the derived category of the cayley planekiem/yeosu/manivel.pdfthe derived category of the cayley...

The derived category of the Cayley plane

Laurent Manivel

Institut FourierCNRS / Grenoble University, France

Yeosu, February 19, 2013

Laurent Manivel (Institut Fourier CNRS / Grenoble University, France)The derived category of the Cayley plane Yeosu, February 19, 2013 1 / 16

Homogeneous spaces

Projective rational homogeneous spaces G/P are well-understood varieties.


Homogeneous spaces


Examples: Projective spaces, Grassmannians, flag varieties,


Homogeneous spaces


Examples: Projective spaces, Grassmannians, flag varieties,classical Grassmannians of isotropic spaces.


Homogeneous spaces



The Chow ring has a basis given by the classes of Schubert varieties, whoseinclusion ordering defines the Bruhat order.


Homogeneous spaces




There is an equivalence of categories

G− homogeneous vector bundles vs P− modules


Homogeneous spaces






Caveat: the parabolic subgroup P is NOT reductive.


Homogeneous spaces






Caveat: the parabolic subgroup P is NOT reductive.Its representations may be complicated.


Homogeneous spaces






Caveat: the parabolic subgroup P is NOT reductive.Its representations may be complicated.The (semi)simple ones have trivial action of the unipotent part.


Homogeneous spaces






Caveat: the parabolic subgroup P is NOT reductive.Its representations may be complicated.The (semi)simple ones have trivial action of the unipotent part.

Main Problem: Understand the derived categories of the G/P.


Derived categories

For X a smooth complex projective variety, let Db(X) denote the derivedcategory of coherent sheaves on X .


Derived categories

For X a smooth complex projective variety, let Db(X) denote the derivedcategory of coherent sheaves on X .Objects = Bounded complexes of coherent sheaves


Derived categories

For X a smooth complex projective variety, let Db(X) denote the derivedcategory of coherent sheaves on X .Objects = Bounded complexes of coherent sheavesMorphisms taken modulo homotopies + localization wrt quasi-isoms


Derived categories


Important invariant:


Derived categories



1 If X is Fano, Db(X) uniquely defines X . Moreover any self-equivalencecomes from Aut(X), Pic(X) and shifts.


Derived categories




2 There exist two non-isomorphic (non-birational) Calabi-Yau manifolds Yand Z such that Db(Y )≃ Db(Z ).


Derived categories




2 There exist two non-isomorphic (non-birational) Calabi-Yau manifolds Yand Z such that Db(Y )≃ Db(Z ).

Example: Projective spaces (Beilinson)

Db(Pn) = 〈O,O(1), . . . ,O(n)〉.


Exceptional collections

This is an example of a full exceptional collection

Db(X) = 〈E1, . . . ,EN〉.

Each object is exceptional and no Ext’s from right to left.




Db(X) = 〈E1, . . . ,EN〉.

Each object is exceptional and no Ext’s from right to left.Strongly exceptional collection if no higher Ext’s from left to right.




Db(X) = 〈E1, . . . ,EN〉.

Each object is exceptional and no Ext’s from right to left.Strongly exceptional collection if no higher Ext’s from left to right.Then the derived category can be encoded in a quiver.




Db(X) = 〈E1, . . . ,EN〉.

Each object is exceptional and no Ext’s from right to left.Strongly exceptional collection if no higher Ext’s from left to right.Then the derived category can be encoded in a quiver.Moreover the images of E1, . . . ,EN in K-theory give a basis.




Db(X) = 〈E1, . . . ,EN〉.


Beilinson’s theorem has been extended to Grassmannians by Kapranov usingresolutions of diagonals (here N is a number of partitions),




Db(X) = 〈E1, . . . ,EN〉.


Beilinson’s theorem has been extended to Grassmannians by Kapranov usingresolutions of diagonals (here N is a number of partitions), and also to quadricsusing Clifford algebras.




Db(X) = 〈E1, . . . ,EN〉.



Conjecture (folklore). Any G/P has a full exceptional collection.




Db(X) = 〈E1, . . . ,EN〉.



Conjecture (folklore). Any G/P has a full exceptional collection.

Conjecture (Catanese). Any G/P has a full strongly exceptional collectioncompatible with the Bruhat order.


Lefschetz decompositions

Let OX (1) be a very ample line bundle on X . A Lefschetz decomposition of thederived category is a semi-orthogonal decomposition

Db(X) = 〈A0,A1(1), . . . ,Am(m)〉

where Am ⊂ ·· · ⊂ A1 ⊂ A0 ⊂ Db(X) are subcategories.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉

where Am ⊂ ·· · ⊂ A1 ⊂ A0 ⊂ Db(X) are subcategories.More interesting if m is big. Not that m < index(X) by adjunction.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉


Conjecture. Any G/P has a minimal Lefschetz decomposition.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉



Theorem (Fonarev 2011). Any Grassmannian has a minimal Lefschetzdecomposition.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉




Theorem (Kuznetsov-Polischuk 2012). Any classical G/P has an exceptionalcollection of the expected length.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉





In particular the images generate the full K-theory.




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉





In particular the images generate the full K-theory. But fullness? Somephantom category? (Recently proved to exist on some special surfaces.)




Db(X) = 〈A0,A1(1), . . . ,Am(m)〉





In particular the images generate the full K-theory. But fullness? Somephantom category? (Recently proved to exist on some special surfaces.)Moreover semisimple bundles are not expected to be enough.


The main result

The Cayley plane is the simplest homogeneous space of exceptional type(except small examples for G2).


The main result


Automorphism group: the adjoint group of type E6.


The main result


Automorphism group: the adjoint group of type E6.Dimension 16, Picard number 1, index 12.


The main result


Automorphism group: the adjoint group of type E6.Dimension 16, Picard number 1, index 12.Minimal equivariant embedding in P(C27).


The main result


Automorphism group: the adjoint group of type E6.Dimension 16, Picard number 1, index 12.Minimal equivariant embedding in P(C27).Chow ring of rank 27 (lines on the cubic surface!)


The main result



Theorem (M., Faenzi-M.)

The derived category of the Cayley plane has a minimal Lefschetzdecomposition, which also gives a strongly exceptional collectionof length 27.


The main result




The derived category of the Cayley plane has a minimal Lefschetzdecomposition, which also gives a strongly exceptional collectionof length 27. This collection is constructed from only two simplehomogeneous vector bundles, of rank 10 and 54.


The main result




The derived category of the Cayley plane has a minimal Lefschetzdecomposition, which also gives a strongly exceptional collectionof length 27. This collection is constructed from only two simplehomogeneous vector bundles, of rank 10 and 54.

Best possible situation!


Severi varieties

The Cayley plane is best known as the largest of the four Severi varieties.


Severi varieties


Theorem (Zak, Lazarsfeld)

Let Σ⊂ PN be smooth n-dimensional, linearly non-degenerate.


Severi varieties




If 3n > 2N−2, then Sec(Σ) = PN .


Severi varieties





If 3n = 2N−2, same conclusion except if Σ is oneof the four Severi varieties


Severi varieties






v2(P2), P

2×P2, G(2,6), E6/P1.


Severi varieties






v2(P2), P

2×P2, G(2,6), E6/P1.

Consequence: a smooth low-codimensional subvariety of PN−1 cannot be aprojection. (The case of surfaces was treated by Severi.)


Plane projective geometry over A

The Severi varieties have dimensions 2,4,8,16.



The Severi varieties have dimensions 2,4,8,16.They can be interpreted as projective planes over the four (complexified)normed algebras

A= R,C,H,O.




A= R,C,H,O.

Consider the Jordan algebra of Hermitian matrices

H3(A) =

{

r z yz s xy x t

}

.




A= R,C,H,O.


H3(A) =

{

r z yz s xy x t

}

.

Three types of nonzero matrices:




A= R,C,H,O.


H3(A) =

{

r z yz s xy x t

}

.


rank three: generic




A= R,C,H,O.


H3(A) =

{

r z yz s xy x t

}

.


rank three: generic

rank two: Det = 0, cubic hypersurface C ⊂ PH3(A)




A= R,C,H,O.


H3(A) =

{

r z yz s xy x t

}

.


rank three: generic

rank two: Det = 0, cubic hypersurface C ⊂ PH3(A)

rank one: Severi variety Σ= Sing(C), and C = Sec(Σ).


Hermitian matrices of rank one “are” Hermitian projections onto lines


Hermitian matrices of rank one “are” Hermitian projections onto lines⇒ we consider Σ as a projective plane over A and denote it AP2.



Facts (Freudenthal-Tits, 50’s):

1. The stabilizer of C is a semisimple complex Lie group G





2. H3(A) is an irreducible representation of G






3. G has only three orbits in PH3(A) (≃ “rank”)







4. In particular Σ is G-homogeneous.








For A=H, Σ=HP2 = G(2,6) and C is the Pfaffian cubic.









For A=O, Σ=OP2 ⊂ P

26 is the Cayley plane









For A=O, Σ=OP2 ⊂ P

26 is the Cayley plane and C is the Cartan cubic(first written down by Elie Cartan in terms of tritangent planes to a smoothcubic surface).


Projective lines over O

Projective lines .



Projective lines .A projective plane is covered by a family of lines parametrized by a dual plane.



Projective lines .A projective plane is covered by a family of lines parametrized by a dual plane.Here lines =OP

1 = Q8 = eight dimensional quadrics,




1 = Q8 = eight dimensional quadrics, parametrized by a copyof OP

2 ⊂ (P26)∨.





2 ⊂ (P26)∨.

P26

∪C = Sec(OP

2)∪

OP2 ⊃ Qp ←−

(P26)∨

∪C∨ = (OP

2)∗

∪p ∈OP

2





2 ⊂ (P26)∨.

P26

∪C = Sec(OP

2)∪

OP2 ⊃ Qp ←−

(P26)∨

∪C∨ = (OP

2)∗

∪p ∈OP

2

Alternative definition :OP

2 is a Severi variety





2 ⊂ (P26)∨.

P26

∪C = Sec(OP

2)∪

OP2 ⊃ Qp ←−

(P26)∨

∪C∨ = (OP

2)∗

∪p ∈OP

2


2 is a Severi variety⇒ its secant variety C is very degenerate, ie the entryloci are big.





2 ⊂ (P26)∨.

P26

∪C = Sec(OP

2)∪

OP2 ⊃ Qp ←−

(P26)∨

∪C∨ = (OP

2)∗

∪p ∈OP

2


2 is a Severi variety⇒ its secant variety C is very degenerate, ie the entryloci are big. If x ∈ C is smooth, its entry locus is a line Qx ≃OP

1 in OP2.


Plane projective geometry over O

Facts : The axioms of a plane projective geometry are satisfied.




1 Two (generic) points in OP2 are connected by a unique line ≃OP

1.





1.

2 Two (generic) such lines meet at a unique point.





1.


The main vector bundle . S = N(−1) = twisted normal bundle of the Cayleyplane:





1.


The main vector bundle . S = N(−1) = twisted normal bundle of the Cayleyplane: globally generated simple homogeneous vector bundle, with

H0(OP2,S) = H3(O).





1.



H0(OP2,S) = H3(O).

Dually, S∨ ⊂OP2×H3(O)∨ is such that S∨p is the linear span of the quadric

Qp.





1.



H0(OP2,S) = H3(O).


Qp. Hence a splittingSym2(S) = O(1)⊕S2.





1.



H0(OP2,S) = H3(O).


Qp. Hence a splittingSym2(S) = O(1)⊕S2.

The bundle S2 is simple of rank 54.


The exceptional collection

Fact : The bundles S and S2 are both exceptional.




Theorem

The derived category Db(OP2) has full exceptional collection

O,S ,S2,O(1),S(1),S2(1),O(2),S(2),S2(2),O(3),S(3), . . . ,O(11),S(11).




Theorem


O,S ,S2,O(1),S(1),S2(1),O(2),S(2),S2(2),O(3),S(3), . . . ,O(11),S(11).

Lefschetz decomposition:

A0 = A1 = A2 = 〈O,S ,S2〉,

A3 = · · ·= A11 = 〈O,S〉.




Theorem


O,S ,S2,O(1),S(1),S2(1),O(2),S(2),S2(2),O(3),S(3), . . . ,O(11),S(11).


A0 = A1 = A2 = 〈O,S ,S2〉,

A3 = · · ·= A11 = 〈O,S〉.

Easy part: exceptionality. Follows from the Borel-Weil-Bott theorem andrepresentation theory of SO10.




Theorem


O,S ,S2,O(1),S(1),S2(1),O(2),S(2),S2(2),O(3),S(3), . . . ,O(11),S(11).


A0 = A1 = A2 = 〈O,S ,S2〉,

A3 = · · ·= A11 = 〈O,S〉.

Easy part: exceptionality. Follows from the Borel-Weil-Bott theorem andrepresentation theory of SO10.

Difficult part: fullness.


Strategy of the proof

Two essential ingredients:




1. Produce more bundles from the exceptional collection.





This requires to construct suitable complexes.






2. Deduce fullness by restricting to a covering family of subvarieties.







This will use the plane geometry over O through restriction to projective lines.







This will use the plane geometry over O through restriction to projective lines.Since O-lines are quadrics their derived category are known from the work ofKapranov.







This will use the plane geometry over O through restriction to projective lines.Since O-lines are quadrics their derived category are known from the work ofKapranov. In fact we will use their hyperplane sections and

Db(Q7) = 〈O,O(1),Σ(1),O(2), . . . ,O(6)〉

where Σ denotes the spinor bundle (rank eight).


First step: a complex

We construct by hand a self-dual long exact sequence

0→ S2(−3)→ S(−2)⊕27→ O(−1)⊕378→ O⊕378→ S

⊕27→ S2→ 0.




0→ S2(−3)→ S(−2)⊕27→ O(−1)⊕378→ O⊕378→ S

⊕27→ S2→ 0.

Twisting by O(3), this implies that S2(3) belongs to the category

D ⊂ Db(OP2)

generated by the exceptional collection.




0→ S2(−3)→ S(−2)⊕27→ O(−1)⊕378→ O⊕378→ S

⊕27→ S2→ 0.


D ⊂ Db(OP2)

generated by the exceptional collection.Then we deduce that many twisted wedge powers of S also belong to D .




0→ S2(−3)→ S(−2)⊕27→ O(−1)⊕378→ O⊕378→ S

⊕27→ S2→ 0.


D ⊂ Db(OP2)

generated by the exceptional collection.Then we deduce that many twisted wedge powers of S also belong to D .

In the end we want to prove that

D = Db(OP2).


Second step: restriction to quadrics

Recall that the bundle S is generated by global sections. A general section sdoes not vanish.



Recall that the bundle S is generated by global sections. A general section sdoes not vanish.The quotient bundle Ss = S/Cs is still globally generated, a general section s′

defines a line through s in H∨, which cuts the Cartan cubic in three points.





Consequence: The zero locus of s′ is the disjoint union of three smoothquadrics of dimension seven.






These quadrics cover the Cayley plane, hence we can control everything byrestricting to them.






These quadrics cover the Cayley plane, hence we can control everything byrestricting to them. We need:

the spinor bundle: S|Q7 ≃ O⊕O(1)⊕Σ.








the Koszul complex: know that enough wedge powers of S belong to D .









Conclusion: An object E in Db(OP2) which is orthogonal to D will restrict to

zero on any of our 7-dim quadrics.










zero on any of our 7-dim quadrics.But since they cover the Cayley plane thisimplies that E = 0.










zero on any of our 7-dim quadrics.But since they cover the Cayley plane thisimplies that E = 0. Hence D = Db(OP

2).


Application to cubic sevenfoldsWhich cubic hypersurfaces are linear sections of the Cartan cubic?



1 dimension ≥ 9 : only singular cubics




2 dimension 8 : some rational cubics





3 dimension ≤ 7 : general cubics






Theorem (Iliev-M.)

A general seven dimensional cubic X can be realized as a linear section of theCartan cubic, in a finite number of different ways.






Theorem (Iliev-M.)


The derived category of X has a semiorthogonal decomposition

Db(X) = 〈AX ,OX , . . . ,OX (5)〉






Theorem (Iliev-M.)



Db(X) = 〈AX ,OX , . . . ,OX (5)〉

where AX is a three-dimensional Calabi-Yau category (Kuznetsov).






Theorem (Iliev-M.)



Db(X) = 〈AX ,OX , . . . ,OX (5)〉

where AX is a three-dimensional Calabi-Yau category (Kuznetsov).The restriction of S to X defines a spherical object in AX .






Theorem (Iliev-M.)



Db(X) = 〈AX ,OX , . . . ,OX (5)〉

where AX is a three-dimensional Calabi-Yau category (Kuznetsov).The restriction of S to X defines a spherical object in AX . By Thomas-Seidel,this induces an auto-equivalence of AX (not extending to Db(X)!).






Theorem (Iliev-M.)



Db(X) = 〈AX ,OX , . . . ,OX (5)〉

where AX is a three-dimensional Calabi-Yau category (Kuznetsov).The restriction of S to X defines a spherical object in AX . By Thomas-Seidel,this induces an auto-equivalence of AX (not extending to Db(X)!).The “dual” section of OP

2 is a Fano of CY type (Iliev-M.).Laurent Manivel (Institut Fourier CNRS / Grenoble University, France)The derived category of the Cayley plane Yeosu, February 19, 2013 16 / 16

the derived category of the cayley planekiem/yeosu/manivel.pdfthe derived category of the cayley...

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