Algebraic vs. topological vector bundles joint with Jean Fasel and Mike Hopkins Aravind Asok (USC) October 27, 2020 Aravind Asok (USC) Algebraic vs. topological vector bundles 1 / 27 Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Notation and the basic question Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Question Can we characterize the image of Φr? Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Notation and the basic question Throughout: k a commutative ring, usually a field, frequently C. X is a smooth k-algebraic variety (Zariski top); Smk - category of such objects an if k = C, X is X(C) as a C-manifold (usual top); alg ∼ Vr (X) - =-classes of algebraic vb of rank r; top ∼ an Vr (X) - =-classes of C-vb of rank r on X If k ⊂ C, set: alg top Φr : Vr (X) −! Vr (X): Question Can we characterize the image of Φr? Aravind Asok (USC) Algebraic vs. topological vector bundles 2 / 27 Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Case. If X not affine: 9 a smooth affine C-variety X~ and a morphism m π : X~ ! X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); the morphism π is not a vector bundle projection; is topologically trivial. Warm-up: topology of smooth algebraic varieties Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a finite CW complex. Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 Case. If X not affine: 9 a smooth affine C-variety X~ and a morphism m π : X~ ! X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); the morphism π is not a vector bundle projection; is topologically trivial. Warm-up: topology of smooth algebraic varieties Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a finite CW complex. Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 the morphism π is not a vector bundle projection; is topologically trivial. Warm-up: topology of smooth algebraic varieties Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a finite CW complex. Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Case. If X not affine: 9 a smooth affine C-variety X~ and a morphism m π : X~ ! X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 Warm-up: topology of smooth algebraic varieties Assume X a smooth C-algebraic variety of dimension d: Theorem The space Xan has the homotopy type of a finite CW complex. Proof. n Case. If X is affine, i.e., X ⊂ C a closed subset; Morse theory (Andreotti–Frankel); in fact, X is a ≤ d-dim’l cell complex Case. If X not affine: 9 a smooth affine C-variety X~ and a morphism m π : X~ ! X that is Zariski locally trivial with fibers isomorphic to C (Jouanolou–Thomason); the morphism π is not a vector bundle projection; is topologically trivial. Aravind Asok (USC) Algebraic vs. topological vector bundles 3 / 27 I = incidence variety of hyperplanes vanishing on a line; π : P](V) ! P(V) is induced by projection (fibers are affine spaces) Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X~; π) with π : X~ ! X Zariski locally trivial with affine space fibers, and X~ smooth affine. Example n We call P](V) the standard Jouanolou device of projective space. If X ,! P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P . Projective varieties Example (Projective space) _ When X = P(V), set P](V) := P(V) × P(V ) n I Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 π : P](V) ! P(V) is induced by projection (fibers are affine spaces) Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X~; π) with π : X~ ! X Zariski locally trivial with affine space fibers, and X~ smooth affine. Example n We call P](V) the standard Jouanolou device of projective space. If X ,! P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P . Projective varieties Example (Projective space) _ When X = P(V), set P](V) := P(V) × P(V ) n I I = incidence variety of hyperplanes vanishing on a line; Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X~; π) with π : X~ ! X Zariski locally trivial with affine space fibers, and X~ smooth affine. Example n We call P](V) the standard Jouanolou device of projective space. If X ,! P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P . Projective varieties Example (Projective space) _ When X = P(V), set P](V) := P(V) × P(V ) n I I = incidence variety of hyperplanes vanishing on a line; π : P](V) ! P(V) is induced by projection (fibers are affine spaces) Aravind Asok (USC) Algebraic vs. topological vector bundles 4 / 27 Example n We call P](V) the standard Jouanolou device of projective space. If X ,! P is a closed subvariety, then we get a Jouanolou device for X by restricting the n standard Jounolou device for P . Projective varieties Example (Projective space) _ When X = P(V), set P](V) := P(V) × P(V ) n I I = incidence variety of hyperplanes vanishing on a line; π : P](V) ! P(V) is induced by projection (fibers are affine spaces) Definition If X is smooth k-algebraic variety, a Jouanolou device for X is (X~; π) with π : X~ ! X Zariski locally trivial with affine space fibers, and X~ smooth affine.