Selecta Mathematica

Items from 1 to 20 out of 24 results

article

Syntactic categories for Nori motives

Luca Barbieri-Viale, Olivia Caramello, Laurent Lafforgue

Selecta Mathematica > 2018 > 24 > 4 > 3619-3648

We give a new construction, based on categorical logic, of Nori’s $$\mathbb Q$$ Q -linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $$\mathbb Q$$ Q . This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $$\mathbb Q$$ Q -linear abelian category of mixed motives...

article

Six-functor-formalisms and fibered multiderivators

Fritz Hörmann

Selecta Mathematica > 2018 > 24 > 4 > 2841-2925

We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract six-functor-formalisms. We also give axioms for Wirthmüller and Grothendieck formalisms (where either $$f^{!}=f^{*}$$ f!=f∗ or $$f_{!}=f_{*}$$ f!=f∗ ) or intermediate formalisms where we have e.g. a natural morphism $$f_{!}\rightarrow f_{*}$$ f!→f∗ . Finally, it is shown that a fibered multiderivator (in particular,...

article

Transition formulas for involution Schubert polynomials

Zachary Hamaker, Eric Marberg, Brendan Pawlowski

Selecta Mathematica > 2018 > 24 > 4 > 2991-3025

The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group $$S_n$$ Sn . Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as $${\hat{\mathfrak S}}_y$$ S^y (to be called involution Schubert polynomials...

article

A Vietoris–Smale mapping theorem for the homotopy of hyperdefinable sets

Alessandro Achille, Alessandro Berarducci

Selecta Mathematica > 2018 > 24 > 4 > 3445-3473

Results of Smale (Proc Am Math Soc 8(3): 604–604, 1957) and Dugundji (Fundam Math 66:223–235, 1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map $$f:X\rightarrow Y$$ f:X→Y with strong connectivity conditions on the fibers is given. We can apply similar techniques to compare the homotopy of spaces living in different categories, for instance an abelian variety...

article

Relative enumerative invariants of real nodal del Pezzo surfaces

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin

Selecta Mathematica > 2018 > 24 > 4 > 2927-2990

The surfaces considered are real, rational and have a unique smooth real $$(-2)$$ (-2) -curve. Their canonical class K is strictly negative on any other irreducible curve in the surface and $$K^2>0$$ K2>0 . For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $$(-2)$$ (-2) -curve...

article

Derived coisotropic structures II: stacks and quantization

Valerio Melani, Pavel Safronov

Selecta Mathematica > 2018 > 24 > 4 > 3119-3173

We extend results about n-shifted coisotropic structures from part I of this work to the setting of derived Artin stacks. We show that an intersection of coisotropic morphisms carries a Poisson structure of shift one less. We also compare non-degenerate shifted coisotropic structures and shifted Lagrangian structures and show that there is a natural equivalence between the two spaces in agreement...

article

$${\mathbb A}^1$$ A1 -curves on affine complete intersections

Xi Chen, Yi Zhu

Selecta Mathematica > 2018 > 24 > 4 > 3823-3834

We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.

article

Dimension of the space of intertwining operators from degenerate principal series representations

Taito Tauchi

Selecta Mathematica > 2018 > 24 > 4 > 3649-3662

Let X be a homogeneous space of a real reductive Lie group G. It was proved by T. Kobayashi and T. Oshima that the regular representation $$C^{\infty }(X)$$ C∞(X) contains each irreducible representation of G at most finitely many times if a minimal parabolic subgroup P of G has an open orbit in X, or equivalently, if the number of P-orbits on X is finite. In contrast to the minimal parabolic case,...

article

Deformation quantisation for unshifted symplectic structures on derived Artin stacks

J. P. Pridham

Selecta Mathematica > 2018 > 24 > 4 > 3027-3059

We prove that every 0-shifted symplectic structure on a derived Artin n-stack admits a curved $$A_{\infty }$$ A∞ deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived Artin n-stack...

article

Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

Christof Geiss, Bernard Leclerc, Jan Schröer

Selecta Mathematica > 2018 > 24 > 4 > 3283-3348

We generalize Lusztig’s nilpotent varieties, and Kashiwara and Saito’s geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac–Moody algebras.

article

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II

Charlotte Chan

Selecta Mathematica > 2018 > 24 > 4 > 3175-3216

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic...

article

Derived coisotropic structures I: affine case

Valerio Melani, Pavel Safronov

Selecta Mathematica > 2018 > 24 > 4 > 3061-3118

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $$\mathbb {P}_n$$ Pn -algebras. Roughly speaking, a coisotropic morphism is given by a $$\mathbb {P}_{n+1}$$ Pn+1 -algebra acting on a $$\mathbb {P}_n$$ Pn -algebra. One of our main results is an identification of the space of such coisotropic structures with the space...

article

Some phenomena in tautological rings of manifolds

Oscar Randal-Williams

Selecta Mathematica > 2018 > 24 > 4 > 3835-3873

We prove several basic ring-theoretic results about tautological rings of manifolds W, that is, the rings of generalised Miller–Morita–Mumford classes for fibre bundles with fibre W. Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitely-generated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological...

article

Some remarks on L-equivalence of algebraic varieties

Alexander I. Efimov

Selecta Mathematica > 2018 > 24 > 4 > 3753-3762

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts (Int J Math 16(1):13–36, 2005, Conjecture 0.3) stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying...

article

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

Alexander Kuznetsov, Evgeny Shinder

Selecta Mathematica > 2018 > 24 > 4 > 3475-3500

We discuss a conjecture saying that derived equivalence of smooth projective simply connected varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in $${\mathbb {P}}^5$$...

article

A 2-categorical extension of Etingof–Kazhdan quantisation

Andrea Appel, Valerio Toledano Laredo

Selecta Mathematica > 2018 > 24 > 4 > 3529-3617

Let $$\mathsf {k}$$ k be a field of characteristic zero. Etingof and Kazhdan (Sel. Math. (N.S.) 2:1–41, 1996) construct a quantisation $$U_\hbar \mathfrak b$$ Uħb of any Lie bialgebra $$\mathfrak b$$ b over $$\mathsf {k}$$ k , which depends on the choice of an associator $$\Phi $$ Φ . They prove moreover that this quantisation is functorial in $$\mathfrak b$$ b (Etingof and Kazhdan in Sel. Math...

article

On 2-Verma modules for quantum $${\mathfrak {sl}_{2}}$$ sl2

Grégoire Naisse, Pedro Vaz

Selecta Mathematica > 2018 > 24 > 4 > 3763-3821

In this paper we study the superalgebra $$A_n$$ An , introduced by the authors in previous work on categorification of Verma modules for quantum $${\mathfrak {sl}_{2}}$$ sl2 . The superalgebra $$A_n$$ An is akin to the nilHecke algebra, and shares similar properties. We also prove a uniqueness result about 2-Verma modules on $$\Bbbk $$ k -linear 2-categories.

article

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

Nathaniel Bottman, Katrin Wehrheim

Selecta Mathematica > 2018 > 24 > 4 > 3381-3443

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-0 effect, and predict its algebraic consequences—geometric composition extends to a curved $$A_\infty $$ A∞ -bifunctor, in particular the associated...

article

Donaldson–Thomas theory of $$[\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1$$ [C2/Zn+1]×P1

Zijun Zhou

Selecta Mathematica > 2018 > 24 > 4 > 3663-3722

We study the relative orbifold Donaldson–Thomas theory of $$[\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1$$ [C2/Zn+1]×P1 . A correspondence is established between the DT theory relative to disjoint union of vertical fibers to quantum multiplication by divisors for the Hilbert scheme of points on $$[\mathbb {C}^2/\mathbb {Z}_{n+1}]$$ [C2/Zn+1] . This determines a correspondence between the...

article

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

Saugata Basu, Cordian Riener

Selecta Mathematica > 2018 > 24 > 4 > 3241-3281

Let $$\mathrm {R}$$ R be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ Rk defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $$d^{O(d)} s^d k^{\lfloor d/2 \rfloor...

Number:
4
Volume:
24
Year:
2018

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Keywords

(OP)FIBERED 2-MULTICATEGORIES (1)
2-VERMA MODULES (1)
BETTI NUMBERS (1)
CATEGORICAL ACTIONS (1)
CATEGORIFICATION (1)
CHOW GROUPS (1)
COMPUTATIONAL COMPLEXITY (1)
CRITICAL POINT RESULT (1)
DEFINABLE GROUPS (1)
DEGENERATE PRINCIPAL SERIES (1)
DERIVATORS (1)
ELLIPTIC ASSOCIATORS (1)
ELLIPTIC POLYLOGARITHMS (1)
EQUIVARIANT COHOMOLOGY (1)
FIBERED DERIVATORS (1)
FRACTIONAL OPERATORS (1)
GODEAUX SURFACE (1)
GROTHENDIECK CONTEXTS (1)
GROTHENDIECK RINGS OF VARIETIES (1)
HODGE STRUCTURES (1)
HOMOTOPY (1)
HYPERSURFACES (1)
INTERTWINING OPERATORS (1)
KRULL–SCHMIDT CATEGORIES (1)
LOG VARIETIES (1)
MODULAR SYMBOLS (1)
MULTIDERIVATORS (1)
MULTIPLE PERIODIC SOLUTIONS (1)
MULTIPLICITY (1)
O-MINIMALITY (1)
QUANTUM $${\MATHFRAK {SL} {2}}$$ SL 2 (1)
RATIONAL CURVES (1)
RATIONAL EQUIVALENCE (1)
REAL SPHERICAL (1)
SIX-FUNCTOR-FORMALISMS (1)
SPHERICAL VARIETY (1)
STABLE SURFACE (1)
SURFACE OF GENERAL TYPE (1)
SYMMETRIC SEMI-ALGEBRAIC SETS (1)
WIRTHMÜLLER CONTEXTS (1)
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Selecta Mathematica

Syntactic categories for Nori motives

Six-functor-formalisms and fibered multiderivators

Transition formulas for involution Schubert polynomials

A Vietoris–Smale mapping theorem for the homotopy of hyperdefinable sets

Relative enumerative invariants of real nodal del Pezzo surfaces

Derived coisotropic structures II: stacks and quantization

$${\mathbb A}^1$$ A1 -curves on affine complete intersections

Dimension of the space of intertwining operators from degenerate principal series representations

Deformation quantisation for unshifted symplectic structures on derived Artin stacks

Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II

Derived coisotropic structures I: affine case

Some phenomena in tautological rings of manifolds

Some remarks on L-equivalence of algebraic varieties

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

A 2-categorical extension of Etingof–Kazhdan quantisation

On 2-Verma modules for quantum $${\mathfrak {sl}_{2}}$$ sl2

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

Donaldson–Thomas theory of $$[\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1$$ [C2/Zn+1]×P1

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

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