arXiv subjects

Dominic Joyce

Publications and source records attributed to Dominic Joyce.

At least 37 records · Page 2Linked to original sources

arXiv2017

A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Pantev, Toen, Vaquié and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${\bf X}$ for $k\in\mathbb Z$, and Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or étale local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.

math.AG↗

arXiv2017

Conjectures on counting associative 3-folds in $G_2$-manifolds

There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,φ,*φ)$ and Calabi-Yau 3-folds $(Y,J,g,ω)$. We can also generalize $(X,φ,*φ)$ to 'tamed almost $G_2$-manifolds' $(X,φ,ψ)$, where we compare $φ$ with $ω$ and $ψ$ with $J$. Associative 3-folds in $X$, a special kind of minimal submanifold, are analogous to $J$-holomorphic curves in $Y$. Several areas of Symplectic Geometry -- Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories -- are built using 'counts' of moduli spaces of $J$-holomorphic curves in $Y$, but give an answer depending only on the symplectic manifold $(Y,ω)$, not on the (almost) complex structure $J$. We investigate whether it may be possible to define interesting invariants of tamed almost $G_2$-manifolds $(X,φ,ψ)$ by 'counting' compact associative 3-folds $N\subset X$, such that the invariants depend only on $φ$, and are independent of the 4-form $ψ$ used to define associative 3-folds. We conjecture that one can define a superpotential $Φ_ψ:{\mathcal U}\toΛ_{>0}$ 'counting' associative $\mathbb Q$-homology 3-spheres $N\subset X$ which is deformation-invariant in $ψ$ for $φ$ fixed, up to certain reparametrizations $Υ:{\mathcal U}\to{\mathcal U}$ of the base ${\mathcal U}=$Hom$(H_3(X;{\mathbb Z}),1+Λ_{>0})$, where $Λ_{>0}$ is a Novikov ring. Using this we define a notion of '$G_2$ quantum cohomology'. These ideas may be relevant to String Theory or M-Theory on $G_2$-manifolds. We also discuss Donaldson and Segal's proposal in arXiv:0902.3239, section 6.2, to define invariants 'counting' $G_2$-instantons on tamed almost $G_2$-manifolds $(X,φ,ψ)$, with 'compensation terms' counting weighted pairs of a $G_2$-instanton and an associative 3-fold, and suggest some modifications to it.

math.DG↗

arXiv2016

Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds

Let $({\bf X},ω_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that $X_{\rm an}$ can be given the structure of a derived smooth manifold ${\bf X}_{\rm dm}$, of real virtual dimension ${\rm vdim}_{\mathbb R}{\bf X}_{\rm dm}=n$. This ${\bf X}_{\rm dm}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological space $X_{\rm an}$. There is a 1-1 correspondence between orientations on $({\bf X},ω_{\bf X}^*)$ and orientations on ${\bf X}_{\rm dm}$. Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented $-2$-shifted symplectic derived $\mathbb C$-schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebro-geometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension, and from purely complex algebraic input, can yield a virtual class of odd real dimension. Now derived moduli schemes of coherent sheaves on a Calabi-Yau 4-fold are expected to be $-2$-shifted symplectic (this holds for stacks). We propose to use our virtual classes to define new Donaldson-Thomas style invariants 'counting' (semi)stable coherent sheaves on Calabi-Yau 4-folds $Y$ over $\mathbb C$, which should be unchanged under deformations of $Y$.

math.AG↗

arXiv2016

Algebraic Geometry over $C^\infty$-rings

If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation $Φ_f:C^\infty(X)^n\to C^\infty(X)$ acting by $Φ_f:(c_1,\ldots,c_n)\mapsto f(c_1,...,c_n)$, and these operations $Φ_f$ satisfy many natural identities. Thus, $C^\infty(X)$ actually has a far richer structure than the obvious $\mathbb R$-algebra structure. We develop a version of algebraic geometry in which rings or algebras are replaced by $C^\infty$-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are $C^\infty$-$schemes$, a category of geometric objects which generalize smooth manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent and coherent sheaves on $C^\infty$-schemes, and $C^\infty$-$stacks$, in particular Deligne-Mumford $C^\infty$-stacks, a 2-category of geometric objects generalizing orbifolds. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems. This paper forms the foundations of the author's new theory of "derived differential geometry", surveyed in arXiv:1206.4207 and in more detail in arXiv:1208.4948, which studies d-manifolds and d-orbifolds, "derived" versions of smooth manifolds and smooth orbifolds. Derived differential geometry has applications to areas of symplectic geometry involving moduli spaces of $J$-holomorphic curves. Many of these ideas are not new: $C^\infty$-rings and $C^\infty$-schemes have long been part of synthetic differential geometry. But we develop them in new directions. This paper is surveyed in arXiv:1104.4951.

math.AG↗

arXiv2016

A generalization of manifolds with corners

In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under much weaker conditions than in $\bf Man^c$. This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of $J$-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners. Our manifolds with g-corners are related to the 'interior binomial varieties' of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874), and to the 'positive log differentiable spaces' of Gillam and Molcho in arXiv:1507.06752.

math.DG↗

arXiv2016

Manifolds with analytic corners

Manifolds with boundary and with corners form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two notions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle ${}^bTX$. The usual definition of smooth structure uses $TX$, as $f:X\to\mathbb{R}$ is defined to be smooth if $\nabla^kf$ exists as a continuous section of $\bigotimes^kT^*X$ for all $k\ge 0$. We define 'manifolds with analytic corners', or 'manifolds with a-corners', with a different smooth structure, in which roughly $f:X\to\mathbb{R}$ is smooth if ${}^b\nabla^kf$ exists as a continuous section of $\bigotimes^k({}^bT^*X)$ for all $k\ge 0$. These are different from manifolds with corners even when $X=[0,\infty)$, for instance $x^α:[0,\infty)\to\mathbb{R}$ is smooth for all real $α\ge 0$ when $[0,\infty)$ has a-corners. Manifolds with a-boundary and with a-corners form categories ${\bf Man}\subset{\bf Man^{ab}}\subset{\bf Man^{ac}}$, with well behaved differential geometry. Partial differential equations on manifolds with boundary may have boundary conditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumann boundary conditions, or (ii) 'at infinity', prescribing the asymptotic behaviour of the solution. We argue that manifolds with corners should be used for (i), and with a-corners for (ii). We discuss many applications of manifolds with a-corners in boundary problems of type (ii), and to singular p.d.e. problems involving 'bubbling', 'neck-stretching' and 'gluing'.

math.DG↗

arXiv2015

A new definition of Kuranishi space

'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a satisfactory definition of Kuranishi space has been the subject of recent debate (see e.g. arXiv:1208.1340, arXiv:1209.4410, arXiv:1510.06849). We propose three new definitions of Kuranishi space: a simple 'manifold' version, '$μ$-Kuranishi spaces', which form an ordinary category $\boldsymbolμ\bf Kur$; a more complicated 'manifold' version, 'm-Kuranishi spaces', which form a weak 2-category $\bf mKur$; and an 'orbifold' version, 'Kuranishi spaces', which form a weak 2-category $\bf Kur$. These are related by an equivalence of categories $\boldsymbolμ{\bf Kur}\simeq{\rm Ho}({\bf mKur})$, where ${\rm Ho}({\bf mKur})$ is the homotopy category of $\bf mKur$, and by a full and faithful embedding ${\bf mKur}\hookrightarrow\bf Kur$. We also define ($μ$-, m-)Kuranishi spaces with boundary, and with corners. We hope our definitions will become accepted as final, replacing previous definitions. Any Fukaya-Oh-Ohta-Ono Kuranishi space $\bf X$ can be made into a compact Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that is, up to isomorphism in ${\rm Ho}({\bf Kur})$). The same holds for topological spaces with Fukaya-Oh-Ohta-Ono 'good coordinate systems', and for McDuff and Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. A compact topological space $\bf X$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and Zehnder can be made into a Kuranishi space $\bf X$ uniquely up to equivalence in $\bf Kur$. This book is surveyed in arXiv:1510.07444.

math.DG↗

arXiv2015

Some new homology and cohomology theories of manifolds and orbifolds

For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex $(MC_*(Y;R),\partial)$ is generated by quadruples $[V,n,s,t]$ satisfying relations, where $V$ is an oriented manifold with corners, $n\in\mathbb N$, and $s:V\to{\mathbb R}^n$, $t:V\to Y$ are smooth with $s$ proper near 0 in ${\mathbb R}^n$. We show that $MH_*(Y;R),MH^*(Y;R)$ satisfy the Eilenberg-Steenrod axioms, and so are canonically isomorphic to conventional (co)homology. The usual operations on (co)homology -- pushforwards $f_*$, pullbacks $f^*$, fundamental classes $[Y]$ for compact oriented $Y$, cup, cap and cross products $\cup,\cap,\times$ -- are all defined and well-behaved at the (co)chain level. Chains $MC_*(Y;R)$ form flabby cosheaves on $Y$, and cochains $MC^*(Y;R)$ form soft sheaves on $Y$, so they have good gluing properties. We also define $compactly$-$supported$ $M$-$cohomology$ $MH^*_{cs}(Y;R)$, $locally$ $finite$ $M$-$homology$ $MH_*^{lf}(Y;R)$ (a kind of Borel-Moore homology), and two variations on the entire theory, $rational$ $M$-($co$)$homology$ and $de$ $Rham$ $M$-($co$)$homology$. All of these are canonically isomorphic to the corresponding type of conventional (co)homology. The reason for doing this is that our M-(co)homology theories are very well behaved at the (co)chain level, and will be better than other (co)homology theories for some purposes, particularly in problems involving transversality. In a sequel we will construct virtual classes and virtual chains for Kuranishi spaces in M-(co)homology, with a view to applications of M-(co)homology in areas of Symplectic Geometry involving moduli spaces of $J$-holomorphic curves.

math.AT↗

arXiv2015

Symmetries and stabilization for sheaves of vanishing cycles

Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $Φ:U\to U$ is an isomorphism with $f\circΦ=f$ and $Φ\vert_X=$id$_X$. Then $Φ$ induces an isomorphism $Φ_*:PV_{U,f}\to PV_{U,f}$. We show that $Φ_*$ is multiplication by det$(dΦ\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\vert_{X^{(3)}}:X^{(3)}\to\mathbb A^1$. (c) If $U,V$ are smooth $\mathbb C$-schemes, $f:U\to\mathbb A^1$, $g:V\to\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $Φ:U\to V$ is an embedding with $f=g\circΦ$ and $Φ\vert_X:X\to Y$ an isomorphism, there is a natural isomorphism $Θ_Φ:PV_{U,f}\toΦ\vert_X^*(PV_{V,g})\otimes_{\mathbb Z_2}P_Φ$, for $P_Φ$ a natural principal $\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $PV_{U,f}$ by $\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690.

math.AG↗

arXiv2015

Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in C^m

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit family of examples found by Joyce, Lee and Tsui, arXiv:0801.3721. If instead $L$ is immersed rather than embedded, the only extra possibility in (a),(b) is $L=Π_1\cupΠ_2$. Our methods, which are new and can probably be used to prove other similar uniqueness theorems, involve $J$-holomorphic curves, Lagrangian Floer cohomology, and Fukaya categories from symplectic topology. When $m=2$, (a) is easy to prove using hyperkahler geometry, and (b) is proved by Lotay and Neves, arXiv:1208.2729.

math.SG↗

arXiv2014

Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist for all time, and $L^\infty=\lim_{t\to\infty}L^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas-Yau conjectures, and discuss related issues. It is a folklore conjecture that there exists a Bridgeland stability condition $(Z,\mathcal P)$ on the derived Fukaya category $D^b\mathcal F(M)$ of $M$, such that an isomorphism class in $D^b\mathcal F(M)$ is $(Z,\mathcal P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with "stable singularities"), $E\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, then there is a unique family $\{(L^t,E^t,b^t):t\in[0,\infty)\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\cong(L,E,b)$ in $D^b\mathcal F(M)$ for all $t$, and $\{L^t:t\in[0,\infty)\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\dots,$ and in graded Lagrangian integral currents we have $\lim_{t\to\infty}L^t=L_1+\cdots+L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{iπϕ_j}$ for $ϕ_1>\cdots>ϕ_n$, and $(L_1,ϕ_1),\ldots,(L_n,ϕ_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\ldots.$

math.DG↗

arXiv2013

A classical model for derived critical loci

Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call "$-1$-shifted symplectic derived schemes". They show that intersections of algebraic Lagrangians in a smooth symplectic $\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3-fold over $\mathbb K$, are also $-1$-shifted symplectic derived schemes. Thus, their theory may have applications in algebraic symplectic geometry, and in Donaldson-Thomas theory of Calabi-Yau 3-folds. This paper defines and studies a new class of spaces we call "algebraic d-critical loci", which should be regarded as classical truncations of $-1$-shifted symplectic derived schemes. They are simpler than their derived analogues. We also give a complex analytic version of the theory, "complex analytic d-critical loci", and an extension to Artin stacks, "d-critical stacks". In the sequels arXiv:1305.6302, arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090 we will define truncation functors from $-1$-shifted symplectic derived schemes or stacks to algebraic d-critical loci or d-critical stacks, and we will apply d-critical loci to motivic and categorified Donaldson-Thomas theory, and to intersections of (derived) complex Lagrangians in complex symplectic manifolds. We will show that the important structures one wants to associate to a derived critical locus -- virtual cycles, perverse sheaves and mixed Hodge modules of vanishing cycles, and motivic Milnor fibres -- can be defined for oriented d-critical loci and oriented d-critical stacks.

math.AG↗

arXiv2012

An introduction to d-manifolds and derived differential geometry

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of the 'derived manifolds' of Spivak (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". For brevity, this survey concentrates mostly on d-manifolds without boundary. A longer and more detailed summary of the book is given in arXiv:1208.4948. Much of differential geometry extends very nicely to d-manifolds and d-orbifolds -- immersions, submersions, submanifolds, transverse fibre products, orientations, etc. Compact oriented d-manifolds and d-orbifolds have virtual classes. There are truncation functors to d-manifolds and d-orbifolds from essentially every geometric structures on moduli spaces used in enumerative invariant problems in differential geometry or complex algebraic geometry, including Fredholm sections of Banach vector bundles over Banach manifolds, the "Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer, Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect obstruction theories in algebraic geometry. Thus, results in the literature imply that many important classes of moduli spaces are d-manifolds or d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic geometry. D-manifolds and d-orbifolds will have applications in symplectic geometry, and elsewhere.

math.DG↗

arXiv2012

D-manifolds, d-orbifolds and derived differential geometry: a detailed summary

This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general overview are advised to start there. We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algebraic geometry' of Toen and Lurie. They are a 2-category truncation of Spivak's 'derived manifolds' (see arXiv:0810.5174, arXiv:1212.1153). The category of manifolds Man embeds in dMan as a full (2-)subcategory. We also define 2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds". Much of differential geometry extends very nicely to d-manifolds and d-orbifolds -- immersions, submersions, submanifolds, transverse fibre products, orientations, orbifold strata, bordism, etc. Compact oriented d-manifolds and d-orbifolds have virtual classes. There are truncation functors to d-manifolds and d-orbifolds from essentially every geometric structure on moduli spaces used in enumerative invariant problems in differential geometry or complex algebraic geometry, including Fredholm sections of Banach vector bundles over Banach manifolds, the "Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer, Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect obstruction theories in algebraic geometry. Thus, results in the literature imply that many important classes of moduli spaces are d-manifolds or d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic geometry. D-manifolds and d-orbifolds will have applications in symplectic geometry, and elsewhere.

math.DG↗

arXiv2012

An introduction to C-infinity schemes and C-infinity algebraic geometry

This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\infty(X) acting by Φ_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations Φ_f satisfy many natural identities. Thus, C^\infty(X) actually has a far richer structure than the obvious R-algebra structure. We explain a version of algebraic geometry in which rings or algebras are replaced by C-infinity rings. As schemes are the basic objects in algebraic geometry, the new basic objects are "C-infinity schemes", a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss "C-infinity stacks", including Deligne-Mumford C-infinity stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on C-infinity schemes and C-infinity stacks, and orbifold strata of Deligne-Mumford C-infinity stacks. This enables us to use the tools of algebraic geometry in differential geometry, and to describe singular spaces such as moduli spaces occurring in differential geometric problems. Many of these ideas are not new: C-infinity rings and C-infinity schemes have long been part of synthetic differential geometry. But we develop them in new directions. In a new book, surveyed in arXiv:1206.4207 and at greater length in arXiv:1208.4948, the author uses C-infinity algebraic geometry to develop a theory of "derived differential geometry", which studies "d-manifolds" and "d-orbifolds", derived versions of smooth manifolds and orbifolds. D-orbifolds will have applications in symplectic geometry, as the geometric structure on moduli spaces of J-holomorphic curves.

math.DG↗

arXiv2010

On manifolds with corners

Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f : X --> Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X x_Z Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way. But we have written it as a separate paper as we believe it is of independent interest.

math.DG↗

arXiv2010

On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian deformations. It is called Hamiltonian stable if in addition the second variation of volume under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in C^n satisfying the extra condition of being Hamiltonian rigid, then for any M,w,g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p in M and locally modelled on tL for small t>0, identifying M near p with C^n near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in C^n shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to T^n, and to (S^1 x S^{n-1})/{1,-1}, and with other topologies, in every compact symplectic 2n-manifold (M,w) with compatible metric g.

math.DG↗

arXiv2010

A theory of generalized Donaldson-Thomas invariants

Let X be a Calabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integers DT^a(t) which count stable sheaves with Chern character a on X, with respect to a Gieseker stability condition t. They are defined only for Chern characters a for which there are no strictly semistable sheaves on X. They have the good property that they are unchanged under deformations of X. Their behaviour under change of stability condition t was not understood until now. This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) are rational numbers, defined for all Chern characters a, and are equal to DT^a(t) if there are no strictly semistable sheaves in class a. They are deformation-invariant, and have a known transformation law under change of stability condition. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M. We compute our invariants in examples, and make a conjecture about their integrality properties. We extend the theory to abelian categories of representations of a quiver with relations coming from a superpotential, and connect our ideas with Szendroi's "noncommutative Donaldson-Thomas invariants" and work by Reineke and others. This book is surveyed in the paper arXiv:0910.0105.

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