Quot schemes and Fourier-Mukai transformation

Indranil Biswas; Umesh V. Dubey; Manish Kumar; A. J. Parameswaran

doi:10.1515/coma-2023-0152

Open Access Published by De Gruyter Open Access September 4, 2023

Quot schemes and Fourier-Mukai transformation

Indranil Biswas , Umesh V. Dubey , Manish Kumar and A. J. Parameswaran

From the journal Complex Manifolds

https://doi.org/10.1515/coma-2023-0152

Abstract

We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.

Keywords: quot scheme; Hilbert scheme; Fourier-Mukai transformation; symmetric product

MSC 2010: 14C05; 14L30

1 Introduction

Fourier-Mukai transformations arise in numerous contexts in algebraic geometry [2,4]. Over time, it has emerged to be an immensely useful concept. Here, we investigate Fourier-Mukai transformations in a particular context, namely, in the set-up of quot schemes. We show that the conservativity of the Fourier-Mukai transformation holds in the following cases:

Let M be an irreducible smooth projective variety over an algebraically closed field k such that dim M ≥ 2 . Denote by Hilb d ( M ) the Hilbert scheme parametrizing the zero-dimensional subschemes of M of length d . Let P M (respectively, P H ) be the projection to M (respectively, Hilb d ( M ) ) of the tautological subscheme S ⊂ M × Hilb d ( M ) . Let E and F be two vector bundles on M such that the vector bundles P H ∗ P M ∗ E and P H ∗ P M ∗ F on Hilb d ( M ) are isomorphic. Then, we show that E and F are isomorphic (see Proposition 2.1).
Let Q M (respectively, Q S ) be the projection to M (respectively, Sym d ( M ) ) of the tautological subscheme S ⊂ M × Sym d ( M ) . Let E and F be two vector bundles on M such that the vector bundles Q S ∗ Q M ∗ E and Q S ∗ Q M ∗ F on Sym d ( M ) are isomorphic. Then, we show that E and F are isomorphic (see Lemma 2.2).
Let C be an irreducible smooth projective curve defined over k . Fix a vector bundle E over C of rank at least two. Let Q d ( E ) denote the quot scheme parametrizing the torsion quotients of E of degree d . There is a tautological quotient Φ C ∗ E ⟶ Q over C × Q d ( E ) , where Φ C : C × Q d ( E ) ⟶ C is the natural projection. Let V and W be vector bundles on C such that the vector bundles Φ Q ∗ ( ( Φ C ∗ V ) ⊗ Q ) and Φ Q ∗ ( ( Φ C ∗ W ) ⊗ Q ) on Q d ( E ) are isomorphic, where Φ Q : C × Q d ( E ) ⟶ Q d ( E ) is the natural projection. Then, we show that E and F are isomorphic (see Proposition 2.3).

We also prove a similar result in the context of vector bundles on curves equipped with a group action (see Section 3).

A key method in our proofs is the Atiyah’s Krull-Schmidt theorem for vector bundles.

2 A Fourier-Mukai transformation

2.1 Vector bundles on Hilbert schemes

Let k be an algebraically closed field. Let M be an irreducible smooth projective variety over k such that dim M ≥ 2 . For any integer d ≥ 2 , let Hilb d ( M ) denote the Hilbert scheme parametrizing the zero-dimensional subschemes of M of length d . We have the natural projections:

M ← p M M × Hilb d ( M ) → p H Hilb d ( M ) .

There is a tautological subscheme

S ⊂ M × Hilb d ( M ) ,

such that for any z ∈ Hilb d ( M ) , the preimage p H − 1 ( z ) is the subscheme z ⊂ M . The restriction of p M (respectively, p H ) to S will be denoted by P M (respectively, P H ).

For any vector bundle E on M , we have the direct image P H ∗ P M ∗ E on S . We note that P H ∗ P M ∗ E is locally free because P H is a finite morphism and P M ∗ E is locally free. Let

(2.1) E ˜ ≔ P H ∗ P M ∗ E

be this vector bundle; its rank is d ⋅ rank ( E ) . It is known that two vector bundles E and F on M are isomorphic if E ˜ and F ˜ are isomorphic [3,5]. We will give a very simple proof of it.

Proposition 2.1

Let E and F be two vector bundles on M such that the corresponding vector bundles E ˜ and F ˜ on Hilb d ( M ) are isomorphic (see (2.1)). Then, E and F are isomorphic.

Proof

Since rank ( E ˜ ) and rank ( F ˜ ) are d ⋅ rank ( E ) and d ⋅ rank ( F ) , respectively, it follows that rank ( E ) = rank ( F ) . Let rank ( E ) = r = rank ( F ) .

Fix a zero-dimensional subscheme Z 0 ⊂ M of length d − 1 . Let Z red 0 = { x 1 , … , x b } ⊂ M be the reduced subscheme for Z 0 . The complement M \ Z red 0 = M \ { x 1 , … , x b } will be denoted by M 0 . Let

(2.2) ι : M 0 ⟶ M

be the inclusion map. We have a morphism

φ : M 0 ⟶ Hilb d ( M )

that sends any x ∈ M 0 to Z 0 ∪ { x } . The pullback φ ∗ E ˜ (respectively, φ ∗ F ˜ ) is isomorphic to ι ∗ E ⊕ V 0 (respectively, ι ∗ F ⊕ V 0 ), where V 0 is a trivial vector bundle on M 0 of rank ( d − 1 ) r and ι is the map in (2.2). The vector bundles φ ∗ E ˜ and φ ∗ F ˜ are isomorphic because E ˜ and F ˜ are isomorphic. So

(2.3) ι ∗ E ⊕ V 0 = ι ∗ F ⊕ V 0 .

There are no nonconstant functions on M 0 (recall that dim M ≥ 2 ). Hence, using [1, p. 315, Theorem 2(i)], from (2.3), it follows that ι ∗ E = ι ∗ F (see [6] for vast generalizations of [1]). Hence, we have

ι ∗ ι ∗ E = ι ∗ ι ∗ F .

But ι ∗ ι ∗ E (respectively, ι ∗ ι ∗ F ) is E (respectively, F ). This completes the proof.□

The line of arguments in Proposition 2.1 works in some other contexts. We will describe two such instances.

2.2 Vector bundles on symmetric product

As mentioned previously, M is an irreducible smooth projective variety of dimension at least two. For any integer d ≥ 2 , let Sym d ( M ) denote the quotient of M d under the action of the symmetric group S d that permutes the factors of the Cartesian product. We recall that Sym d ( M ) is a normal projective variety. There is a tautological subscheme

(2.4) S ⊂ M × Sym d ( M )

parametrizing all ( z , y ) ∈ M × Sym d ( M ) such that z ∈ y . Let

Q M : S ⟶ M and Q S : S ⟶ Sym d ( M )

be the natural projections. For any vector bundle E on M , the direct image

E ^ ≔ Q S ∗ Q M ∗ E

on Sym d ( M ) is locally free because Q S is a finite morphism and Q M ∗ E is locally free.

Lemma 2.2

Let E and F be two vector bundles on M such that the corresponding vector bundles E ^ and F ^ on Sym d ( M ) are isomorphic. Then, E and F are isomorphic.

Proof

Fix any z 0 = { x 1 , … , x d − 1 } ∈ Sym d − 1 ( M ) (repetitions are allowed). Let

ι : M 0 ≔ M \ z 0 ↪ M

be the inclusion map. We have a morphism

ϕ : M \ z 0 ⟶ Sym d ( M ) , x ⟼ { x , z } .

First, note that ϕ ∗ E ^ = ϕ ∗ F ^ because E ^ = F ^ . Evidently, we have ϕ ∗ E ^ = ( ι ∗ E ) ⊕ O M 0 ⊕ ( d − 1 ) ⋅ rank ( E ) and ϕ ∗ F ^ = ( ι ∗ F ) ⊕ O M 0 ⊕ ( d − 1 ) ⋅ rank ( F ) . Now, the argument in the proof of Proposition 2.1 goes through without any changes.□

2.3 Vector bundles on quot scheme

Let C be an irreducible smooth projective curve defined over k . Fix a vector bundle E over C of rank at least two. Fix an integer d ≥ 1 . Let Q d ( E ) denote the quot scheme parametrizing the torsion quotients of E of degree d . Let

(2.5) Φ C : C × Q d ( E ) ⟶ C and Φ Q : C × Q d ( E ) ⟶ Q d ( E )

be the natural projections. There is a tautological quotient

(2.6) Φ C ∗ E ⟶ Q

over C × Q d ( E ) whose restriction to any C × { Q } , where Q ∈ Q d ( E ) , is the quotient of E represented by Q .

Given a vector bundle V on C , we have the direct image

(2.7) F ( V ) ≔ Φ Q ∗ ( ( Φ C ∗ V ) ⊗ Q ) ⟶ Q d ( E ) ,

where Φ Q and Φ C are the projections in (2.5), and Q is the quotient in (2.6); this F ( V ) is a vector bundle because the support of Q is finite over Q d ( E ) .

Proposition 2.3

Let V and W be vector bundles on C such that the corresponding vector bundles F ( V ) and F ( W ) are isomorphic (see (2.7)). Then, V and W are isomorphic.

Proof

Since rank ( F ( V ) ) and rank ( F ( W ) ) are d ⋅ rank ( V ) and d ⋅ rank ( W ) respectively, from the given condition that F ( V ) and F ( W ) are isomorphic, we conclude that rank ( V ) = rank ( W ) . Let r denote rank ( V ) = rank ( W ) .

Let

(2.8) β : P ( E ) ⟶ X

be the projective bundle parametrizing the hyperplanes in the fibers of E . So P ( E ) = Q 1 ( E ) . For any z ∈ β − 1 ( x ) ⊂ P ( E ) , if H ( z ) ⊂ E x is the corresponding hyperplane, then the element of Q 1 ( E ) for z represents the quotient sheaf E ⟶ E x ∕ H ( z ) of E . For any z ∈ P ( E ) , the quotient sheaf map from E to the torsion quotient E x ∕ H ( z ) of E of degree 1 corresponding to z will be denoted by z .

Fix d − 1 distinct points x 1 , … , x d − 1 of X . Fix points y i ∈ β − 1 ( x i ) , 1 ≤ i ≤ d − 1 , where β is the projection in (2.8). The complement P ( E ) \ { y 1 , … , y d − 1 } will be denoted by P . Let

(2.9) ι : P ↪ P ( E )

be the inclusion map.

Note that the subset { y 1 , … , y d − 1 } defines a point of Q d − 1 ( E ) representing the quotient ⊕ j = 1 d − 1 y j of E ; this point of Q d − 1 ( E ) will be denoted by y . We have a morphism

Ψ : P ⟶ Q d ( E ) , z ⟼ z ⊕ y ;

recall that P ( E ) = Q 1 ( E ) and both y and z are the quotients of E .

Now, the vector bundle Ψ ∗ F ( V ) (respectively, Ψ ∗ F ( W ) ) is isomorphic to ( ι ∗ ( ( β ∗ V ) ⊗ O P ( E ) ( 1 ) ) ) ⊕ A (respectively, ( ι ∗ ( ( β ∗ W ) ⊗ O P ( E ) ( 1 ) ) ) ⊕ A ), where ι and β are the maps in (2.9) and (2.8), respectively, and A is a trivial vector bundle on P of rank r ( d − 1 ) ; the tautological line bundle on P ( E ) is denoted by O P ( E ) ( 1 ) .

Since V and W are isomorphic, we conclude that ( ι ∗ ( ( β ∗ V ) ⊗ O P ( E ) ( 1 ) ) ) ⊕ A and ( ι ∗ ( ( β ∗ V ) ⊗ O P ( E ) ( 1 ) ) ) ⊕ A are isomorphic. As there are no nonconstant functions on P , it follows that ( ι ∗ β ∗ V ) ⊗ O P ( E ) ( 1 ) and ( ι ∗ β ∗ W ) ⊗ O P ( E ) ( 1 ) are isomorphic. This implies that ι ∗ β ∗ V and ι ∗ β ∗ W are isomorphic.

The direct image ι ∗ ι ∗ β ∗ V (respectively, ι ∗ ι ∗ β ∗ W ) is β ∗ V (respectively, β ∗ W ). Hence, we conclude that β ∗ V and β ∗ W are isomorphic. So β ∗ β ∗ V = V is isomorphic to β ∗ β ∗ W = W .□

3 Action of group on a curve

Let C be an irreducible smooth projective curve, and let Γ be a finite group acting faithfully on C . Consider the quotient curve

(3.1) f : C ⟶ Y ≔ C ∕ Γ .

For any vector bundle V on Y , the pullback f ∗ V is a Γ -equivariant vector bundle on C .

The order of the group Γ is denoted by d . We have a morphism

(3.2) ρ : Y ⟶ Sym d ( C )

that sends any y ∈ Y to the element of Sym d ( C ) given by the scheme-theoretic inverse image f − 1 ( y ) , where f is the map in (3.1). To describe ρ explicitly, let { z 1 , z 2 , … , z n } be the reduced inverse image f − 1 ( y ) red . Then,

ρ ( y ) = ∑ i = 1 n b i z i ,

where b i is the order of the isotropy subgroup Γ z i ⊂ Γ of z i for the action of Γ on C . Note that ρ is an embedding.

The action of Γ on C produces an action of Γ on Sym d ( C ) . The action of any γ ∈ Γ sends any ( x 1 , … , x d ) ∈ Sym d ( C ) to ( γ ( x 1 ) , … , γ ( x d ) ) . We have

(3.3) ρ ( Y ) ⊂ Sym d ( C ) Γ .

We note that Sym d ( C ) is an irreducible smooth projective variety of dimension d . As in (2.4),

(3.4) S ⊂ C × Sym d ( C )

is the tautological subscheme parametrizing all ( c , x ) ∈ C × Sym d ( C ) such that c ∈ x . Let

(3.5) Q C : S ⟶ C and Q S : S ⟶ Sym d ( C )

be the natural projections. For any vector bundle E on C of rank r , the direct image

(3.6) E ^ ≔ Q S ∗ Q C ∗ E

is a vector bundle on Sym d ( C ) of rank d r .

We will describe an alternative construction of the vector bundle E ^ in (3.6). For 1 ≤ i ≤ d , let

p i : C d ⟶ C

be the projection to the i -th factor. Let

(3.7) P : C d ⟶ Sym d ( C )

be the quotient map for the action of the symmetric group S d that permutes the factors of C d . The action of S d on C d lifts to the vector bundle

E [ d ] ≔ ⊕ i = 1 d p i ∗ E ⟶ C d .

The action of S d on E [ d ] produces an action of S d on P ∗ E [ d ] , where P is the projection in (3.7). The vector bundle E ^ in (3.6) coincides with the S d -invariant part

( P ∗ E [ d ] ) S d ⊂ P ∗ E [ d ] .

The actions of Γ on C and Sym d ( C ) (see (3.3)) together produce a diagonal action of Γ on C × Sym d ( C ) . This action of Γ on C × Sym d ( C ) preserves the subscheme S in (3.4). For this action of Γ on S , the projections Q C and Q S in (3.5) are evidently Γ -equivariant.

Now, let E be a Γ -equivariant vector bundle on C . Since the projections Q C and Q S in (3.5) are Γ -equivariant, the vector bundle E ^ in (3.6) is also Γ -equivariant. From (3.3), it now follows that the vector bundle

(3.8) ρ ∗ E ^ ⟶ Y

is equipped with an action of Γ over the trivial action of Γ on Y .

Proposition 3.1

Let E and F be vector bundles on Y such that the corresponding Γ -equivariant vector bundles ρ ∗ f ∗ E ^ and ρ ∗ f ∗ F ^ on Y are isomorphic. Then, E and F are isomorphic.

Proof

The vector bundle f ∗ E has a natural action of Γ because it is pulled back from C ∕ Γ . The action of Γ on f ∗ E produces an action of Γ on f ∗ f ∗ E over the trivial action of Γ on Y . Similarly, Γ acts on f ∗ f ∗ F .

Consider Q S − 1 ( ρ ( Y ) ) ⊂ S , where Q S and ρ are the maps in (3.5) and (3.2), respectively. Let

Q C ′ ≔ Q C ∣ Q S − 1 ( ρ ( Y ) ) : Q S − 1 ( ρ ( Y ) ) ⟶ C

be the restriction of the map Q C in (3.5). It is straightforward to check that this map Q C ′ is an isomorphism. So we have the commutative diagram

(3.9)where the horizontal maps are isomorphisms. Moreover, all the maps in (3.9) are Γ -equivariant with Γ acting trivially on Y and ρ ( Y ) . Therefore, from (3.9), we conclude that there are isomorphisms

(3.10) f ∗ f ∗ E ⟶ ∼ ρ ∗ f ∗ E ^ and f ∗ f ∗ F ⟶ ∼ ρ ∗ f ∗ F ^

as Γ -equivariant vector bundles.

Since the Γ -equivariant vector bundles ρ ∗ f ∗ E ^ and ρ ∗ f ∗ F ^ are isomorphic, from (3.10), it follows that

(3.11) f ∗ f ∗ E ⟶ ∼ f ∗ f ∗ F

as Γ -equivariant vector bundles

Next, we will show that

(3.12) ( f ∗ f ∗ E ) Γ = E and ( f ∗ f ∗ F ) Γ = F .

To prove (3.12), first note that the action of Γ on C produces an action of Γ on f ∗ O C . The projection formula gives that

f ∗ f ∗ E ⟶ ∼ E ⊗ ( f ∗ O C ) .

The action of Γ on f ∗ O C and the trivial action of Γ on E together produce an action of Γ on E ⊗ ( f ∗ O C ) . The aforementioned isomorphism between f ∗ f ∗ E and E ⊗ ( f ∗ O C ) is evidently Γ -equivariant. Since ( f ∗ O C ) Γ = O Y , we conclude that (3.12) holds.

Finally, the proposition follows from (3.11) and (3.12).□

4 Alternative constructions

Let C be a smooth projective curve over k and E a vector bundle on C . Unlike in Section 2.3, E can be a line bundle; we no longer assume rank ( E ) to be at least two. As mentioned earlier, Q d ( E ) denotes the quot scheme that parametrizes the torsion quotients of E of degree d . Let

(4.1) γ : Q d ( E ) ⟶ Sym d ( C )

be the natural Chow morphism.

For any vector bundle V on C , consider the vector bundle F ( V ) on Q d ( E ) constructed in (2.7). We will describe its direct image γ ∗ F ( V ) on Sym d ( C ) , where γ is the map in (4.1).

For every 1 ≤ j ≤ d , let φ j : C d ⟶ C be the projection to the j -th factor. Take a vector bundle V on C . We have the vector bundle

(4.2) V ≔ ⊕ j = 1 d φ j ∗ ( V ⊗ E ) ⟶ C d .

The symmetric group S d acts on C d by permuting the factors of the tensor product (see Section 2.2). The corresponding quotient is Sym d ( C ) . As in (3.7), let

(4.3) P : C d ⟶ C d ∕ S d = Sym d ( C )

be the quotient map. The action of S d on C d has a natural lift to an action of S d on the vector bundle V in (4.2). This action of S d on V produces an action of S d on the direct image P ∗ V , where P is the projection in (4.3).

Lemma 4.1

The direct image γ ∗ F ( V ) on Sym d ( C ) , where F ( V ) and γ are as in (2.7) and (4.1), respectively, is naturally identified with the S d -invariant part

( P ∗ V ) S d ⊂ P ∗ V

for the aforementioned action of S d on P ∗ V .

Proof

There is a natural homomorphism

ϖ : F ( V ) ⟶ P ∗ V .

It is straightforward to check that ϖ ( F ( V ) ) ⊂ ( P ∗ V ) S d ⊂ P ∗ V and that the resulting homomorphism F ( V ) ⟶ ( P ∗ V ) S d is an isomorphism.□

Let

(4.4) Ψ C : C × Sym d ( C ) ⟶ C and Ψ S : C × Sym d ( C ) ⟶ Sym d ( C )

be the natural projections.

Consider ( Id C × γ ) ∗ Q on C × Sym d ( C ) , where Q is the sheaf in (2.6) and γ is the map in (4.1). Given a vector bundle V on C , we have the direct image

G ( V ) ≔ Ψ S ∗ ( Ψ C ∗ V ⊗ ( Id C × γ ) ∗ Q ) ⟶ Sym d ( C ) ,

where Ψ C and Ψ S are projections in (4.4).

Proposition 4.2

For any vector bundle V on C, there is a natural isomorphism

γ ∗ F ( V ) ≃ G ( V ) ,

where F ( V ) is constructed in (2.7).

Proof

Consider the following commutative diagram

Now, using the aforementioned commutative diagram, we can obtain the required isomorphism as follows:

□ γ ∗ F ( V ) = γ ∗ Φ Q ∗ ( ( Φ C ∗ V ) ⊗ Q ) ≃ Ψ S ∗ ( Id × γ ) ∗ ( ( Φ C ∗ V ) ⊗ Q ) = Ψ S ∗ ( Id × γ ) ∗ ( ( Id × γ ) ∗ ( Ψ C ∗ V ) ⊗ Q ) ≃ Ψ S ∗ ( ( Ψ C ∗ V ) ⊗ ( Id × γ ) ∗ Q ) (projection formula) = G ( V ) .

indranil29@gmail.com

Funding information: No funding was received for this manuscript.
Conflict of interest: There is no conflict of interests regarding this manuscript.

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Received: 2023-04-03

Accepted: 2023-08-09

Published Online: 2023-09-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Quot schemes and Fourier-Mukai transformation

Abstract

1 Introduction

2 A Fourier-Mukai transformation

2.1 Vector bundles on Hilbert schemes

Proposition 2.1

Proof

2.2 Vector bundles on symmetric product

Lemma 2.2

Proof

2.3 Vector bundles on quot scheme

Proposition 2.3

Proof

3 Action of group on a curve

Proposition 3.1

Proof

4 Alternative constructions

Lemma 4.1

Proof

Proposition 4.2

Proof

References

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