A twisted nonabelian Hodge correspondence

A twisted nonabelian Hodge correspondence Alberto García-Raboso University of Toronto March 15, 2014 Workshop on Algebraic Varieties Fields Institute 1 / 28

Outline 1 The (classical) nonabelian Hodge theorem 2 Twisted vector bundles The descent-type definition Adding connections Adding Higgs fields Main theorem (the cocycle picture) 3 Towards a cocycle-free version Codifying twisted vector bundles Codifying connections and Higgs fields Main theorem 4 Sketch of proof The Hodge correspondence for gerbes Generalization to other groups 2 / 28

The (classical) nonabelian Hodge theorem Connections and Higgs fields X smooth complex projective variety. E, F holomorphic vector bundles of rank n on X. Flat connections Higgs fields C-linear map r : E!E 1 X satisfying the Leibniz rule: r(fv) =v df + f rv O X -linear map : F!F 1 X. Curvature C(r): Curvature C( ): E r! E 1 r X! E 2 X F! F 1 X! F 2 X Flatness: C(r) =0. Flatness: C( )=0. 3 / 28

The (classical) nonabelian Hodge theorem Connections and Higgs fields X smooth complex projective variety. E, F holomorphic vector bundles of rank n on X. Flat -connections C-linear map r : E!E 1 X satisfying the -twisted Leibniz rule: r(fv) = v df + f rv =1 Flat connections =0 Higgs fields Curvature C(r): E r! E 1 X r! E 2 X Flatness: C(r) =0. 3 / 28

The (classical) nonabelian Hodge theorem Families of flat -connections 4 / 28

The (classical) nonabelian Hodge theorem Families of flat -connections 4 / 28

The (classical) nonabelian Hodge theorem Higgs bundles and local systems Theorem (Simpson 92) There is a fully faithful functor E : v.b. of rank n on X (E, r) r : flat connection,! (F, ) F : v.b. of rank n on X : Higgs field with essential image given by those semistable Higgs bundles with A Higgs bundle (F, ch 1 (F) [!] dim CX 1 = ch 2 (F) [!] dim CX 2 =0 8F 0 F: ) is semistable if (F 0 ) F 0 1 X =) µ(f 0 ) apple µ(f) Also: G linear algebraic group / C: Tannaka duality a version for principal G-bundles. 5 / 28

Twisted vector bundles The descent-type definition Choose: a cover U = {U i } i2i of X. Proposition A vector bundle on X is a collection E = {E i } i2i, g = {g ij } i,j2i of vector bundles E i on U i and isomorphisms g ij : E j U ij!e i U ij satisfying g ii = id Ei, g ij = g 1 ji and g ij g jk g ki =id Ei 6 / 28

Twisted vector bundles The descent-type definition Choose: a cover U = {U i } i2i of X, and a 2-cocycle = { ijk }2Ž 2 (U, O X ); let =[ ] 2 H2 (X, O X ). Definition An -twisted vector bundle on X is a collection E = {E i } i2i, g = {g ij } i,j2i of vector bundles E i on U i and isomorphisms g ij : E j U ij!e i U ij satisfying g ii = id Ei, g ij = g 1 ji and g ij g jk g ki = ijk id Ei 6 / 28

Twisted vector bundles Three remarks Torsion of 2 H 2 (X, O X ): E, g - -twisted vector bundle of rank n + det(g ij )det(g jk )det(g ki )= n ijk m det g 2 Č 1 (U, O X ) and Projectivizing a twisted vector bundle: Lifting P n Čech det g = n E, g - -twisted vector bundle of rank n + PE, Pg - honest P n 1 -bundle 1 -bundles: H 1 X, GL n (O X )! H 1 X, PGL n (O X )! H 2 X, O X 7 / 28

Twisted vector bundles Adding connections Choose: Let: satisfy:! = {! ij }2Č1 (U, 1 X ), and F = {F i }2Č0 (U, 2 X ). E, g -twisted vector bundle, and r = {r i } i2i, where r i is a connection on E i C(r i )=F i 2 (U i, End OX (E i ) 2 X ), and r i g ij r j g 1 ij =! ij 2 (U ij, End OX (E i ) 1 X ). Compatibility conditions: df i =0 F i F j = d! ij ()! ik =! ij +! jk + d log ijk,!, F 2 Ž 2 (U, dr X ) i dr X ho := d log X! 1 d X! 2 d X! 8 / 28

Twisted vector bundles Adding Higgs fields Choose: Let: satisfy:! 0 = {! ij 0 }2Č1 (U, 1 X ), and F 0 = {F 0 i }2Č 0 (U, 2 X ). F, g 0 0 -twisted vector bundle, and = { i } i2i, where i is a (fat) Higgs field on F i C( i )=Fi 0 2 (U i, End OX (F i ) 2 X ), and i gij 0 j(gij 0 ) 1 =! ij 0 2 (U ij, End OX (F i ) 1 X ). Compatibility conditions: F 0 i F 0 j =0! 0 ik =!0 ij +!0 jk () 0,! 0, F 0 2 Ž 2 (U, Dol X ) i Dol X ho := 0 X! 1 0 X! 2 0 X! 9 / 28

Twisted vector bundles A little bookkeeping... Definition 2 Ž 2 U, O X U-G m -gerbe over X E, g Basic vector bundle on,!, F 2 Ž 2 U, dr X U-G m -gerbe with flat connection over X E, g, r Basic vector bundle on,!, F 0,! 0, F 0 2 Ž 2 U, Dol X Higgs U-G m -gerbe over X F, g 0, Basic vector bundle on 0,! 0, F 0 10 / 28

Twisted vector bundles Main theorem (the cocycle picture) Theorem (G-R) Given a U-G m -gerbe with flat connection over X,,!, F, there is a Higgs U-G m -gerbe over X, 0,! 0, F 0, such that we have a fully faithful functor Basic v.b. E, g, r of rank n on,!, F,! Basic v.b. F, g 0, of rank n on 0,! 0, F 0 Conversely, given a Higgs U-G m -gerbe over X, 0,! 0, F 0, there is a U-G m -gerbe with flat connection over X,,!, F, such that the same conclusion holds. 11 / 28

Towards a cocycle-free version Why reformulate? Four reasons: Is this statement independent of choices? What is the essential image? Why is it natural to allow for central twistings? ot a particularly beautiful statement... 12 / 28

Towards a cocycle-free version Principal G-bundles G linear algebraic group / C (e.g., GL n, PGL n, SL n ). Definition 1: locally a cartesian product I A fiber bundle P! X of complex-analytic manifolds, with fiber G I equipped with a reduction of structure group to G Aut G. Definition 2: total space with an action of the group I A complex-analytic manifold P I equipped with a free, proper, holomorphic right G-action () X := P/G is a complex-analytic manifold). Definition 3: morphism from the base to the classifying stack I A complex-analytic manifold X I equipped with a holomorphic map X! BG. P p X [P] EG ' BG 13 / 28

Towards a cocycle-free version Principal G-bundles G linear algebraic group / C (e.g., GL n, PGL n, SL n ). Definition 2 : total space with an action of the group I A complex-analytic manifold P I equipped with a holomorphic right G-action () X := [P/G] is a complex-analytic 1-stack). Definition 2 : total space with an action of the group I A complex-analytic 1-stack P I equipped with a holomorphic right G-action. Definition 3 : morphism to the classifying stack I A complex-analytic 1-stack P I equipped with a holomorphic map X! BG. P p X [P] EG ' BG 14 / 28

Towards a cocycle-free version Principal G-bundles G linear algebraic group / C (e.g., GL n, PGL n, SL n ). Definition (Principal G-bundle, or G-torsor) Bun G := G St ' St /BG Objects: 0 Bun G (X) = H 1 (X, G). Morphisms: P X G-equiv. Q Map G (P, Q) ' Map /BG (X, Y) Y ' Map(X, Y) h Map(X,BG),[P] [P] BG [Q] 15 / 28

Towards a cocycle-free version Principal BA-bundles A abelian linear algebraic group / C (e.g., G m, G a, µ n ) ) BA abelian group stack / C. Definition (Principal BA-bundle, or A-gerbe) Bun BA := BA St ' St /B2 A Objects: 0 Bun BA (X) = H 2 (X, A). Morphisms: X X BA-equiv. B 2 A Y Map BA ( X, Y) ' Map /B A(X, Y) 2 Y ' Map(X, Y) h Map(X,B 2 A), ikolaus, Schreiber, Stevenson, Principal 1-bundles - General theory, arxiv:1207.0248 16 / 28

Two G m -gerbes: Towards a cocycle-free version Vector bundles on G m -gerbes 2 H 2 (X, O X ) 1! G m! GL n! PGL n! 1 X p X B 2 G m BGL n p BPGL n B 2 G m Proposition Basic vector bundles of rank n on X ' Map BGm X, BGL n ' Map(X, BPGL n ) h Map(X,B 2 G m), 17 / 28

Towards a cocycle-free version The de Rham stack Definition X dr := (X X )^ X The de Rham stack of X codifies connections: ppal.g-bundles on X Map(X dr, BG) ' with flat connection Map(X dr, B 2 Gm-gerbes over X G m ) '! H with flat connection 2 X, dr X Proposition Basic vector bundles of rank n on (X dr ) ' Map BGm (X dr ), BGL n ' Map(X dr, BPGL n ) h Map(X dr,b 2 G m), 18 / 28

Towards a cocycle-free version The Dolbeault stack Definition X Dol := (TX )^0 X The Dolbeault stack of X codifies Higgs fields: ppal.g-bundles on X Map(X Dol, BG) ' with Higgs field Map(X Dol, B 2 G m ) '{Higgs G m-gerbes over X}! H 2 X, Dol X Proposition Basic vector bundles of rank n on 0(X Dol ) ' Map BGm 0(X Dol), BGL n ' Map(X Dol, BPGL n ) h Map(X Dol,B 2 G m), 0 19 / 28

Towards a cocycle-free version Main theorem Theorem (G-R) Given 2 Map(X dr, B 2 G m ) there exists 0 2 Map(X Dol, B 2 G m ) such that we have a fully faithful functor Map BGm (X dr ), BGL n,! Map BGm 0(X Dol), BGL n Given 0 2 Map(X Dol, B 2 G m ) there is 2 Map(X dr, B 2 G m ) such that the same conclusion holds. 20 / 28

Sketch of proof The obvious approach Map BGm (X dr ), BGL n Map(X dr, BPGL n ) Map(X dr, B 2 G m ) nah ' '? Map(X Dol, BPGL n ) ss,0 Map(X Dol, B 2 G m ) Map BGm 0(X Dol), BGL n pss 0 21 / 28

Sketch of proof Hodge correspondence for gerbes Proposition (G-R) If A ' G m a F, we have Map X dr, B 2 A ' Map X Dol, B 2 A For A = G a, classical abelian Hodge theory. For A = F, the connection and Higgs data are trivial (f =0). For A = G m : Map X dr, B 2 G m Map X Dol, B 2 G m ' ' Map X dr, BG m 6' Map X Dol, BG m 22 / 28

Sketch of proof Torsion to the rescue Map BGm (X dr ), BGL n e Map(X dr, BPGL n ) Map(X dr, B 2 G m ) Map(X dr, B 2 µ n ) nah ' ' Map(X Dol, B 2 µ n ) Map(X Dol, BPGL n ) ss,0 Map(X Dol, B 2 G m ) Map BGm 0(X Dol), BGL n pss e 0 0 23 / 28

Sketch of proof Some homotopy theory... Map BGm (X dr ), BGL n e Map(X dr, BPGL n ) Map(X dr, B 2 G m ) Map(X dr, B 2 µ n ) Map(X Dol, B 2 µ n ) Map BGm (X dr ), BGL Map(X Dol, BPGL n ) ss,0 n ' Map(X Dol, B 2 G m ) a Map Bµn e(x dr ), BSL n Map(X dr, BG m ) Map BGm 0(X Map(X pss dr, Bµ n ) liftings Dol), BGL n 24 / 28

Sketch of proof Some homotopy theory... Map BGm (X dr ), BGL n 0(X pss Dol), BGL n ' Map BGm a Map Bµn e0(x Dol ), BSL n Map(X Dol, BG m ) Map(X dr, BPGL n ) Map(X Map(X dr, B 2 G m ) Dol, Bµ n ) liftings nah ' ' Map(X dr, B 2 µ n ) Map(X Dol, B 2 µ n ) Map(X Dol, BPGL n ) ss,0 Map(X Dol, B 2 G m ) e Map BGm 0(X Dol), BGL n pss e 0 0 25 / 28

Sketch of proof The last condition becomes evident Map BGm (X dr ), BGL n 0(X ss Dol), BGL n ' Map BGm a Map Bµn e0(x Dol ), BSL n Map(X Dol, BG m ) 0 Map(X dr, BPGL n ) Map(X Map(X dr, B 2 G m ) Dol, Bµ n ) Map(X dr, B 2 µ n ) tnah Map(X Dol, B 2 µ n ) Map BGm (X dr ), BGL Map(X Dol, BPGL n ) ss,0 n ' Map(X Dol, B 2 G m ) a Map Bµn e(x dr ), BSL n Map(X dr, BG m ) Map BGm liftings ' 0(X Map(X pss dr, Bµ n ) liftings Dol), BGL n 26 / 28

Sketch of proof Generalization to other groups The master diagram for GL n : 1 µ n SL n PGL n 1 1 G m GL n PGL n 1 ( ) n det G m G m GL n -bundles twisted by 2 H 2 (X, G m ). 2 im H 2 (X,µ n )! H 2 (X, G m ). They induce untwisted PGL n -bundles. SL n -bundles twisted by lifts of to H 2 (X,µ n ) in the proof. 27 / 28

Sketch of proof Generalization to other groups The master diagram for GL n : 1 µ n SL n PGL n 1 1 G m GL n PGL n 1 ( ) n det G m G m G m! GL n with G m Z(GL n ). det GL n! G m I det is surjective, I det Gm is surjective, and I ker (det Gm ) contains no tori. 27 / 28

Sketch of proof Generalization to other groups The master diagram for H (connected linear algebraic group / C): 1 A 0 H 0 K 1 1 A H K 1 apple A apple G r m A! H with A Z(H). H apple! G r m I apple is surjective, I apple A is surjective, and I ker (apple A ) contains no tori. G r m 28 / 28

Sketch of proof Generalization to other groups The master diagram for H (connected linear algebraic group / C): 1 A 0 H 0 K 1 1 A H K 1 apple A apple G r m G r m H-bundles twisted by 2 H 2 (X, A). 2 im H 2 (X, A 0 )! H 2 (X, A). They induce untwisted K-bundles. H 0 -bundles twisted by lifts of to H 2 (X, A 0 ) in the proof. 28 / 28