Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities

By Arend Bayer, Emanuele Macrì, Yukinobu Toda. 2011

88 citations recorded by the source corpus.

We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a BogomolovGieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples. Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5d70bb9f69a3001020929ac4eda325a8ea5d2c8

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Citations in the corpus

On stability conditions for the quintic threefold (2018, methodology): “Under the framework of [4,5,7], the heart A of the stability condition is constructed by ‘double-tilting’ Coh(X).”
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, methodology): “In [11], the first two authors and Yukinobu Toda, also based on discussions with Aaron Bertram, proposed a general approach towards the construction of stability conditions on a smooth projective threefold X .”
Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture (2011, methodology): “A Bogomolov-Gieseker-type inequality on Chern classes of “tilt-stable” objects in the derived category of a threefold was conjectured in [BMT11] in the context of constructing Bridgeland stability conditions.”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “…limit t → 0 up to rescaling, we obtain a vector w 0 · H,β with components If we similarly take the limit t → + ∞ , we obtain a vector w ∞· H,β with components We will also use the following two observations several times; for Lemma 9.3 see, e.g., [8, Section 7.2]; for Lemma 9.4 see [6, Lemma 5.9].”
Stability conditions on Fano threefolds of Picard number 1 (2015, methodology): “Based on the previous works [1], [2] and [3] on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D(X).”
A generalized Bogomolov–Gieseker inequality for the smooth quadric threefold (2013, background): “); 14J30, 18E30 (Secondary). Key words and phrases. Bridgeland stability conditions, Derived category, BogomolovGieseker inequality. 1 2 BENJAMIN SCHMIDT threefolds is due to Bayer, Macrì and Toda in [BMT14]. It was conﬁrmed to work for P3 in [Mac12] and for principally polarized abelian threefolds ”