Overdetermined systems of PDEs: formal theory and applications Boris Kruglikov UiT the Arctic University of Norway Lecture 1: Formal Theory (d'apres D. Spencer, S. Sternberg, D. Quillen, V. Guillemin, J.-P. Serre, M. Kuranishi, H. Goldschmidt, B. Malgrange, J.-F. Pommaret, A. Vinogradov, V. Lychagin et al) Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Jet Spaces For a bundle π : E ! M consider the equivalence relation on 0 0 k+1 1 Γ(E): s ' s iff s − s 2 mx · Γ(E), where mx ⊂ C (M) is the k k max. ideal of x 2 M. The equiv. classes jxs ≡ the jet-space J π. In local coordinates (xi; uj) on E the Taylor expansion of sections i j s : x 7! u(x) yields coordinates (x ; uσ) with jσj ≤ k. This defines the jet-lift jk : Γ(E) ! Γ(J kπ), s 7! jks. We have the projections J 1π !···! J kπ ! J k−1π !···! J 0π = E ! M: k Choose a point ak 2 J π with πk;l(ak) = al for k > l, a0 = a, π(a) = x. Let F = Ker(daπ) ⊂ TaE, T = TxM. Then we identify k ∗ V (ak) = Ker(dak πk;k−1) = S T ⊗ F . Compactification: a manifold E and jets of m-dim submanifolds M ⊂ E. The last projection π : J 0 = E ! M does not exist and k ∗ Ker(dak πk;k−1) = S T ⊗ F , where T = TaM, F = TaE=TaM. Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Cartan distribution Cartan distribution is given by the formula −1 k C(ak) = spanfL(ak+1): ak+1 2 πk+1;k(ak)g ⊂ Tak J π; k k+1 k where L(ak+1) = Tak (j s) for ak+1 = jx s, ak = jxs. In local coordinates we have: (k) X j C(ak) = hD = @ i + u @ j ;@ j : jτj = ki: i x σ+1i uσ uτ jσj<k We have: C(ak) = L(ak+1) ⊕ V (ak) and dak πk;k−1C(ak) = L(ak). On J 1π we have integrable distribution X j C(a1) = L(a1) = hDi = @ i + u @ j i; x σ+1i uσ (1) i where Di = Di is the operator of total derivative along x . Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Differential equations Geometrically differential equation of order k is a submanifold k k−1 i Ek ⊂ J π that submerses on J π. We let Ei = J π for i < k (i−k) i and Ei = Ek ⊂ J π be the (i − k)-th prolongation. The s-th prolongations of Ek = ff = 0g has defining equations Dτ f = 0, jτj = s, where Dτ = Di1 ··· Dis for τ = (i1; : : : ; is). 1 Thus a differential equation is a co-filtered manifold E = fEigi=0. It is called compatible (sometimes said involutive) if πi+1;i : Ei+1 !Ei is a submersion for i ≥ k. Cartan distribution of E is the distribution CE = T E\C for each jet-level i. Ex. Complete system of equations of order k: k−1 uσ = fσ(j u); jσj = k: For this system E the Cartan distribution CE is horizontal, and the compatibility is equivalent to its Frobenius-integrability, i.e. Difτ+1j = Djfτ+1i for all i < j and τ with jτj = k − 1. Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Symbolic module Let 1M = M × R be the trivial one-dimensional bundle. If E is a ∗ linear system, then E = Hom(E; 1M ) is a D-module, i.e. a module over the algebra of differential operators Diff(1M ; 1M ). Both the algebra and the module are filtered. Passing to the 1 i graded objects at a point get: the ring ST = ⊕i=0S T of polynomial functions on T ∗M and the symbolic module gr(E∗). For nonlinear systems the symbols are bundles over E defined as i ∗ gi = Ker(dπi;i−1 : T Ei ! T Ei−1) ⊂ S T ⊗ F . In fact, starting from the symbol gk of Ek define the symbolic 1 i ∗ system fgigi=0 by: gi = S T ⊗ F for i < k and (i−k) i−k ∗ i ∗ gi = gk := (S T ⊗ gk) \ (S T ⊗ F ): ∗ ∗ The dual module g = ⊕gi is naturally a module over R = ST , and is called the symbolic module ME . Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Spencer δ-cohomology For a module M over the ring ST Koszul complex is defined by 0 M M ⊗ T M ⊗ Λ2T M ⊗ Λ3T ::: Dualizing it over R (but not over R!) we get the Spencer δ-complex, in each gradation 2 ∗ 3 ∗ 0 ! gk ! gk−1 ⊗ T ! gk−2 ⊗ Λ T ! gk−3 ⊗ Λ T ! ::: j ∗ i;j Its cohomology at the term gi ⊗ Λ T is denoted by H (g), and also Hi;j(E) if g is the symbolic system of E. The Spencer cohomology Hi;j(g) are dual to the Koszul homology ∗ ∗ Hi;j(g ), and if g = ST=I for an ideal I this also equals to Hi−1;j+1(I). Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Iterpretation of δ-cohomology Cohomology H∗;0(g) is supported in gradation 0: H0;0(g) = F and Hi;0(g) = 0 for i > 0. Cohomology H∗;1(g) counts generators of the module g∗. For the equation E: Hi;1(E) is the number of defining differential equations of order i, so in our setup it is non-zero only for i = k. Cohomology H∗;2(g) counts compatibility conditions of E. In fact, the curvature (structure tensor) of the distribution CE at a point ak, that splits pointwise CE (ak) = L(ak+1) ⊕ gk(ak), is an element 2 ∗ Ξ 2 V (ak−1) ⊗ Λ CE (ak). Its restriction to a horizontal plane 2 ∗ L(ak+1) ⊂ CE (ak) gives ΞL(ak+1) 2 gk−1 ⊗ Λ T . −1 Now change of ak+1 2 πk+1;k(ak) results in change of this tensor ∗ 2 ∗ k−1;2 by Im(δ : gk ⊗ T ! gk−1 ⊗ Λ T ), whence Wk(ak) 2 H (E). i;2 System E of order k is involutive if Wk ≡ 0, H (E) = 0 8 i ≥ k. Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Variations of Spencer cohomology When the system is not compatible (≡ involutive after prolongation), this is achieved by prolongation-projection, resulting in a smaller equation E~ ⊂ E with the same amount of solutions. Consider the partial case of Lie equation E: the linear system equations LX (q) = 0 for some geometric structure q. In other words, E describes the infinitesimal symmetries of q. In this case for classical geometries, after re-numeration gi = gi+1 we get (after prolongation-projection) the Lie algebra of symmetries g = g−1 ⊕ g0 ⊕ g1 ⊕ ::: as a symbolic system. A generalization of this is related to filtered geometries, when g = g−ν ⊕ ::: g−1 ⊕ g0 ⊕ g1 ⊕ ::: . A partial case of this is given by the class of parabolic geometries. Another generalization arises in super-geometry. The Spencer δ-cohomology is defined as the cohomology of the super-complex: 2 ∗ 3 ∗ 0 ! gk ! gk−1 ⊗ T ! gk−2 ⊗ Λ T ! gk−3 ⊗ Λ T ! ::: Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Characteristic variety From computational viewpoint let f = (f 1; : : : ; f r) be the defining (non-linear) operators of order k for E, and let i X X i,σ j `(f )(v) = Pj vσ jσ|≤k j i,σ be the linearization of the components, Pj depending on ak. ∗ i P i,σ For p = (p1; : : : ; pn) 2 T M denote Pj (p) = jσj=k Pj pσ, where pσ = pi1 ··· pik for a multi-index σ = (i1; : : : ; ik). The symbol of f at ak is 0 1 1 1 P1 (p) ··· Pm(p) B . .. C smblf (p) = @ . A r r P1 (p) ··· Pm(p) Assume E is (over)determined, in na¨ıveterms: dim F = m < r. The affine/projective/complex characteristic variety is ∗ Char(E; ak) = fp 2 Tx M : rank(smblf (p)) < mg: Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Relation to the symbolic module ∗ Recall that ME = g is the symbolic module. In the language of commutative algebra Char(E) = supp(ME ). Every projective module over the Noetherian ring R = ST can be realized as a module of sections of some sheaf. For the module ME the characteristic sheaf K is defined in such a way and evaluation at p 2 T ∗ of the stalk is the kernel of the operator m r smblf (p) considered as a map from R to R . For p 2 Char(E) the rank of evaluation belongs to [1; m]. ∗ k Covector p 2 T is characteristic iff p ⊗ v 2 gk n 0 for some v 2 F . In particular, the system E is of finite type dim g < 1 iff CharC(E) = 0 (in projective setting ;). Boris Kruglikov (Tromsø) Srni-2019 Overdetermined systems of PDEs: Lecture 1 Dimensions of the solution spaces Cartan's theory of characters gives Cauchy data that uniquely determines solutions: in the analytic context of the Cartan-K¨ahler theorem the general local solution depends on sd functions of d arguments, . , s1 functions of 1 arguments, s0 constants. These numbers have no invariant meaning except for the Cartan genre d (maximal i with si 6= 0) and Cartan integer σ = sd. In modern terms these can be defined as follows: C X d = dim Charaff(E); σ = d · deg Σ; C where Charproj(E) = [Σ is the decomposition of the projective characteristic variety into irreducible components.