D-manifolds, a newtheory of derived

differential geometry.

Dominic Joyce, Oxford

Cambridge, April, 2010.

work in progress,

chapters of a book on it

may be downloaded from

people.ox.ac.uk/∼joyce/dmanifolds.html.

See also arXiv:0910.3518

and arXiv:1001.0023.

These slides available at

people.ox.ac.uk/∼joyce/talks.html.1

1. IntroductionMany important areas in both dif-ferential and algebraic geometry in-volve forming ‘moduli spaces’M ofsome geometric objects, and then‘counting’ the points in M to getan ‘invariant’ I(M) with interestingproperties, for example Donaldson,Seiberg–Witten, Gromov–Wittenand Donaldson–Thomas invariants.Taking the ‘invariant’ to be a vec-tor space, category, . . . , rather thana number, Floer homology theo-ries, contact homology, Symplec-tic Field Theory, and Fukaya cat-egories also fit in this framework.

2

All these ‘invariants’ theories havesome common features:• You start with some geometricalspace X you want to study.• You define a moduli space M ofauxiliary geometric objects E on X.• This M is a topological space,hopefully compact and Hausdorff,but generally not a manifold – itmay have bad singularities.• Nevertheless, M behaves as if itis a compact, oriented manifold ofknown dimension k. One definesa virtual class [M]vir in Hk(M;Q),which ‘counts’ the points in M.• This [M]vir is then independentof choices in the construction, de-formations of X etc., and containsinteresting information.

3

Methods for defining [M]vir vary. Ingood cases,with generic initial dataM is smooth. Otherwise, we proveM has some extra geometric struc-ture G, and use G to define [M]vir.• In algebraic geometry problemsM is a scheme or Deligne–Mumfordstack with obstruction theory.• In areas of symplectic geometrybased on moduli of J-holomorphiccurves – Gromov–Witten theory, La-grangian Floer cohomology, Sym-plectic Field Theory, Fukaya cate-gories – there are two main geo-metric structures: Kuranishi spaces(Fukaya–Oh–Ohta–Ono) and poly-folds (Hofer–Wysocki–Zehnder).

4

2. D-manifolds and d-orbifoldsI will describe a new class of geo-metric objects I call d-manifolds —‘derived’ smooth manifolds. Someproperties of d-manifolds:• They form a strict 2-categorydMan. That is, we have objects X,the d-manifolds, 1-morphisms f , g :X → Y , the smooth maps, and also2-morphisms η : f ⇒ g.• Smooth manifolds embed into d-manifolds as a full (2)-subcategory.• There are also 2-categories dManb,dManc of d-manifolds with bound-ary and with corners, and orbifoldversions dOrb, dOrbb, dOrbc of these,d-orbifolds.

5

• Many concepts of differential ge-ometry extend nicely to d-manifolds:submersions, immersions, orienta-tions, submanifolds, transverse fi-bre products, cotangent bundles, . . . .• Almost any moduli space usedin any enumerative invariant prob-lem over R or C has a d-manifoldor d-orbifold structure, natural upto equivalence. There are trunca-tion functors to d-manifolds and d-orbifolds from structures currentlyused – C-schemes with obstructiontheories, Kuranishi spaces, polyfolds.• Virtual classes/cycles/chains canbe constructed for compact orientedd-manifolds and d-orbifolds.

6

So, d-manifolds and d-orbifolds pro-vide a unified framework for study-ing enumerative invariants and mod-uli spaces. They also have otherapplications, and are interesting andbeautiful in their own right.D-manifolds and d-orbifolds arerelated to other classes of spacesalready studied, in particular to theKuranishi spaces of Fukaya–Oh–Ohta–Ono in symplectic geometry,and to David Spivak’s derived man-ifolds, from Jacob Lurie’s ‘derivedalgebraic geometry’ programme.

7

2.1. Kuranishi spacesKuranishi spaces were defined byFukaya–Ono 1999 and Fukaya–Oh–Ohta–Ono 2009 as the geometricstructure on moduli spaces M ofJ-holomorphic curves in symplecticgeometry. A Kuranishi space is lo-cally modelled on the zeroes s−1(0)of a smooth section s of a vectorbundle E → V over an orbifold V .The theory has a lot of problems,and is basically incomplete.My starting point for this projectwas to find the ‘right’ definition ofKuranishi space. I claim that thisis: a Kuranishi space is (should re-ally be) a d-orbifold with corners.

8

2.2. Derived manifoldsDerived manifolds were defined byDavid Spivak (Duke Math. J. 153,2010), a student of Jacob Lurie. Alot of my ideas are stolen from Spi-vak. D-manifolds are much simplerthan derived manifolds. D-manifoldsare a 2-category, using Hartshorne-level algebraic geometry. Derivedmanifolds are an ∞-category, anduse very advanced and scary tech-nology – homotopy sheaves, Bous-feld localization, . . . .D-manifolds are a 2-category trun-cation of derived manifolds. I claimthat this truncation remembers allthe geometric information of im-portance to symplectic geometers,and other real people.

9

2.3. Why should dMan be a2-category?

Here are two reasons why any classof ‘derived manifolds’ should be (atleast) a 2-category. Firstly, one prop-erty we want of dMan is that it con-tains manifolds Man as a subcate-gory, and if X,Y, Z are manifoldsand g : X → Z, h : Y → Z aresmooth then a fibre product W =X ×g,Z,h Y should exist as in dMan,characterized by a universal prop-erty in dMan, and should be a d-manifold of ‘virtual dimension’

vdimW = dimX + dimY − dimZ.

Note that g, h need not be trans-verse, and vdimW may be negative.

10

Consider the case X = Y = ∗, thepoint, Z = R, and g, h : ∗ 7→ 0.If dMan were an ordinary categorythen as ∗ is a terminal object, theunique fibre product ∗×0,R,0∗ wouldbe ∗. But this has virtual dimension0, not −1. So dMan must be somekind of higher category.Secondly, two approximations fordMan are C-schemes X with obstruc-tion theory, and quasi-smooth dg-schemes. Both of these include a‘cotangent complex’ in Db coh(X)concentrated in two degrees −1,0.It seems reasonable to capture thebehaviour of such complexes in a2-category.

11

3. The definition of d-manifoldsAlgebraic geometry (based on alge-bra and polynomials) has excellenttools for studying singular spaces –the theory of schemes.In contrast, conventional differen-tial geometry (based on smooth realfunctions and calculus) deals wellwith nonsingular spaces – manifolds– but poorly with singular spaces.There is a little-known theory ofschemes in differential geometry,C∞-schemes, going back to Law-vere, Dubuc, Moerdijk and Reyes,. . . in synthetic differential geome-try in the 1960s-1980s. This will bethe foundation of our d-manifolds.

12

3.1. C∞-ringsLet X be a manifold, and C∞(X)the set of smooth functions c : X→R. Then C∞(X) is an R-algebra,by adding and multiplying smoothfunctions. But there are many moreoperations on C∞(X), e.g. if c :X → R is smooth then exp(c) : X →R is smooth, giving exp : C∞(X)→C∞(X), algebraically independent ofaddition and multiplication.Let f : Rn → R be smooth. DefineΦf : C∞(X)n→ C∞(X) by

Φf(c1, . . . , cn)(x)=f(c1(x), . . . , cn(x)

)for all x ∈ X. Addition comes fromf : R2 → R, f : (c1, c2) 7→ c1 + c2,multiplication from (c1, c2) 7→ c1c2.

13

Definition. A C∞-ring is a set C togetherwith n-fold operations Φf : Cn → C for allsmooth maps f : Rn→ R, n > 0, satisfyingthe following conditions:Let m,n > 0, and fi : Rn → R for i =1, . . . ,m and g : Rm → R be smooth func-tions. Define h : Rn→ R by

h(x1, . . . , xn) = g(f1(x1, . . . , xn), . . . , fm(x1 . . . , xn)),

for (x1, . . . , xn) ∈ Rn. Then for all c1, . . . , cnin C we have

Φh(c1, . . . , cn) =

Φg(Φf1(c1, . . . , cn), . . . ,Φfm(c1, . . . , cn)).

Also defining πj : (x1, . . . , xn) 7→ xj for j =1, . . . , n we have Φπj : (c1, . . . , cn) 7→ cj.A morphism of C∞-rings is φ : C→ D withΦf φn = φΦf : Cn→ D for all smooth f :Rn → R. Write C∞Rings for the categoryof C∞-rings.

14

Then C∞(X) is a C∞-ring for anymanifold X, and from C∞(X) wecan recover X up to isomorphism.If f : X → Y is smooth then f∗ :C∞(Y ) → C∞(X) is a morphismof C∞-rings. This gives a full andfaithful functor F : Man→ C∞Ringsop

by F : X 7→ C∞(X), F : f 7→ f∗.Thus, we think of manifolds as ex-amples of C∞-rings, and C∞-ringsas generalizations of manifolds. Butthere are many more C∞-rings thanmanifolds, e.g. C0(X) is a C∞-ringfor any topological space X.

15

3.2. C∞-schemesWe can now develop the whole ma-chinery of scheme theory in alge-braic geometry, replacing rings oralgebras by C∞-rings throughout —see my arXiv:1001.0023.We obtain a category C∞Sch of C∞-schemes X = (X,OX), which aretopological space X equipped witha sheaf of C∞-rings OX locally mod-elled on the spectrum of a C∞-ring.If X is a manifold, define a C∞-scheme X = (X,OX) by OX(U) =C∞(U) for all open U ⊆ X. Thisdefines a full and faithful embed-ding Man → C∞Sch.

16

We also define vector bundles, co-herent sheaves coh(X) and quasi-coherent sheaves qcoh(X), and thecotangent sheaf T ∗X on X. Thenqcoh(X) is an abelian category.Some differences with conventionalalgebraic geometry:• affine schemes are Hausdorff. Noneed to introduce etale topology.• partitions of unity exist subordi-nate to any open cover of a (nice)C∞-scheme X.• C∞-rings such as C∞(Rn) are notnoetherian as R-algebras. Causesproblems with coherent sheaves:coh(X) is not closed under kernels,so not an abelian category.

17

3.3. The 2-category of d-spacesWe define d-manifolds as a 2-subcategoryof a larger 2-category of d-spaces. Theseare ‘derived’ versions of C∞-schemes.

Definition. A d-space is a is a quintu-ple X = (X,O′X, EX, ıX, X) where X =(X,OX) is a separated, second countable,locally fair C∞-scheme, O′X is a secondsheaf of C∞-rings on X, and EX is a qua-sicoherent sheaf on X, and ıX : O′X → OXis a surjective morphism of sheaves of C∞-rings whose kernel IX is a sheaf of squarezero ideals in O′X, and X : EX → IX isa surjective morphism in qcoh(X), so wehave an exact sequence of sheaves on X:

EXX //O′X

ıX //OX // 0.

18

A 1-morphism f : X → Y is a triple f =(f, f ′, f ′′), where f = (f, f ]) : X → Y is amorphism of C∞-schemes and f ′ : f−1(O′Y )→ O′X, f ′′ : f∗(EY ) → EX are sheaf mor-phisms such that the following commutes:

f−1(EY )f ′′

f−1(Y )

//f−1(O′Y )f ′

f−1(ıY )

//f−1(OX)f ]

// 0

EXX //O′X

ıX //OX // 0.

Let f , g : X → Y be 1-morphisms withf = (f, f ′, f ′′), f = (g, g′, g′′). Suppose f =g. A 2-morphism η : f ⇒ g is a morphism

η : f−1(ΩO′Y)⊗f−1(O′Y ) OX −→ EX

in qcoh(X), where ΩO′Yis the sheaf of

cotangent modules of O′Y , such that g′ =f ′+ X η ΠXY and g′′ = f ′′+ η f∗(φY ),for natural morphisms ΠXY , φY .

Theorem 1.This defines a strict 2-categorydSpa. All fibre products exist in dSpa.

19

We can map C∞Sch into dSpa by tak-

ing a C∞-scheme X = (X,OX) to the d-

space X = (X,OX,0, idOX ,0), with exact

sequence

0 0 //OXidOX //OX // 0.

This embeds C∞Sch, and hence manifolds

Man, as discrete 2-subcategories of dSpa.

For transverse fibre products of manifolds,

the fibre products in Man and dSpa agree.

The use of square zero extensions in defin-

ing dSpa seems to be key in defining a

good 2-category, and in nicely truncating

Spivak’s ∞-category. I’m not sure why

this works.

20

3.4. The 2-subcategory of d-manifolds

Definition. A d-space X is a d-manifold

of dimension n ∈ Z if X may be covered by

open d-subspaces Y equivalent in dSpa to

a fibre product U×WV , where U ,V ,W are

manifolds without boundary and dimU +

dimV − dimW = n. We allow n < 0.

Think of a d-manifold X=(X,O′X, EX, ıX, X)

as a ‘classical’ C∞-scheme X, with extra

‘derived’ data O′X, EX, ıX, X.

Write dMan for the full 2-subcategory of

d-manifolds in dSpa. It is not closed under

fibre products in dSpa, but we can say:

Theorem 2. All fibre products of the form

X ×Z Y with X,Y d-manifolds and Z a

manifold exist in the 2-category dMan.

21

4. Properties of d-manifolds4.1. Gluing by equivalencesA 1-morphism f : X → Y in dMan

is an equivalence if there exist a 1-morphism g :Y →X and 2-morphismsη : g f ⇒ idX and ζ : f g ⇒ idY .Theorem3. Let X, Y be d-manifolds,∅ 6= U ⊆ X, ∅ 6= V ⊆ Y open d-submanifolds, and f : U → V anequivalence. Suppose the topolog-ical space Z = X ∪U=V Y made bygluing X,Y using f is Hausdorff.Then there exists a d-manifold Z,unique up to equivalence, open X, Y⊆ Z with Z = X ∪ Y , equivalencesg : X → X and h : Y → Y , and a2-morphism η : g|U ⇒ h f.

22

Equivalence is the natural notionof when two objects in dMan are‘the same’. In Theorem 3, Z is apushout XqidU ,U ,f

Y in dMan. Theo-rem 3 generalizes to gluing familiesof d-manifolds Xi : i ∈ I by equiva-lences on double overlaps Xi ∩ Xj,with (weak) conditions on tripleoverlaps Xi ∩Xj ∩Xk.This is very useful for proving ex-istence of d-manifold structures onmoduli spaces.

23

4.2. Virtual vector bundlesVector bundle and cotangent bun-dles have good 2-category general-izations. Let X be a C∞-scheme.Define a 2-category vqcoh(X) ofvirtual quasicoherent sheaves to haveobjects morphisms φ : E1 → E2 inqcoh(X). If φ : E1 → E2 and ψ :F1→ F2 are objects, a 1-morphism(f1, f2) : φ → ψ is morphisms f j :Ej → Fj in qcoh(X) for j = 1,2with ψf1 = f2φ. If (f1, f2), (g1, g2)are 1-morphisms φ→ ψ, a 2-morphismη : (f1, f2) ⇒ (g1, g2) is a mor-phism η : E2→ F1 in qcoh(X) withg1 = f1 + η φ and g2 = f2 + ψ η.

24

Call φ : E1 → E2 a virtual vectorbundle on X of rank k ∈ Z if X maybe covered by open U ⊆ X suchthat φ|U : E1|U → E2|U is equiva-lent in the 2-category vqcoh(U) toψ : F1→ F2, where F1,F2 are vec-tor bundles on U with rankF2 −rankF1 = k. Write vvect(X) forthe full 2-subcategory of virtual vec-tor bundles on vqcoh(X).If X is a d-manifold, it has a natu-ral virtual cotangent bundle T ∗X invvect(X), of rank vdimX.If f : X → Y is a 1-morphism indMan, there is a natural 1-morphismΩf : f∗(T ∗Y )→ T ∗X in vvect(X).

25

Then f is etale (a local equivalence)if and only if Ωf is an equivalence invvect(X). Similarly, f is an immer-sion or submersion if Ωf is surjec-tive or injective in a suitable sense.If φ : E1 → E2 lies in vvect(X) wecan define a line bundle Lφ on X

analogous to the ‘top exterior power’of φ : E1→ E2. So for a d-manifoldX, LT ∗X is a line bundle on X whichwe think of as ΛtopT ∗X. An ori-entation on X is an orientation onthe line bundle LT ∗X. Orientationshave the properties one would ex-pect from the manifold case.

26