic Variety ◽  
Differential Operators ◽  
Module Theory ◽  
Rings Of Differential Operators ◽  
Complex Algebraic Variety ◽  
Free Modules

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

scholarly journals Remarks on endomorphisms and rational points

2011 ◽  
Vol 147 (6) ◽  
pp. 1819-1842 ◽  
Author(s):  
E. Amerik ◽  
F. Bogomolov ◽  
M. Rovinsky
Keyword(s):  
Fixed Point ◽  
Number Field ◽  
Algebraic Variety ◽  
Rational Points ◽  
Potential Density ◽  
Cubic Fourfold

AbstractLet X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

Exceptional Moufang Quadrangles of Type F4

1999 ◽  
Vol 51 (2) ◽  
pp. 347-371 ◽  
Author(s):  
Bernhard Mühlherr ◽  
Hendrik Van Maldeghem
Keyword(s):  
Fixed Point ◽  
Generalized Quadrangle ◽  
Geometric Construction

AbstractIn this paper, we present a geometric construction of the Moufang quadrangles discovered by Richard Weiss (see Tits & Weiss [18] or Van Maldeghem [19]). The construction uses fixed point free involutions in certain mixed quadrangles, which are then extended to involutions of certain buildings of type F4. The fixed flags of each such involution constitute a generalized quadrangle. This way, not only the new exceptional quadrangles can be constructed, but also some special type of mixed quadrangles.

scholarly journals On the algebraicity of the zero locus of an admissible normal function

2013 ◽  
Vol 149 (11) ◽  
pp. 1913-1962 ◽  
Author(s):  
Patrick Brosnan ◽  
Gregory Pearlstein
Keyword(s):  
Galois Group ◽  
Algebraic Variety ◽  
Hodge Structure ◽  
Normal Function ◽  
Mixed Hodge Structure ◽  
Nilpotent Orbits ◽  
Smooth Complex ◽  
Complex Algebraic Variety ◽  
Zero Locus

AbstractWe show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

On Orbit Closures of Symmetric Subgroups in Flag Varieties

2000 ◽  
Vol 52 (2) ◽  
pp. 265-292 ◽  
Author(s):  
Michel Brion ◽  
Aloysius G. Helminck
Keyword(s):  
Fixed Point ◽  
Borel Subgroup ◽  
Reductive Group ◽  
Double Coset ◽  
Restriction Map ◽  
Flag Varieties ◽  
Parabolic Subgroups ◽  
Involutive Automorphism ◽  
Orbit Closures ◽  
Geometric Analogue

AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

scholarly journals PROJECTIVE GENERATION FOR EQUIVARIANT -MODULES

2021 ◽  
Author(s):  
G. BELLAMY ◽  
S. GUNNINGHAM ◽  
S. RASKIN
Keyword(s):  
Reductive Group ◽  
Derived Category ◽  
Independent Interest ◽  
Affine Variety ◽  
Projective Generator ◽  
Analogous Statement ◽  
Finite Dimensional ◽  
Smooth Affine

AbstractWe investigate compact projective generators in the category of equivariant "Image missing"-modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest.

scholarly journals Geometric motivic Poincaré series of quasi-ordinary singularities

2010 ◽  
Vol 149 (1) ◽  
pp. 49-74 ◽  
Author(s):  
HELENA COBO PABLOS ◽  
PEDRO D. GONZÁLEZ PÉREZ
Keyword(s):  
Algebraic Variety ◽  
Poincaré Series ◽  
Explicit Description ◽  
Hypersurface Singularity ◽  
Grothendieck Ring ◽  
Rational Form ◽  
Poincare Series ◽  
Newton Polyhedra ◽  
Complex Algebraic Variety

AbstractThegeometric motivic Poincaré seriesof a germ (S, 0) of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through (S, 0). Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when (S, 0) is an irreducible germ ofquasi-ordinary hypersurface singularityin terms of the Newton polyhedra of thelogarithmic jacobian ideals. These ideals are determined by thecharacteristic monomialsof a quasi-ordinary branch parametrizing (S, 0).

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