Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

By Arend Bayer, Emanuele Macrì, Yukinobu Toda. Journal of Algebraic Geometry, 2011

148 citations recorded by the source corpus.

<p>We apply a conjectured inequality on third Chern classes of stable two-term complexes on threefolds to Fujita’s conjecture. More precisely, the inequality is shown to imply a Reider-type theorem in dimension three which in turn implies that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript upper X Baseline plus 6 upper L"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>6</mml:mn> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K_X + 6L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is very ample when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is ample, and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5 upper L"> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">5L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is very ample when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript upper X"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is trivial.</p>

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e463a16dca52074d4e961e6ff06c0820b04a2af7

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Citations in the corpus

Stability conditions on Kuznetsov components (2017, methodology): “n be achieved in two steps: ﬁrst we start with ordinary slope-stability forcoherent sheaves, andtilttoobtain anewheart Cohβ(X); wethen useaweakstability condition on Cohβ(X), called tilt-stability in [BMT14], and tilt again to arrive at a situation where our general method applies; see Section 6. In”
On stability conditions for the quintic threefold (2018, methodology): “To give the accurate statement, we introduce some notions from [4,5,7] and briefly summarize the construction of stability conditions on a quintic threefold.”
A generalized Bogomolov–Gieseker inequality for the smooth quadric threefold (2013, background): “Then we obtain G = O(−1)[3] ⊕O[1]⊕4−c O[1].”
Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “Assuming a conjecture on ‘Bogomolov–Gieseker type inequalities’, Bayer, Macrì and Toda [8] construct Bridgeland stability conditions on D coh.”
Stability conditions on Fano threefolds of Picard number 1 (2015, methodology): “Combined with the results in [1] and [2], we get an open subset of geometric stability conditions on Stab(X).”
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, methodology): “It was shown that this conjecture would imply the existence of Bridgeland stability conditions,2 and, in the companion paper [6], a version of an open case of Fujita’s conjecture, on the very ampleness of adjoint line bundles on threefolds.”