Homological projective duality

(Alexander Kuznetsov!)

Beilinson

DpPn´1q “@O,Op1q, . . . ,Opn ´ 1q

D

“@Dpptq, Dpptq, . . . , Dpptq

D.

The notation x y means a semi-orthogonal decomposition:

§ They are semi-orthogonal: Ext˚pOpiq,Opjqq “ 0 for i ą j ,

§ Opiq is exceptional: R HompOpiq,Opiqq “ C. id. Equivalently,

Dpptq Ñ DpPn´1q

C ÞÑ Opiq

is an embedding,

§ The sheaves O,Op1q, . . . ,Opn ´ 1q generate DpPn´1q.

Sketch proofThe first two conditions are simple cohomology computations.

Generation: Since O,Op1q, . . . ,Opn ´ 1q are exceptional andsemi-orthogonal the Gram-Schmidt process shows that any objectof DpPn´1q is an extension of a piece in their span and a piece intheir orthogonal.

Thus we want to show their orthogonal is zero.

Use the Koszul resolution associated to the section

Opx0,x1,...,xn´1q // Op1q‘n

with zero locus one point tpu. Shows that Op is in the span ofO,Op1q, . . . ,Opn ´ 1q for any point p P Pn´1.

Thus any object F in the orthogonal has

F |p – R HompOp, F qrn ´ 1s “ 0

for any p P Pn´1. Therefore F “ 0 by the Nakayama lemma.

Family version

Fix

§ X “ PpE q πÝÑ B , a Pn´1-bundle, where

§ E is a rank n vector bundle over B .

Orlov proved that

DpX q “A

DpBq, DpBqp1q, . . . , DpBqpn ´ 1qE

.

Here the ith copy of DpBq is embedded via π˚p ¨ q b Opiq, whereOp´1q ãÑ π˚E is the tautological bundle on X .

Blow ups

Blow up in Z Ă X of codimension-n (both smooth):

E

p

��

� � j // BlZ X

π

��Z

� � i // X

Orlov also proved

DpBlZ X q “A

DpX q, DpZ qp´E q, . . . , DpZ qp´pn ´ 1qE qE

.

Here

§ DpX q is included via π˚ and DpZ q by j˚p˚

§ p : E Ñ Z is a Pn´1-bundle

§ its Op1q line bundle is Op´E q

so this is an analogue of his projective bundle result.

Universal hyperplaneSet up:

§ variety and line bundle pX ,OX p1qq§ basepoint free linear system V Ď H0pOX p1qq

Equivalently

§ map f : X Ñ PpV ˚q such that§ im f not contained in any hyperplane

Over the dual projective space PpV q we have the universal familyof hyperplanes H Ñ PpV q:

H :“␣

px , sq : spxq “ 0(

Ă X ˆ PpV q.

Discriminant locus is classical projective dual X _ Ă PpV q of X .

Given a linear subspace L Ď V with annihilator LK Ď V ˚, set

XLK :“ X ˆPpV ˚q PpLKq,

HL :“ H ˆPpV q PpLq.

Projective duality

Baselocus XLK of L Ď H0pOX p1qq is contained in every fibre ofHL Ñ PpLq, giving a diagram

XLK ˆ PpLq

p

��

� � j // HL

π

��

� � ι // X ˆ PpLq

ρyyttttttttttt

XLK� � i // X

Notice that

§ π has general fibre P`´2 p` :“ dim Lq§ over XLK the fibre is PpLq “ P`´1

E.g. ` “ 2 then HL Ñ X is blow up in codimension-two XLK Ă X .

Homological projective duality I

Suppose that dim XLK “ dim X ´ `.

The above diagram gives an inclusion of the derived category ofXLK into that of the universal hypersurface HL over the linearsystem:

j˚p˚ : D`XLK

˘ãÝÑ DpHLq.

π˚ is also full and faithful. Together these give a semi-orthogonaldecomposition

DpHLq “A

D`XLK

˘, π˚DpX qp0, 1q, . . . , π˚DpX qp0, ` ´ 1q

E.

pi , jq is twist by OX piq b OPpV qpjq (restricted to HL Ă X ˆ PpLq).

The “interesting part” of the derived category

Want to make DpHq smaller; more comparable to DpX q.

Cut it down to its “interesting part” (Kuznetsov).

Standard example: start with DpPn´1q “@O, . . . ,Opn ´ 1q

D.

Restrict to degree-d hypersurface H Ă X , then Opdq, . . . ,Opn ´ 1qremains exceptional semi-orthogonal collection on restriction to H.

Define “interesting part” of DpHq to be its right orthogonal:

CH :“@OHpdq,OHpd ` 1q, . . . ,OHpn ´ 1q

DK

“␣E P DpHq : R HompOpiq, E q “ 0 for i “ d , . . . , n ´ 1

(.

So

DpHq “@CH ,OHpdq,OHpd ` 1q, . . . ,OHpn ´ 1q

D.

Amazingly, CH is always a fractional Calabi-Yau category.

Examples

Interesting examples include pd “ 2, n even) and pd “ 3, n “ 6q:§ Even dimensional quadrics. CH is generated by two spinor

bundles which are exceptional and orthogonal to each other;

CH – Dppt \ ptq.

Families ù double covers of linear systems of quadrics.

§ Cubic fourfolds. CH is the derived category of a K3 surface,noncommutative in general.

2-dimensional CY category RHompE , F q˚ – RHompF , E qr2s,deformation of DpK3q.

“Explains” the Beauville-Donagi holomorphic symplectic form onthe Fano variety F pHq of lines in H. F pHq is a moduli space ofobjects πCH

pILq P CH so inherits Mukai’s symplectic structure.

More later!

Families

Put categories CH Ă DpHq together over the P`H0pOPn´1pdq

˘

family of all Hs. So:

§ Set V “ H0pOPn´1pdqq§ For L Ď V , with universal hypersurface HL Ă Pn´1 ˆ PpLq,

CHL:“

@DpPpLqqpd , 0q, DpPpLqqpd`1, 0q, . . . , DpPpLqqpn´1, 0q

DK

so that DpHLq is

@CHL

, DpPpLqqpd , 0q, DpPpLqqpd ` 1, 0q, . . . , DpPpLqqpn ´ 1, 0qD

§ Putting L “ V gives CH, the HP dual of pPn´1,Opdqq.

Soon we will refine HPD I by replacing DpHLq with CHL.

Lefschetz collections

For general pX ,OX p1qq Kuznetsov replaces the Beilinsondecomposition with a (rectangular) Lefschetz decomposition:

§ an admissible subcategory A Ď DpX q§ generating a semi-orthogonal decomposition

DpX q “@A,Ap1q, . . . ,Api ´ 1q

D.

(Above example is pX ,OX p1qq “ pPn´1,OPn´1pdqq andA “

@O,Op1q, . . . ,Opd ´ 1q

Dwith i “ n{d . If d ­ |n can use

non-rectangular Lefschetz collection.)

Setting CH :“@Ap1q,Ap2q, . . . ,Api ´ 1q

DK Ă DpHq (forH P |OX p1q|) to be the interesting part of DpHq gives

DpHq “@CH ,Ap1q,Ap2q, . . . ,Api ´ 1q

D.

Lefschetz collections and families

Similarly for X Ñ PpV ˚q and L Ď V we set

CHL:“

@Ap1q b DpPpLqq, . . . , Api ´ 1q b DpPpLqq

DK

so that DpHLq is

@CHL

, DpPpLqqpd , 0q, DpPpLqqpd ` 1, 0q, . . . , DpPpLqqpn ´ 1, 0qD

Putting L “ V gives CH, the HP dual of X Ñ PpV q with the(rectangular) Lefschetz decomposition DpX q “

@A, . . . ,Api ´ 1q

D.

(Kuznetsov asks for CH to be geometric: i.e. to have aDpPpV qq-linear equivalence CH – DpY q for some Y Ñ PpV q.)

Picturing HPD I

DpHLq “@D`XLK

˘, π˚DpX qp0, 1q, . . . , π˚DpX qp0, ` ´ 1q

D

Api ´1,1q

Api ´1,

`´1qAp1,`´1q

Ap1, 1q

Ap0,`´1q

Ap0, 1q

D`XLK

˘

Picturing interesting part of DpHLq

CHL“@Ap1q b DpPpLqq, . . . , Api ´ 1q b DpPpLqq

DK

p0, 1q

p1, 0q pi ´2,0q pi ´1,0q

pi ´1,1q

pi ´1,

`´1qp1, `´1q

p1, 1q

p0, `´1q

D`XLK

˘

HPD IIProjecting (“mutating”) the grey boxes into the white boxes infirst column we find

Theorem (Kuznetsov)

§ If ` ą i thenCHL

“@D`XLK

˘, Ap0, 1q, Ap0, 2q, . . . , Ap0, ` ´ iq

D

§ If ` “ i then CHL– D

`XLK

˘

§ If ` ă i thenD`XLK

˘“@CHL

, Ap1, 0q, Ap2, 0q, . . . , Api ´ `, 0qD

Looks like a duality now! X Ñ PpV ˚q and its HP dualCH{DpPpV qq have similar size and are on the same footing.

Passing to a codimension-` linear section,

§ lose the first ` copies of Ap ¨ q§ gain (restriction to XLK of) the PpLq family of categories CH

as H runs through the hyperplanes containing XLK

Example I (with Addington)

Cubic fourfold X Ă P5. Hodge diamond

11

0 1 21 1 011

Generically H2,2pX ,Zq “ xh2y, but special X have an extra classT P H2,2

primpX ,Zq.

Hassett determined when the integral Hodge structurexh2, T yK Ă H4pX ,Zq is isometric to (Tate twist of) H2

primpS ,Zqfor some polarised K3 surface S .

Kuznetsov: DpX q –@CX ,O,Op1q,Op2q

Dwhere CX is a

noncommutative K3 and sometimes really – DpK3q.

We showed (more-or-less) that X is Hassett ðñ Kuznetsov.

Cubic fourfolds containing a planeCubic fourfold X Ă P5 containing a plane P Ă X Ă P5.Defines another plane

P2 :“␣3-planes P Ă P3 Ă P5

(

Such a 3-plane intersects X in a singular cubic surface P Y Q.

ñ P2 family of quadric surfaces Q; in fact

BlP X Ñ P2

is a quadric surface fibration, generic fibre P1 ˆ P1, singular fibres(cone over a conic) over discriminant sextic curve Ă P2.

S :“␣rulings of fibres

(

����P2

is the double cover corresponding to the 2 objects in Cfibre.(Branched over discriminant sextic curve.)

S Q s parameterises the sheaves ι˚IL on X , where L is any line inthe ruling corresponding to the point s P S .

Projecting into CX Ă DpX q, find S is a moduli space of objects inCX . Gives

Theorem (Kuznetsov)

Universal object on S ˆ X gives an equivalenceDpSq „ÝÑ CX Ă DpX q.

(Modulo Brauer class issues.)

We use this description and deformation theory to reach all otherdivisors of special cubic fourfolds (where a different description ofCX – DpK3q holds).

Example II (with Calabrese)

Two ways to get a Calabi-Yau 3-fold from a pencil of cubic 4-folds:

§ X “ X3,3 Ă P5 is baselocus of the pencil, i.e. the intersectionof two cubic fourfolds,

§ Y Ñ P1 is (noncommutative) K3 fibration associated to theuniversal cubic fourfold over the pencil P1.

Choose special cubic fourfolds so Y geometric (commutative).Then HPD predicts

DpX q “ DpY q.

Sheaf on Y ù sheaf on each cubic fourfold fibre ù sheaf onbaselocus X (contained in each fibre)

Derived (auto)equivalences of CY 3-folds

Pencil special ñ X , Y both singular.

For cubic fourfolds containing a plane we were able to resolvecrepantly to give non-birational Calabi-Yau 3-folds pX , pY with

DppX q – DppY q.

For cubic fourfolds with an ODP we were able to resolve crepantlyto give birational Calabi-Yau 3-folds pX , pY with an autoequivalence

DppX q ý .

Example III (with Segal)

The Grassmannian of 2-planes in C2n,

Gr p2, 2nq Ă P`Λ2C2n

˘

has classical projective dual the Pfaffian variety of degenerate2-forms (rank ď 2n ´ 2) on C2n:

Pf p2n ´ 2, 2nq Ă P`Λ2pC2nq˚˘.

Pf is a (singular) degree-n hypersurface␣ω P Λ2pC2nq˚ : ω^n “ 0

(.

Kuznetsov conjectures one can pass to a “categorical crepantresolution” so that Pf Ñ P˚ becomes HP dual to Gr Ñ P.

Taking L Ă Λ2C2n such that PpLKq Ă P`Λ2pC2nq˚

˘misses the

singularities of Pf, would get relation

D`Gr X PpLq

˘ÐÑ D

`Pf X PpLKq

˘

Quintic 3-fold

We prove this under some conditions by different methods(“LG models” or matrix factorisations)

Taking ` “ dim L “ 40, for instance, gives full and faithfulembeddings

D`Pf X P4

˘ãÝÑ D

`Gr X P39

˘.

LHS is a quintic 3-fold.Beauville shows the general quintic 3-fold is Pfaffian.

RHS is Fano 11-fold, linear section of Gr(2,10).