scholarly journals Localizing virtual structure sheaves for almost perfect obstruction theories

2020 ◽  
Vol 8 ◽  
Author(s):  
Young-Hoon Kiem ◽  
Michail Savvas
Keyword(s):  
Obstruction Theory ◽  
Euler Characteristics ◽  
Moduli Stacks ◽  
Wall Crossing ◽  
Wall Crossing Formula ◽  
Gysin Maps

Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

scholarly journals Localizing virtual fundamental cycles for semi-perfect obstruction theories

2018 ◽  
Vol 29 (04) ◽  
pp. 1850032 ◽  
Author(s):  
Young-Hoon Kiem
Keyword(s):  
Algebraic Geometry ◽  
Local Existence ◽  
Obstruction Theory ◽  
Compatibility Conditions ◽  
Euler Characteristics ◽  
Smoothness Assumption ◽  
Fundamental Class ◽  
Derived Algebraic Geometry ◽  
Wall Crossing ◽  
Invent Math

Recently, Chang and Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization of Graber–Pandharipande [T. Graber and R. Pandharipande, Localization of virtual cycles, Invent. Math. 135(2) (1999) 487–518], the cosection localization [Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013) 1025–1050] and their combination [H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964–986], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [Y. Jiang and R. Thomas, Virtual signed Euler characteristics, preprint (2014), arXiv:1408.2541] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

On the Moduli Space of a Spherical Polygonal Linkage

1999 ◽  
Vol 42 (3) ◽  
pp. 307-320 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Level Sets ◽  
Morse Function ◽  
Great Circle ◽  
Winding Number ◽  
Wall Crossing ◽  
Polygonal Linkage ◽  
Spherical Linkages ◽  
Wall Crossing Formula

AbstractWe give a “wall-crossing” formula for computing the topology of the moduli space of a closed n-gon linkage on 𝕊2. We do this by determining the Morse theory of the function ρn on the moduli space of n-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first (n − 1) side-lengths are fixed. We obtain a Morse function on the (n − 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of ρn are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ρn at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Seiberg-Witten invariants, the topological degree and wall crossing formula

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Simple Proof ◽  
Topological Degree ◽  
Witten Invariant ◽  
Homotopy Invariance ◽  
Finite Dimensional ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractFollowing S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b 1 = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals General wall crossing formula

1995 ◽  
Vol 2 (6) ◽  
pp. 797-810 ◽  
Author(s):  
T. J. Li ◽  
A. Liu
Keyword(s):  
Wall Crossing ◽  
Wall Crossing Formula

scholarly journals Wall-crossing in genus-zero hybrid theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Emily Clader ◽  
Dustin Ross
Keyword(s):  
Projective Space ◽  
Hybrid Model ◽  
Type Theory ◽  
Singularity Theory ◽  
Complete Intersections ◽  
Genus Zero ◽  
Witten Theory ◽  
Wall Crossing ◽  
Wall Crossing Formula ◽  
Hybrid Theory

Abstract The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11].

scholarly journals Instantons, Topological Strings, and Enumerative Geometry

2010 ◽  
Vol 2010 ◽  
pp. 1-70 ◽  
Author(s):  
Richard J. Szabo
Keyword(s):  
Moduli Spaces ◽  
Gauge Theories ◽  
Topological String ◽  
Two Dimensions ◽  
Enumerative Geometry ◽  
Topological String Theory ◽  
Euler Characteristics ◽  
Wall Crossing ◽  
Supersymmetric Gauge ◽  
Torsion Free

We review and elaborate on certain aspects of the connections between instanton counting in maximally supersymmetric gauge theories and the computation of enumerative invariants of smooth varieties. We study in detail three instances of gauge theories in six, four, and two dimensions which naturally arise in the context of topological string theory on certain noncompact threefolds. We describe how the instanton counting in these gauge theories is related to the computation of the entropy of supersymmetric black holes and how these results are related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the computation of their Euler characteristics are also elucidated.

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