scholarly journals On Weil homomorphism in locally free sheaves over structured spaces

2017 ◽  
Vol 50 (1) ◽  
pp. 28-41
Author(s):  
Ewa Falkiewicz ◽  
Wiesław Sasin
Keyword(s):  
Euler Characteristic ◽  
Linear Connection ◽  
Characteristic Classes ◽  
Free Sheaf ◽  
Locally Free Sheaf

Abstract Inspired by the work of Heller and Sasin [1], we construct in this paper Weil homomorphism in a locally free sheaf W of ϕ-fields [2] over a structured space. We introduce the notion of G-consistent, linear connection on this sheaf, what allows us to clearly define Chern, Pontrjagin and Euler characteristic classes. We also show proper equalities between those classes.

scholarly journals Nested Hilbert Schemes and the nested $q,t$-Catalan series

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mahir Bilen Can
Keyword(s):  
General Formula ◽  
Euler Characteristic ◽  
Hilbert Scheme ◽  
Negative Integer ◽  
Hilbert Schemes ◽  
Free Sheaf ◽  
Integer Coefficients ◽  
Positive Polynomial ◽  
Locally Free Sheaf ◽  
International Audience

International audience In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series. Dans cet article, nous étudions les espaces tangents du schéma de Hilbert emboité lisse $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ de points du plan, et donnons une formule générale pour le calcul de la caractéristique d’Euler d’un faisceau $\mathbb{T}^2$-équivariant localement libre sur $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. En appliquant notre resultat a un faisceau particulier, nous conjecturons que le résultat est un polynôme en$q$ et $t$ à coefficents positifs ou nuls. Nous appelons ce polynôme conjecturalement positif la “série de $q; t$-Catalan emboîtée”, car il a de nombreuses propriétés (conjecturées) similaires à celles de la série de $q; t$-Catalan.

scholarly journals A pathological case of the C1 conjecture in mixed characteristic

2018 ◽  
Vol 167 (01) ◽  
pp. 61-64 ◽  
Author(s):  
INDER KAUR
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Projective Curve ◽  
Free Sheaf ◽  
Smooth Projective Curve ◽  
Mixed Characteristic ◽  
Locally Free Sheaf ◽  
Fixed Rank ◽  
Pathological Case

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

Berezinian, of a locally free sheaf

2004 ◽  
pp. 51-52
Author(s):  
Steven Duplij ◽  
Joshua Feinberg ◽  
Moshe Moshe ◽  
Soon-Tae Hong ◽  
Omer Faruk Dayi ◽  
...  
Keyword(s):  
Free Sheaf ◽  
Locally Free Sheaf

scholarly journals MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

2019 ◽  
Vol 236 ◽  
pp. 251-310 ◽  
Author(s):  
MARC LEVINE
Keyword(s):  
Euler Characteristic ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Reductive Group ◽  
Characteristic Classes ◽  
Euler Characteristics ◽  
Symmetric Powers ◽  
Maximal Tori ◽  
General Calculus

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

scholarly journals The resolution property of algebraic surfaces

2011 ◽  
Vol 148 (1) ◽  
pp. 209-226 ◽  
Author(s):  
Philipp Gross
Keyword(s):  
Toric Varieties ◽  
Coherent Sheaf ◽  
Algebraic Surfaces ◽  
Two Dimensional ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Excellent Ring ◽  
Resolution Property

AbstractWe prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

Rost’s Chain Lemma

2019 ◽  
pp. 119-143
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Inductive Assumption ◽  
Roots Of Unity ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Special Case

This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a field containing 1/𝓁 and all 𝓁th roots of unity. It fixes an integer 𝑛 ≥ 2 and an 𝑛-tuple (𝑎1, ..., 𝑎𝑛) of units in 𝑘, such that the symbol ª = {𝑎1, ..., 𝑎𝑛} is nontrivial in the Milnor 𝐾-group 𝐾𝑀 𝑛(𝑘)/𝓁. The chapter produces the statement of the Chain Lemma by first proving the special case 𝑛 = 2. The notion of an 𝓁-form on a locally free sheaf over 𝑆 is then introduced, before the chapter shows how 𝓁-forms may be used to define elements of 𝐾𝑀 𝑛(𝑘(𝑆))/𝓁.

scholarly journals Smoothing Pairs Over Degenerate Calabi–Yau Varieties

2020 ◽  
Author(s):  
Kwokwai Chan ◽  
Ziming Nikolas Ma
Keyword(s):  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Tangent Sheaf ◽  
Perfect Complex

Abstract We apply the techniques developed in [2] to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi–Yau variety $X$. In particular, if $X$ is a projective Calabi–Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the results in [ 8], that the pair $(X,\mathfrak{F})$ is formally smoothable when $\textrm{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

scholarly journals Higher rank K-theoretic Donaldson-Thomas Theory of points

2021 ◽  
Vol 9 ◽  
Author(s):  
Nadir Fasola ◽  
Sergej Monavari ◽  
Andrea T. Ricolfi
Keyword(s):  
Partition Function ◽  
Elliptic Genus ◽  
Obstruction Theory ◽  
Free Sheaf ◽  
Crucial Step ◽  
Quot Scheme ◽  
Locally Free Sheaf ◽  
Higher Rank ◽  
Definition Of ◽  
Critical Structure

Abstract We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

scholarly journals On the torsion of the first direct image of a locally free sheaf

2015 ◽  
Vol 65 (1) ◽  
pp. 101-136
Author(s):  
Andrei Teleman
Keyword(s):  
Direct Image ◽  
Free Sheaf ◽  
Locally Free Sheaf
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