scholarly journals Decomposition of degenerate Gromov–Witten invariants

2020 ◽  
Vol 156 (10) ◽  
pp. 2020-2075
Author(s):  
Dan Abramovich ◽  
Qile Chen ◽  
Mark Gross ◽  
Bernd Siebert
Keyword(s):  
Moduli Stacks ◽  
Smooth Family ◽  
Decomposition Formula ◽  
Singular Fibre

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals Kac Polynomials for Canonical Algebras

2017 ◽  
Vol 2019 (13) ◽  
pp. 3981-4003
Author(s):  
Pierre-Guy Plamondon ◽  
Olivier Schiffmann
Keyword(s):  
Finite Field ◽  
Dimension Vector ◽  
Indecomposable Representations ◽  
Moduli Stacks ◽  
Fixed Dimension ◽  
Canonical Algebra

Abstract We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

PRIMARY PROCESS STUDIES BY MONOISOTOPIC PHOTOSENSITIZATION ON THE REACTIONS OF Hg 6(3P1) ATOMS WITH ALKYL CHLORIDES

1964 ◽  
Vol 42 (9) ◽  
pp. 2056-2064 ◽  
Author(s):  
J. K. S. Wan ◽  
O. P. Strausz ◽  
W. F. Allen ◽  
H. E. Gunning
Keyword(s):  
Mercury Vapor ◽  
Quantum Yields ◽  
Primary Process ◽  
Reaction Parameters ◽  
Specific Nature ◽  
Electrodeless Discharge ◽  
Discharge Source ◽  
Bond Scission ◽  
Decomposition Formula ◽  
Process Studies

The specific nature of the primary process in the reaction of 202Hg 6(3P1) atoms, photoexcited in natural mercury vapor by a cool 202Hg electrodeless discharge source, with CH3Cl has been examined in detail. Primary C–Cl bond scission occurs with unit efficiency. Quantum yields (φ) for the two primary modes of decomposition[Formula: see text]were found to have values of 0.71 (b) and 0.29 (a). The effect of various reaction parameters on the 202Hg enrichment in the calomel product has been investigated and the importance of isotopic mercury depletion in the reaction zone demonstrated by the use of intermittent illumination.A brief study of the reaction of ethyl, n-propyl, i-propyl, t-butyl, and n-amyl chlorides has revealed a relation between the molecular structure of the alleyl chloride and the efficiency of the monoisotopic route (a) to calomel formation. Thus, while the reactions of all the normal alkyl chlorides have φa values between 0.29 and 0.32, φa (isopropyl chloride) is only 0.22 and φa (t-butyl chloride) is 0.17.

scholarly journals Néron models and limits of Abel–Jacobi mappings

2010 ◽  
Vol 146 (2) ◽  
pp. 288-366 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Normal Function ◽  
Motivic Cohomology ◽  
Normal Functions ◽  
Geometric Setting ◽  
Neron Models ◽  
A Finite Group ◽  
Intermediate Jacobian ◽  
Néron Models ◽  
Singular Fibre ◽  
Finite Singularity

AbstractWe show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators onK-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

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