Derived Differential Geometry lecture courseHere are the course materials for a 14 hour graduate lecture
course I gave in Oxford in summer term 2015, and then again at a
graduate summer school in August 2015 at the Centre for Quantum
Geometry of Moduli spaces at Aarhus, Denmark. You can download the
slides of the lectures at the bottom of the page.
Derived Differential Geometry is the study of derived smooth manifolds
and orbifolds, where “derived” is in the sense of the Derived Algebraic
Geometry of Jacob Lurie  and ToŽn-Vezzosi [15,16]. There are by now
several different models for (higher) categories of derived manifolds
and derived orbifolds: the “derived manifolds” of Spivak , the
“d-manifolds” and “d-orbifolds” of Joyce [2,3,6], the “derived
manifolds” of Borisov-Noel [9,10], and the “M-Kuranishi spaces” and
“Kuranishi spaces” of Joyce . For derived manifolds without
boundary, these are all known to be roughly equivalent, at least at the
level of homotopy categories.
Actually, a prototype version of derived orbifolds has been used
for many years in the work of Fukaya, Oh, Ohta and Ono [7,11,12] as
their “Kuranishi spaces”, a geometric structure on moduli spaces of
J-holomorphic curves in symplectic geometry, but it was not understood
until recently that these are part of the world of derived geometry –
Derived manifolds are a (higher) category of geometric spaces
which include ordinary smooth manifolds, but also many more singular
objects – for instance, if X,Y are embedded submanifolds of a
Z, then the intersection X ∩ Y has the structure of a derived
of dimension dim X +
dim Y – dim Z (which can be negative), and
any closed subset S of Rn
be given the structure of a derived
manifold of dimension n – 1.
They have a “derived” geometric structure,
which is difficult to define, but means that derived manifolds have
their own differential geometry, and in many ways behave as well (and
sometimes better) than smooth manifolds.
One reason derived manifolds and derived orbifolds are important
is that many moduli spaces (families of isomorphism classes of
geometric objects) in differential geometry, and in complex algebraic
geometry, can be given the structure of derived manifolds and derived
orbifolds. For example, any moduli space of solutions of a
nonlinear elliptic partial differential equation on a compact manifold
is a derived manifold. Also, compact, oriented derived manifolds and
derived orbifolds have “virtual classes”, generalizing the fact that a
compact, oriented n-dimensional
manifold X has a fundamental
in its top-dimensional homology group Hn(X;Z). This means that
manifolds and orbifolds have applications in enumerative invariant
problems (e.g. Donaldson invariants, Gromov-Witten invariants,
Seiberg-Witten invariants, Donaldson-Thomas invariants, . . .), and
generalizations such as Floer homology theories and Fukaya categories.
This lecture course will define and discuss 2-categories of
derived manifolds and derived orbifolds, and their applications to
moduli spaces of solutions of nonlinear elliptic partial differential
equations including J-holomorphic
curves, and “counting” problems in
differential geometry, and complex algebraic geometry.
We explain two different ways to define these 2-categories,
firstly using C∞-rings, C∞-schemes and C∞-algebraic
following Spivak , Joyce [2,3,6] and Borisov-NoŽl [9,10], or
secondly using an “atlas of charts” (Kuranishi neighbourhoods) approach
in , which is based on Fukaya et al. [7,11,12]. (See reading list
Lecture 1: Different
spaces in algebraic geometry: schemes,
stacks, higher stacks, derived stacks. Basics of category theory,
categories, functors. The Yoneda Lemma. Schemes as functors AlgK →
Sets, and stacks as functors AlgK → Groupoids. Grothendieck’s approach
to moduli spaces as ‘representable functors’.
Lecture 2: What is derived
geometry? Derived schemes and stacks.
Commutative differential graded algebras, examples. Bťzout’s Theorem
and derived Bťzout’s Theorem. Patching together local models in derived
geometry. Fibre products in ordinary categories; why we need higher
categories in derived geometry. Outlook on derived geometry.
Versions of derived manifolds due to Spivak, Borisov-NoŽl, and myself.
Sample properties of derived manifolds X: tangent
obstruction spaces OxX, d-transversality,
existence of d-transverse
fibre products in the 2-category dMan.
Application to moduli problems,
existence of d-manifold and d-orbifold structures on many moduli spaces.
Lecture 3: C∞-algebraic geometry. C∞-rings and their
modules, the cotangent module. Sheaves on topological spaces. C∞-schemes. Differences
with ordinary algebraic geometry.
Lecture 4: 2-categories, d-spaces and d-manifolds. Strict and
weak 2-categories. Fibre products in 2-categories. Differential graded C∞-rings, and the
2-category of square zero dg C∞-rings.
D-spaces, a kind of derived C∞-scheme,
which are essentially schemes over square zero dg C∞-rings, and form a
2-category dSpa, which
contains C∞-schemes and
manifolds as full (2-)subcategories. Existence of fibre products in dSpa, and gluing d-spaces by
equivalences on open d-subspaces. D-manifolds, a kind of derived
manifold, as d-spaces X locally of the
form U ◊WV
Lecture 5: Differential-geometric description of d-manifolds.
The O(s) and O(s2)
notation. Standard model d-manifolds SV,E,s, 1-morphisms SV',f,f', and 2-morphisms SΛ.
Tangent spaces and obstruction spaces of d-manifolds. Criteria for when
a standard model 1-morphism SV',f,f',is ťtale or an
Lecture 6: M-Kuranishi spaces, a simple way to define an
(ordinary) category of derived manifolds, using an ‘atlas of charts’
approach. M-Kuranishi neighbourhoods and their morphisms. M-coordinate
changes. The sheaf property of morphisms of M-Kuranishi neighbourhoods.
Definition of M-Kuranishi spaces, and their morphisms. Composition of
morphisms, proof of existence using the sheaf property. Manifolds as
M-Kuranishi spaces. Tangent and obstruction spaces.
Lecture 7: Orbifolds. Why orbifolds should form a 2-category.
Examples. Hilsum-Skandalis morphisms of quotient orbifolds. Orbifold
charts, 1-morphisms, 2-morphisms, coordinate changes. Stacks on
topological spaces. The stack property of the 2-categories of orbifold
charts on open sets S in X. Definition of the weak
2-category of orbifolds Orb by
an ‘atlas of charts approach’. Composition of 1-morphisms defined using
the stack property.
Lecture 8: Kuranishi spaces, following my arXiv:1409.6908, ß4.
Kuranishi neighbourhoods, 1-morphisms coordinate changes, and
2-morphisms of coordinate changes on a topological space X.
Criteria for a 1-morphism to be a coordinate change. The stack property
of the 2-categories of Kuranishi neighbourhoods on open sets S in X. Definition of the weak
2-category of Kuranishi spaces Kur
by an ‘atlas of charts approach’. Composition of 1-morphisms defined
using the stack property. Kuranishi spaces with trivial orbifold groups
give a weak 2-category of derived manifolds KurtrG equivalent to dMan.
Lecture 9: Differential geometry of derived manifolds and
orbifolds. Orbifold groups, tangent spaces and obstruction spaces of
derived orbifolds. (Weak) immersions, (weak) embeddings and derived
submanifolds of derived manifolds and orbifolds. Embedding derived
manifolds into manifolds, the Whitney Embedding Theorem, necessary and
sufficient conditions for a derived manifold to be principal (covered
by a single Kuranishi neighbourhood). (Weak) submersions. Orientations
on derived manifolds.
Lecture 10: D-transversality (or strong d-transversality) for
: X→ Z, h : Y→ Z in dMan or dOrb, as conditions for existence of
a fibre product W
= X g,Z,h
Y (or for W
to exist and be a manifold). g a weak
submersion (or a submersion) ensures g,h
d-transverse (or strongly d-transverse). Sketch proof of the existence
of d-transverse fibre products. Orientations on fibre products.
Lecture 11: Derived manifolds and orbifolds with boundary, and
with corners. Manifolds with corners X,
boundaries ∂X, k-corners Ck(X), and the corner functor. Tangent
bundles TX and b-tangent
Conditions for existence of transverse fibre products; manifolds with
generalized corners. Kuranishi spaces with corners as clean way to
define derived orbifolds with corners. Differential geometry of
Kuranishi spaces with corners.
Lecture 12: Bordism groups Bn(Y) and derived bordism groups dBn(Y) for a manifold Y, and the isomorphism Bn(Y) ≅ dBn(Y). Virtual classes in homology for
compact oriented derived manifolds and orbifolds (without boundary).
the problem of forming virtual chains for compact oriented derived
orbifolds with corners, and its application in Lagrangian Floer
Lecture 13: Existence of derived manifold and orbifold
structures on moduli spaces. Moduli spaces of solutions of nonlinear
elliptic p.d.e.s. ‘Truncation functors’ from other geometric
structures: FOOO Kuranishi spaces, polyfolds, C-schemes with perfect
obstruction theories, quasi-smooth derived C-schemes, derived C-schemes
with –2-shifted symplectic structures. A conjectural approach to moduli
spaces in differential geometry using Grothendieck-style ‘representable
Lecture 14: Moduli spaces of J-holomorphic
curves in symplectic geometry. J-holomorphic
curves with marked points, Deligne-Mumford stable curves. Moduli spaces
are compact oriented Kuranishi spaces. Virtual classes for these define
 D. Joyce, ‘An introduction to C∞-schemes
 D. Joyce, ‘An introduction to
d-manifolds and derived differential
geometry’, pages 230-281 in L. Brambila-Paz et al. ‘Moduli spaces’,
L.M.S. Lecture Notes 411, C.U.P., 2014. arXiv:1206.4207. (Survey.)
 D. Joyce, ‘D-manifolds,
d-orbifolds and derived differential
geometry: a detailed summary’, arXiv:1208.4948.
 D. Joyce, ‘A new definition of
Kuranishi space’, arXiv:1409.6908.
 D. Joyce, ‘Algebraic Geometry
over C∞-rings’, arXiv:1001.0023, 2010.
 D. Joyce, ‘D-manifolds and
d-orbifolds: a theory of derived
differential geometry’, preliminary version of book (2012), 768
 K. Fukaya, ‘Floer homology of
Lagrangian submanifolds’, arXiv:1106.4882. (Survey.)
 E.J. Dubuc, ‘C∞-schemes’,
Amer. J. Math. 103 (1981), 683-690.
Even further reading:
 D. Borisov, ‘Derived manifolds and Kuranishi models’, arXiv:1212.1153.
 D. Borisov and J. NoŽl, ‘Simplicial
approach to derived
differential manifolds’, arXiv:1112.0033.
 K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, ‘Lagrangian intersection
Floer theory - anomaly and obstruction’, Parts I and II.
A.M.S./International Press, 2009.
 K. Fukaya and K. Ono, ‘Arnold
Conjecture and Gromov-Witten
invariant’, Topology 38 (1999), 933-1048.
 J. Lurie, ‘Derived Algebraic
Geometry V: Structured spaces’, arXiv:0905.0459.
 D.I. Spivak, ‘Derived smooth
manifolds’, Duke Mathematical Journal
153 (2010), 55-128. arXiv:0810.5174.
 B. ToŽn, ‘Derived Algebraic
 B. ToŽn and G. Vezzosi, ‘Homotopical
Algebraic Geometry II:
Geometric Stacks and Applications’, Mem. Amer. Math. Soc. 193
no. 902. math.AG/0404373.
PDF files to download:
Slides of lectures 1 and 2
Slides of lectures 3 and 4
Slides of lectures 5 and 6
Slides of lectures 7 and 8
Slides of lectures 9 and 10
Slides of lectures 11 and 12
Slides of lectures 13 and 14