scholarly journals Stability conditions in families

2021 ◽  
Author(s):  
Arend Bayer ◽  
Martí Lahoz ◽  
Emanuele Macrì ◽  
Howard Nuer ◽  
Alexander Perry ◽  
...  
Keyword(s):  
Infinite Series ◽  
Moduli Spaces ◽  
Stability Conditions ◽  
Hodge Conjecture ◽  
Semistable Reduction ◽  
The Family ◽  
Hyperkähler Manifolds ◽  
Bridgeland Stability Conditions ◽  
Support Property ◽  
Cubic Fourfold

AbstractWe develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

scholarly journals Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions

2021 ◽  
Vol 31 (1) ◽  
pp. 13-73
Author(s):  
Georg Oberdieck ◽  
Dulip Piyaratne ◽  
Yukinobu Toda
Keyword(s):  
Discriminant Function ◽  
Stability Conditions ◽  
Full Support ◽  
Wall Crossing ◽  
Rank One ◽  
Bridgeland Stability Conditions ◽  
Support Property ◽  
The Stability ◽  
Definition Of ◽  
Stability Manifold

We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.

scholarly journals Wall-crossing in genus zero Landau–Ginzburg theory

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Dustin Ross ◽  
Yongbin Ruan
Keyword(s):  
Moduli Spaces ◽  
Homogeneous Polynomial ◽  
Genus Zero ◽  
Quantum Invariants ◽  
Rational Parameter ◽  
The Family ◽  
Wall Crossing

AbstractWe study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial

scholarly journals Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds II

2016 ◽  
Vol 27 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Antony Maciocia ◽  
Dulip Piyaratne
Keyword(s):  
Type Inequality ◽  
Stability Conditions ◽  
Coherent Sheaves ◽  
Rank One ◽  
Bridgeland Stability Conditions ◽  
Abelian Categories

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov–Gieseker (BG) type inequality. This is done by showing certain Fourier–Mukai transforms (FMTs) give equivalences of abelian categories which are double tilts of coherent sheaves.

scholarly journals Moduli space holography and the finiteness of flux vacua

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Thomas W. Grimm
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Boundary Data ◽  
Hodge Theory ◽  
Two Dimensional ◽  
Hodge Conjecture ◽  
String Compactifications ◽  
Finiteness Result ◽  
Dimensional Gravity ◽  
Dimension One

Abstract A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

scholarly journals On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

2020 ◽  
pp. 2050034
Author(s):  
Salvatore Floccari ◽  
Lie Fu ◽  
Ziyu Zhang
Keyword(s):  
Recent Result ◽  
Moduli Spaces ◽  
Betti Number ◽  
Defect Group ◽  
Crepant Resolution ◽  
Tate Conjecture ◽  
Chow Motive ◽  
Crepant Resolutions ◽  
The Difference ◽  
Cubic Fourfold

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

scholarly journals Families of explicitly isogenous Jacobians of variable-separated curves

2011 ◽  
Vol 14 ◽  
pp. 179-199 ◽  
Author(s):  
Benjamin Smith
Keyword(s):  
Infinite Series ◽  
Moduli Space ◽  
Number Fields ◽  
Isomorphism Type ◽  
High Genus ◽  
The Family

AbstractWe construct six infinite series of families of pairs of curves (X,Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs (f,g) of polynomials such that f(x1)−g(x2) is reducible.Supplementary materials are available with this article.

scholarly journals Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 969-975 ◽  
Author(s):  
Gulsah Ozdemir ◽  
Yilmaz Simsek
Keyword(s):  
Infinite Series ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Functional Equations ◽  
The Other ◽  
The Family

The purpose of this paper is to construct generating functions for the family of the Fibonacci and Jacobsthal polynomials. Using these generating functions and their functional equations, we investigate some properties of these polynomials. We also give relationships between the Fibonacci, Jacobsthal, Chebyshev polynomials and the other well known polynomials. Finally, we give some infinite series applications related to these polynomials and their generating functions.

scholarly journals Systolic inequalities for K3 surfaces via stability conditions

2021 ◽  
Author(s):  
Yu-Wei Fan
Keyword(s):  
Stability Condition ◽  
Symplectic Geometry ◽  
K3 Surfaces ◽  
Stability Conditions ◽  
Triangulated Categories ◽  
K3 Surface ◽  
Bridgeland Stability Conditions ◽  
Systolic Inequality

AbstractWe introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that $$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$ sys ( σ ) 2 ≤ C · vol ( σ ) holds for any stability condition on $$\mathcal {D}^b\mathrm {Coh}(X)$$ D b Coh ( X ) . This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.

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