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 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

Rhetoric and Dutch Translation Theory (1750-1820)1

Target ◽  
1993 ◽  
Vol 5 (1) ◽  
pp. 55-69 ◽  
Author(s):  
Luc G. Korpel
Keyword(s):  
The Netherlands ◽  
Functional Approach ◽  
European Countries ◽  
Translation Theory ◽  
Dutch Version ◽  
Crucial Step ◽  
Rhetorical Tradition ◽  
Dutch Translation ◽  
Theory Of Language ◽  
Definition Of

Abstract Due to the influence of rhetoric, Dutch translation theory between 1750 and 1820, like translation theory in other Western European countries in those days, was primarily concerned with the effect of a translation on the Dutch public and the verbal appearance of the Dutch version. This functional approach was reinforced by the definition of translation as interpretatio, imitatio or exercitatio. The translational technique which follows from this prospective orientation is one of adaptation, correction and improvement. By the end of the period, Dutch translation theory seemed to be moving away slowly from the rhetorical tradition, as a result of two major changes: (1) a growing concern as of ± 1780 for fidelity to the verbal aspects of the original within the interpretatio-approach, and (2) a decrease in the popularity of imitatio as a creative technique after 1800. Unlike Germany, translation theory in The Netherlands had not made the crucial step towards a new theory of language before 1820.

Biorobotics researcher: To be or not to be?

2001 ◽  
Vol 24 (6) ◽  
pp. 1054-1054 ◽  
Author(s):  
Carolina Chang
Keyword(s):  
Common Ground ◽  
Crucial Step ◽  
Definition Of

Much confusion exists within the robotics and the biology communities regarding the definition of biorobotics and the aims and strategies that characterize this approach. Not even the basic criteria for identifying biorobotic research are being applied consistently. Barbara Webb has taken a crucial step towards setting a common ground from which biorobotic systems can be described, analyzed, and compared.

scholarly journals Comments on open string with ‘massive’ boundary term

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190748
Author(s):  
Arkady A. Tseytlin
Keyword(s):  
Scattering Amplitudes ◽  
Partition Function ◽  
Gauge Field ◽  
Path Integral ◽  
Open String ◽  
One Dimensional ◽  
Conformally Invariant ◽  
Additional Constant ◽  
The One ◽  
Definition Of

We discuss possible definition of open string path integral in the presence of additional boundary couplings corresponding to the presence of masses at the ends of the string. These couplings are not conformally invariant implying that as in a non-critical string case one is to integrate over the one-dimensional metric or reparametrizations of the boundary. We compute the partition function on the disc in the presence of an additional constant gauge field background and comment on the structure of the corresponding scattering amplitudes.

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 $H^2(X,\mathcal{O}_X) = 0$.

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