P𝑟−1 𝑏 ∈𝑌 P𝑟−2 𝑏 ∈ 𝐵\𝑌
(3.3.10)
and its restriction to the preimage of 𝑌 is the projectivization of the normal bundle N𝑌|𝐵 ' E |𝑌, hence𝑌e' P(E |𝑌).
Theorem 3.3.5. (Orl03, Proposition 2.10) Let the notation be the one of Diagram 3.3.9. Then there exists the following Lefschetz semiorthogonal decomposition for 𝐷𝑏(𝑀):
𝐷𝑏(𝑀) =h𝑗∗𝑝∗𝐷𝑏(𝑌), 𝑞∗𝐷𝑏(𝐵) ⊗𝐿 , . . . , 𝑞∗𝐷𝑏(𝐵) ⊗𝐿⊗(𝑟−1)i. (3.3.11)
3.4 Homological projective duality and related constructions
Introduced by Kuznetsov in (Kuz07), Homological Projetive Duality (HPD) is one of the most useful tools for the manipulation of de-rived categories of coherent sheaves and semiorthogonal decpomposi-tion. Given a smooth variety 𝑋 and a morphism 𝑓 : 𝑋 −→ P(𝑉) to a projective space, the main idea is to construct the derived category of the universal hyperplane sectionHP(𝑉), 𝑓 associated to 𝑓, and determine a subategory C ⊂ 𝐷𝑏(HP(𝑉), 𝑓) compatible with a given semiorthogo-nal decomposition of 𝐷𝑏(𝑋 ×P(𝑉)). If C is geometrical, i.e. if there exists a variety 𝑌 such that 𝐷𝑏(𝑌) ' C, we say that 𝑌 is the homo-logically projective dual to 𝑋. Moreover, linear sections of 𝑋 and 𝑌 by mutually orthogonal hyperplanes share interesting properties. The
scope of homological projecitve duality has been pushed much further by introducing the notion of categorical joins, allowing to prove de-rived equivalences of intersection of varieties of the type described in (OR17; BCP20; Man17). In this section we will give a brief survey of HPD.
3.4.1 Universal hyperplane sections and HPD
Let us first define the notion of universal hyperplane section, which will be useful also for different purposes in Chapter 10.
Definition 3.4.1. Let 𝑋 be a smooth variety and𝑉∨ ⊂ 𝐻0(𝑋 ,O𝑋(1)) a vector space of dimension𝑛+1. Fix a morphism 𝑓 : 𝑋 −→ P(𝑉). We calluniversal hyperplane sectionof𝑋with respect to 𝑓 the fiber product H𝑋 , 𝑓 :=𝑋 ×P(𝑉) 𝐹(1, 𝑛, 𝑉).
Remark 3.4.2. The variety 𝐹(1, 𝑛, 𝑉) is often called incidence quadric of P(𝑉) × P(𝑉∨). In fact, one has 𝐹(1, 𝑛, 𝑉) = {(𝑥 , 𝑦) ∈ P(𝑉) × P(𝑉∨) : 𝑦(𝑥) = 0} which characterizes such flag variety as a quadric hypersurface in P(𝑉) ×P(𝑉∨). Note that for any vector space 𝑉, the flag variety 𝐹(1, 𝑛, 𝑉) is the universal hyperplane section of P(𝑉). Let us momentarily specialize Definition 3.4.1 to the case where𝑉∨= 𝐻0(𝑋 ,O𝑋(1)), so that 𝑓 is a proper embedding. In that case, one gets a more geometric characterization of the universal hyperplane sec-tion:
H𝑋 ={(𝑥 , 𝜎) ∈ 𝑋 ×P(𝐻0(𝑋 ,O𝑋(1))) |𝜎(𝑥) =0}. (3.4.1) Now, suppose that 𝑋 has a Lefschetz semiorthogonal decomposition
with respect to an ample line bundle O𝑋(1) which satisfies O𝑋(1) = 𝑓∗OP(𝑉)(1):
𝐷𝑏(𝑋) =hA0,A1(1), . . . ,A𝑚−1(𝑚−1)i (3.4.2) By a simple application of Theorem 3.3.1, one has the following semiorthogonal decomposition:
𝐷𝑏(𝑋×P(𝑉∨)) =hA0(0)𝐷𝑏(P(𝑉∨)), . . . ,A𝑚−1(𝑚−1)𝐷𝑏(P(𝑉∨))i. (3.4.3) This semiorthogonal decomposition is 𝐷𝑏(P(𝑉∨))-linear by construc-tion.
Definition 3.4.3. Let 𝑋 be a smooth variety and𝑉∨ ⊂ 𝐻0(𝑋 ,O𝑋(1)) a vector space of dimension𝑛. Fix a morphism 𝑓 : 𝑋 −→P(𝑉)and suppose there exists a Lefschetz decomposition as in Equation 3.4.2. Supppose 𝑛 > 𝑚. Then we callhomological projective dual categoryto𝐷𝑏(𝑋)the categoryCappearing in the following semiorthogonal decomposition:
𝐷𝑏(H𝑋 , 𝑓) =hC,A1(1)𝐷𝑏(P(𝑉∨)), . . . ,A𝑚−1(𝑚−1)𝐷𝑏(P(𝑉∨))i. (3.4.4) The existence of C is a simple matter of definition, as it appears in Equation 3.4.4 as semiorthogonal complement of the blocks of the formA𝑖(𝑖)𝐷𝑏(P(𝑉∨)). However, the existence of the semiorthogonal decomposition 3.4.4 is a nontrivial statement (Kuz07, Lemma 5.3).
While C can be an interesting object by its own right, we are mainly
interested in the situation when C is geometrical, i.e. it is the derived category of coherent sheaves of a variety:
Definition 3.4.4. Let 𝑓 : 𝑋 −→ P(𝑉) be a morphism defined on a smooth variety 𝑋 to a projective spaceP(𝑉) as above and let C be the HPD category of𝑋 with respect to 𝑓. Let𝑌 be a variety with a morphism 𝑓 :𝑌 −→ P(𝑉∨). We say that𝑌 ishomologically projective dualto 𝑋 if there exists an equivalence of categories 𝜙 : 𝐷𝑏(𝑌) −→ C which is P(𝑉∨)-linear.
As we mentioned above, homological projective duality behaves well under the action of taking linear sections. Let us consider anadmissible linear subspace 𝐿 ⊂𝑉∨, i.e. a subspace such that 𝑋𝐿 := 𝑋×P(𝑉)P(𝐿⊥) and𝑌𝐿 :=𝑌×P(𝑉∨)P(𝐿⊥) have dimension respectively dim(𝑋) −dim(𝐿) and dim(𝑌) +dim(𝐿) −𝑛−1. Then, one has the following theorem:
Theorem 3.4.5. (Kuz07, Theorem 6.3) Let 𝑓 : 𝑋 −→P(𝑉)and𝑔 :𝑌 −→
P(𝑉∨) be as above, with𝑌 homologically projective dual to 𝑋. Let 𝑋 have a semirothogonal decomposition as in Equation 3.4.2 where we call A𝑘 = h𝔞𝑘, . . . ,𝔞𝑚−1ifor0 ≤ 𝑘 ≤ 𝑚−1, once defined𝔞𝑘 to be the right orthogonal ofA𝑘+1inA𝑘. Then the following statements are true:
1. 𝑌 is smooth and it admits the following semiorthogonal decompo-sition for some positive integer𝑙:
𝐷𝑏(𝑌) =hB𝑙−1(1−𝑙), . . . ,B1(−1),B0i (3.4.5) where we defined the blockB𝑘 =h𝔞0, . . . ,𝔞𝑛−𝑘−2i.
2. For every admissible linear subspace 𝐿 ⊂ 𝑉∨ of dimension𝑟 one
has the following semiorthogonal decompositions:
𝐷𝑏(𝑋𝐿) = hC𝐿,A𝑟(1), . . . ,A𝑚−1(𝑚−𝑟)i
𝐷𝑏(𝑌𝐿) = hB𝑙−1(𝑛+1−𝑟 −𝑙), . . . ,B𝑛−𝑟(−1),C𝐿i
(3.4.6)
3.4.2 HPD for projective bundles
An interesting setting where to apply Theorem 3.4.5 can be given by choosing 𝑋 =P(E) whereE is a vector bundle over a smooth projective variety 𝐵. Let𝑉∨ ⊂ 𝐻0(𝐵,E) be a space of sections generating E, so that we have a surjective morphism of vector bundles on 𝐵:
𝑉∨⊗ O −→ E −→0. (3.4.7) Hence, we can construct a morphism of projective varieties:
𝑓 : 𝑋 −→P(𝑉). (3.4.8)
Then, one can find a semiorthogonal decomposition for 𝑋 by Theorem 3.3.1, and this gives rise to a (naturally P(𝑉)-linear) semiorthogonal decompposition for 𝑋 × P(𝑉). It turns out (Kuz07, Section 8) that a nice semiorthogonal decomposition can also be found for H𝑋 , 𝑓, which proves that the homological projective dual category of 𝑋 is always geometric and it is the derived category of another projective bun-dle.
Lemma 3.4.6. (Kuz07, Lemma 8.1) Let E −→ 𝐵be a vector bundle of rank 𝑟 over a smooth projective variety of dimension 𝑛 and let 𝑉∨ ⊂ 𝐻0(𝐵,E) be an 𝑁-dimensional vector space of sections which generates E. Call𝑋 =P(E) and 𝑓 : 𝑋 −→ P(𝑉) as above. Then the homological
projective dual variety of 𝑋 has codimension𝑟 insideP(𝑉) × 𝐵and it is given by𝑌 =P(E⊥), where we call:
E⊥ =ker(𝑉 ⊗ O −→ E)∨. (3.4.9) An accurate choice of rank and dimension of the space of sections can provide examples of homologically projective self-dual projective bundles.
Proposition 3.4.7. The projective bundleP(∧3𝑇(−3)) overP5is homo-logically projective self-dual.
Proof. Let us fix P5 = P(𝑉6). The vector bundle ∧3𝑇(−3) has rank 10, it is globally generated and it has a space of global sections 𝐻0(P5,∧3𝑇(−3)) = ∧3𝑉∨
6. This comes from the fact that the dual Euler sequence reads:
0−→ O (−1) −→𝑉∨
6 ⊗ O −→𝑇(−1) −→0 (3.4.10) and its third symmetric power gives:
0−→ O (−3) −→𝑉∨
6 ⊗ O (−2) −→ ∧2𝑉∨
6 ⊗ O (−1) −→
−→ ∧3𝑉∨
6 ⊗ O −→ ∧3𝑇(−3) −→0
(3.4.11) from which we get the surjection
∧3𝑉∨
6 ⊗ O −−→ ∧𝑔 3𝑇(−3) −→0 (3.4.12) with kernel ∧2𝑇(−2).
Hence, 𝑔 gives rise to a morphism 𝑓 :P(∧3𝑇(−3)) −→P(∧3𝑉∨
6). In light of Equation 3.4.12, an application of Lemma 3.4.6 yields that the homological projective dual of P(∧3𝑇(−3)) is ∧2𝑇∨(2), but one has
∧2𝑇∨(2) ' ∧3𝑇(2) ⊗det(𝑇∨) ' ∧3𝑇(−3).
3.5 Derived equivalence and birational
We are interested in the following two cases:
◦ If 𝑓∗
Observe that any birational map 𝜇 fitting in a diagram like 3.5.1 such that X1 and X2 have trivial canonical bundle is a 𝐾-equivalence.
A natural question, justified by a multitude of positive examples and by the lack of counterexamples, is to ask whether 𝐾-equivalence implies 𝐷-equivalence, and whether a 𝐾-inequality implies an embedding of categories. In fact, this is known as the 𝐷 𝐾 conjecture, which we state here below:
Conjecture 3.5.1. (BO02; Kaw02) The following statements are true: