Stability manifolds of varieties with finite Albanese morphisms

For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Néron–Severi rank one, and show that they are connected and contractible.

[1]  Christopher D. Hacon,et al.  Characterization of abelian varieties , 1999, math/9903184.

[2]  S. Mukai.  Semi-homogeneous vector bundles on an abelian variety , 1978 .

[3]  Yuchen Liu.  Stability conditions on product varieties , 2019, Journal für die reine und angewandte Mathematik (Crelles Journal).

[4]  Yukinobu Toda,et al.  Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[5]  Arend Bayer,et al.  The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds , 2014, 1410.1585.

[6]  Arend Bayer,et al.  Derived automorphism groups of K3 surfaces of Picard rank $1$ , 2013, 1310.8266.

[7]  Shinnosuke Okawa,et al.  Nonexistence of semiorthogonal decompositions and sections of the canonical bundle , 2015, 1508.00682.

[8]  Verzekeren Naar Sparen,et al.  Cambridge , 1969, Humphrey Burton: In My Own Time.

[9]  Yan Soibelman,et al.  Stability structures, motivic Donaldson-Thomas invariants and cluster transformations , 2008, 0811.2435.

[10]  D. Huybrechts,et al.  Stability conditions for generic K3 categories , 2006, Compositio Mathematica.

[11]  A. Bertram,et al.  Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture , 2011, Journal of Algebraic Geometry.

[12]  D. Arcara,et al.  Bridgeland-stable moduli spaces for $K$-trivial surfaces , 2007, Journal of the European Mathematical Society.

[13]  A. Polishchuk.  Phases of Lagrangian-invariant objects in the derived category of an abelian variety , 2012, 1203.2300.

[14]  A. Polishchuk.  Abelian Varieties, Theta Functions and the Fourier Transform , 2003 .

[15]  K. Hensel.  Journal für die reine und angewandte Mathematik , 1892 .

[16]  A. Maciocia,et al.  Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds , 2013, 1304.3887.

[17]  A. Polishchuk,et al.  Gluing Stability Conditions , 2009, 0902.0323.

[18]  Antony Maciocia,et al.  Fourier-Mukai transforms for K3 and elliptic fibrations , 1999 .

[19]  Mart'i Lahoz,et al.  Chern degree functions , 2021, Communications in Contemporary Mathematics.

[20]  Einzelwerken Muster,et al.  Invent , 2021, Encyclopedic Dictionary of Archaeology.

[21]  T. Bridgeland.  Stability conditions on triangulated categories , 2002, math/0212237.

[22]  B. M. Fulk.  MATH , 1992 .

[23]  Naoki Koseki.  Stability conditions on threefolds with nef tangent bundles , 2018, 1811.03267.

[24]  Chunyi Li.  The space of stability conditions on the projective plane , 2016, 1611.02087.

[25]  Arend Bayer.  A short proof of the deformation property of Bridgeland stability conditions , 2016, Mathematische Annalen.

[26]  Tom Bridgeland.  Equivalences of triangulated categories and Fourier-Mukai transforms , 1998 .

[27]  Chunyi Li.  On stability conditions for the quintic threefold , 2018, Inventiones mathematicae.

[28]  Constant families of t-structures on derived categories of coherent sheaves , 2006, math/0606013.

[29]  D. Huybrechts,et al.  The geometry of moduli spaces of sheaves , 1997 .

[30]  X. Roulleau.  Fano surfaces with 12 or 30 elliptic curves , 2010, 1001.4855.