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Large-Volume Stability Condition

A large-volume stability condition in this algebraic-geometric setting is a Bridgeland stability condition whose central charge is prescribed using a large multiple tH of an ample divisor class. For a smooth projective complex threefold X and a real numerical divisor class B, write ch^B=e^(-B)ch. The ordinary charge is

Z_(B,tH)(E)=-ch_3^B(E)+(t^2)/2H^2ch_1^B(E)+i(tHch_2^B(E)-(t^3)/6H^3ch_0(E)).
(1)

Its standard double-tilt heart first uses the twisted sheaf slope and then the slope

nu_(B,tH)(E)=(tHch_2^B(E)-t^3H^3ch_0(E)/6)/(t^2H^2ch_1^B(E)),
(2)

with value +infty at zero denominator. Both torsion pairs use strictly positive smallest slope and nonpositive largest slope.

For K_X=O_X, the square-root-Todd charge is

Z_(B,tH)^(LV)(E)=-int_Xe^(-B-itH)ch(E)sqrt(td(X)),
(3)

where sqrt(td(X))=1+c_2(X)/24.

See also

Bridgeland Stability Condition, Canonical Bundle, Chern Character, Harder-Narasimhan Filtration, Tilted Heart, Todd Class

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References

Bayer, A. "Polynomial Bridgeland Stability Conditions and the Large Volume Limit." Geom. Topol. 13, 2389-2425, 2009. https://doi.org/10.2140/gt.2009.13.2389.Bayer, A.; Macrì, E.; and Stellari, P. "The Space of Stability Conditions on Abelian Threefolds, and on Some Calabi-Yau Threefolds." Invent. Math. 206, 869-933, 2016. https://doi.org/10.1007/s00222-016-0665-5.Bayer, A.; Macrì, E.; and Toda, Y. "Bridgeland Stability Conditions on Threefolds I: Bogomolov-Gieseker Type Inequalities." J. Alg. Geom. 23, 117-163, 2014. https://doi.org/10.1090/S1056-3911-2013-00617-7.

Cite this as:

Weisstein, Eric W. "Large-Volume Stability Condition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Large-VolumeStabilityCondition.html

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