Derived categories and cubic persurfaces Paolo Stellari - PowerPoint PPT Presentation

Derived categories and cubic hy- Derived categories and cubic persurfaces Paolo Stellari hypersurfaces Paolo Stellari Roma, February 2011

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari Outline

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari 3 -folds 2 Geometry Outline Derived categories Bridgeland stability conditions Irrationality

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari 3 -folds 2 Geometry Outline Derived categories Bridgeland stability conditions Irrationality 4 -folds 3 Geometry Derived categories Categorical Torelli theorem Fano varieties of lines Rationality

Aim Derived categories The aim of the talk is to propose a ‘categorical’ treatment for and cubic hy- persurfaces some fundamental (often unknown) geometric properties of Paolo Stellari smooth (complex) hypersurfaces of degree 3 The Y ⊆ P n + 1 . geometric setting 3 -folds We will study cubic 3 -fold ( n = 3) and cubic 4 -fold ( n = 4). Geometry Derived categories Bridgeland stability conditions For example: Irrationality 4 -folds Geometry Rationality/irrationality of those varieties; Derived categories Categorical Torelli theorem Torelli type theorems; Fano varieties of lines Rationality Geometric description of the Fano varieties of lines of those cubics.

� � � The definition Derived Let A be an abelian category (e.g., mod - R , right categories and cubic hy- R -modules, R an ass. ring with unity, and Coh ( X ) ). persurfaces Define C b ( A ) to be the (abelian) category of bounded Paolo Stellari complexes of objects in A . In particular: The geometric Objects: setting → M p − 1 d p − 1 → M p d p 3 -folds → M p + 1 − M • := {· · · − − − − − → · · · } Geometry Derived categories Morphisms: sets of arrows f • := { f i } i ∈ Z making Bridgeland stability conditions Irrationality commutative the following diagram 4 -folds Geometry Derived categories d i − 2 d i − 1 d i + 1 d i M • � M i − 1 M • � M i + 1 � M i � · · · Categorical Torelli M • M • · · · theorem Fano varieties of lines f i − 1 f i f i + 1 Rationality d i − 2 d i − 1 d i d i + 1 L • � L i − 1 L • � L i + 1 � · · · L • � L i L • · · ·

The definition For a complex M • ∈ C b ( A ) , its i -th cohomology is Derived categories and cubic hy- persurfaces H i ( M • ) := ker ( d i ) Paolo Stellari im ( d i − 1 ) ∈ A . The geometric setting A morphism of complexes is a quasi-isomorphism (qis) if it 3 -folds induces isomorphisms on cohomology. Geometry Derived categories Bridgeland stability conditions Irrationality Definition 4 -folds The bounded derived category D b ( A ) of the abelian Geometry Derived categories category A is such that: Categorical Torelli theorem Fano varieties of Objects: Ob ( C b ( A )) = Ob ( D b ( A )) ; lines Rationality Morphisms: (very) roughly speaking, obtained ‘by inverting qis in C b ( A ) ’.

Semi-orthogonal decompositions Derived Suppose we have a sequence of full triangulated categories and cubic hy- subcategories T 1 , . . . , T n ⊆ D b ( X ) := D b ( Coh ( X )) , where X persurfaces is smooth projective, such that: Paolo Stellari Hom D b ( X ) ( T i , T j ) = 0, for i > j , The geometric setting For all K ∈ D b ( X ) , there exists a chain of morphisms in 3 -folds Geometry D b ( X ) Derived categories Bridgeland stability conditions Irrationality 0 = K n → K n − 1 → . . . → K 1 → K 0 = K 4 -folds Geometry with cone ( K i → K i − 1 ) ∈ T i , for all i = 1 , . . . , n . Derived categories Categorical Torelli theorem Fano varieties of lines This is a semi-orthogonal decomposition of D b ( X ) : Rationality D b ( X ) = � T 1 , . . . , T n � .

Derived categories and Fano varieties Derived categories Theorem (Bondal–Orlov) and cubic hy- persurfaces Let X be a smooth projective complex Fano variety and Paolo Stellari assume that Y is a smooth projective variety such that The geometric D b ( X ) ∼ = D b ( Y ) . setting 3 -folds Then X ∼ Geometry = Y . Derived categories Bridgeland stability conditions Irrationality Thus, if Y is a cubic hypersurface as above, then D b ( Y ) is a 4 -folds Geometry too strong invariant. Derived categories Categorical Torelli theorem Fano varieties of Question lines Rationality Does some ‘piece’ in a semi-orthogonal decomposition of D b ( Y ) behave nicely?

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari 3 -folds 2 Geometry The geometric Derived categories setting Bridgeland stability conditions 3 -folds Geometry Irrationality Derived categories Bridgeland stability conditions Irrationality 4 -folds 3 4 -folds Geometry Geometry Derived categories Derived categories Categorical Torelli theorem Categorical Torelli theorem Fano varieties of lines Fano varieties of lines Rationality Rationality

First properties Derived Let Y ⊆ P 4 be a smooth cubic 3-fold. The following are categories and cubic hy- persurfaces classical results: Paolo Stellari Torelli Theorem (Clemens–Griffiths, Tyurin) The geometric Let Y 1 and Y 3 be cubic 3-folds. Then Y 1 ∼ = Y 2 if and only if setting 3 -folds the intermediate Jacobians ( J ( Y 1 ) , Θ 1 ) and ( J ( Y 2 ) , Θ 2 ) are Geometry isomorphic. Derived categories Bridgeland stability conditions Irrationality 4 -folds Geometry Theorem (Clemens–Griffiths) Derived categories Categorical Torelli theorem Cubic 3-folds are not rational. Fano varieties of lines Rationality Use that J ( Y ) does not decompose as direct sum of Jacobians of curves.

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari 3 -folds 2 Geometry The geometric Derived categories setting Bridgeland stability conditions 3 -folds Geometry Irrationality Derived categories Bridgeland stability conditions Irrationality 4 -folds 3 4 -folds Geometry Geometry Derived categories Derived categories Categorical Torelli theorem Categorical Torelli theorem Fano varieties of lines Fano varieties of lines Rationality Rationality

The decomposition Derived Let Y ⊆ P 4 be a smooth cubic 3-fold. categories and cubic hy- persurfaces Paolo Stellari Theorem (Kuznetsov) The The derived category D b ( Y ) has a semi-orthogonal geometric setting decomposition 3 -folds Geometry Derived categories D b ( Y ) = � T Y , O Y , O Y ( 1 ) � . Bridgeland stability conditions Irrationality 4 -folds Geometry The subcategory T Y is highly non-trivial and cannot be the Derived categories Categorical Torelli derived category of a smooth projective variety. theorem Fano varieties of lines T Y ∼ Indeed the Serre functor S T Y is such that S 3 = [ 5 ] . So T Y Rationality is a so called Calabi–Yau category of fractional dimension 5 3 .

Categorical Torelli Derived categories and cubic hy- persurfaces Question (Kuznetsov) Paolo Stellari Given two cubic 3-folds Y 1 and Y 2 , is it true that Y 1 ∼ = Y 2 if and only if T Y 1 ∼ The = T Y 2 ? geometric setting 3 -folds Geometry Derived categories Theorem (Bernardara–Macr` ı–Mehrotra–S.) Bridgeland stability conditions Irrationality The answer to the above question is positive. 4 -folds Geometry Derived categories Categorical Torelli theorem Idea: realize the Fano variety of lines of Y i as moduli space Fano varieties of lines of stable objects according to a Bridgeland stability Rationality condition on T Y i .

Outline Derived categories The geometric setting and cubic hy- 1 persurfaces Paolo Stellari 3 -folds 2 Geometry The geometric Derived categories setting Bridgeland stability conditions 3 -folds Geometry Irrationality Derived categories Bridgeland stability conditions Irrationality 4 -folds 3 4 -folds Geometry Geometry Derived categories Derived categories Categorical Torelli theorem Categorical Torelli theorem Fano varieties of lines Fano varieties of lines Rationality Rationality

The definition Derived categories and cubic hy- A stability condition on a triangulated category T is a pair persurfaces σ = ( Z , P ) where Paolo Stellari The Z : K ( T ) → C is a linear map called central charge geometric (similar to the slope for sheaves); setting 3 -folds Geometry P ( φ ) ⊂ T are full additive subcategories for each φ ∈ R Derived categories Bridgeland stability ( semistable objects of phase φ ) conditions Irrationality 4 -folds satisfying some compatibilities. Geometry Derived categories Categorical Torelli theorem The minimal objects in P ( φ ) are called stable objects . Fano varieties of lines Rationality Stab ( T ) is the space parametrizing stability conditions on T .