Height of exceptional collections and Hochschild cohomology of quasiphantom categories

We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective varietyX, a graded vector space which controls the restriction morphism from the Hochschild cohomology of X to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define a new invariant (the height) and show that the orthogonal to an exceptional collection of height h in the derived category of a smooth projective varietyX has the same Hochschild cohomology asX in degrees up to h 2. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions for the fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.

[1]  Christian Bohning,et al.  Determinantal Barlow surfaces and phantom categories , 2012, 1210.0343.

[2]  E. Shinder,et al.  Exceptional collections of line bundles on the Beauville surface , 2012, 1210.3339.

[3]  V. Alexeev,et al.  Derived categories of Burniat surfaces and exceptional collections , 2012, 1208.4348.

[4]  Christian Bohning,et al.  On the derived category of the classical Godeaux surface , 2012, 1206.1830.

[5]  A. Kuznetsov.  Base change for semiorthogonal decompositions , 2007, Compositio Mathematica.

[6]  A. Kuznetsov.  Hochschild homology and semiorthogonal decompositions , 2009, 0904.4330.

[7]  Birgit Richter.  HOMOTOPY ALGEBRAS AND THE INVERSE OF THE NORMALIZATION FUNCTOR , 2006 .

[8]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[9]  M. Larsen,et al.  Grothendieck ring of pretriangulated categories , 2004, math/0401009.

[10]  P. Seidel.  Homological mirror symmetry for the quartic surface , 2003, math/0310414.

[11]  L. Grunenfelder,et al.  COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE , 2002, math/0212124.

[12]  V. Drinfeld.  DG quotients of DG categories , 2002, math/0210114.

[13]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[14]  A. Bondal,et al.  HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES , 1994 .

[15]  Bernhard Keller,et al.  Deriving DG categories , 1994 .

[16]  Graeme Segal,et al.  Configuration-spaces and iterated loop-spaces , 1973 .