The weight filtration for real algebraic varieties

The weight filtration for real algebraic varieties joint work with Clint McCrory, University of Georgia Adam Parusiński Université de Nice Sophia Antipolis Bill Bruce 60 and Terry Wall 75 An international workshop in Singularity Theory, its Applications and Future Prospects Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 1 / 21

1 Construction of weight spectral sequence 2 Construction of geometric filtration 3 Additive Invariants 4 Applications Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 2 / 21

We construct for real algebraic varieties on H cl ( ; Z 2) an analog of Deligne s weight filtration : 0 = W k 1 H k (X) W k H k (X) W 0 H k (X) = H cl k (X;Z 2). We give two different constructions of a spectral sequence that defines this filtration. C. McCrory, A. Parusiński, The weight filtration for real algebraic varieties, Topology of Stratified Spaces, MSRI Pub. 58, Cambridge University Press, New York, 2011, 121 160. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 3 / 21

Weight spectral sequence To each real algebraic variety X we associate a functorial spectral sequence E r p,q(x) = H cl p+q(x, Z 2 ). The associated filtration is the weight filtration: 0 = W k 1 H k (X) W k H k (X) W 0 H k (X) = H cl k (X; Z 2). Remark : In general the weight spectral sequence does not collapse at E 2. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 4 / 21

Construction 1 : Resolution of singularities and homological algebra. In 2002 B. Totaro proposed to introduce the weight filtration on the (co)homology of real algebraic varieties using the work of F. Guillén and V. Navarro Aznar, by analogy with Deligne s weight filtration for complex varieties as generalized by H. Gillet and C. Soulé. Recipe: 1 Define a functor on non-singular projective varieties. 2 Show that this functor satisfies some axioms. 3 By Guillén and Navarro Aznar this functor extends to singular not necessarily compact varieties by means of resolution of singularities. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 5 / 21

Construction 1. 1 To M non-singular compact we associate the (co)chain complex with canonical filtration. M (C cl (M, Z 2),F can ) Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 6 / 21

Construction 1. 1 To M non-singular compact we associate the (co)chain complex with canonical filtration. M (C cl (M, Z 2),F can ) 2 Blow-up property. Verify that for any blow-up X = Bl C (X) X, X,C projective non-singular, E the exceptional divisor, we have the short exact sequences 0 H k (E; Z 2 ) H k (C; Z 2 ) H k ( X; Z 2 ) H k (X; Z 2 ) 0 3 By [G., N. A.] this functor extends to arbitrary real algebraic varieties X WC (X) and WC (X) is a filtered chain complex well-defined up to a filtered quasi-isomorphism. The associated spectral sequence is the weight spectral sequence. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 6 / 21

Construction 2. For arbitrary real algebraic variety X we define a functorial filtration 0 G k C k (X) G 0 C k (X) = C k (X) on the complex of semi-algebraic chains C k (X) := C cl k (X; Z 2). Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 7 / 21

Construction 2. For arbitrary real algebraic variety X we define a functorial filtration 0 G k C k (X) G 0 C k (X) = C k (X) on the complex of semi-algebraic chains C k (X) := C cl k (X; Z 2). Let c C k (X). We resolve the singularities of the Zariski closure of c. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 7 / 21

Construction 2. For arbitrary real algebraic variety X we define a functorial filtration 0 G k C k (X) G 0 C k (X) = C k (X) on the complex of semi-algebraic chains C k (X) := C cl k (X; Z 2). Let c C k (X). We resolve the singularities of the Zariski closure of c. So assume Y = c Z is smooth compact and locally c is defined in Y by monomial inequalites. Then, c G k+p C k (X) if, locally, c is a combination of chains with at most p-corners as singularities. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 7 / 21

Construction 2. Theorem The filtration G is functorial for proper regular morphisms and represents the Guillén, Navarro Aznar extension X WC (X). Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 8 / 21

Construction 2. Theorem The filtration G is functorial for proper regular morphisms and represents the Guillén, Navarro Aznar extension X WC (X). What does it mean that for X singular compact that α W k H k (X; Z 2 ) (α is pure )? Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 8 / 21

Construction 2. Theorem The filtration G is functorial for proper regular morphisms and represents the Guillén, Navarro Aznar extension X WC (X). What does it mean that for X singular compact that α W k H k (X; Z 2 ) (α is pure )? Answer 1: α is the image by a regular map of the fundamental class of a connected component of a compact real algebraic set. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 8 / 21

Construction 2. Theorem The filtration G is functorial for proper regular morphisms and represents the Guillén, Navarro Aznar extension X WC (X). What does it mean that for X singular compact that α W k H k (X; Z 2 ) (α is pure )? Answer 1: α is the image by a regular map of the fundamental class of a connected component of a compact real algebraic set. Answer 2: α can be represented by an arc-symmetric set (in the sense of K. Kurdyka). Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 8 / 21

Generalized Euler characteristics for real algebraic varieties A generalized Euler characteristic associates to each real algebraic variety a number e(x) so that 1. e(x \ Y ) = e(x) e(y ) if Y X is a closed subvariety 2. e(x Y ) = e(x)e(y ). 3. e(x) = e(y ) if X isomorphic to Y. Example: Euler characteristic with compact support χ c (X) = i ( 1) i dimh i c (X;k) = i ( 1) i dimh cl i (X;k) (The only homeomorphism invariant generalized Euler characteristic) Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 9 / 21

Virtual Poincaré polynomial Theorem There is a unique generalized Euler caracteristic β(x)(u) = i β i (X)u i, such that β i (X) = dim Z2 H i (X, Z 2 ) for X non-singular and compact. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 10 / 21

Virtual Poincaré polynomial Theorem There is a unique generalized Euler caracteristic β(x)(u) = i β i (X)u i, such that β i (X) = dim Z2 H i (X, Z 2 ) for X non-singular and compact. Definition : β q (X) = p ( 1) p dim Z2 Ẽ 2 p,q, where Ẽ2 p,q = E1 q,p +2q. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 10 / 21

Virtual Poincaré polynomial Note: deg β(x) = dimx, β dim X > 0, β(x) = 0 X =. Theorem Let e : Var G be an additive invariant defined on real algebraic sets such that for any pair X,Y of C diffeomorphic nonsingular compact real algebraic sets we have e(x) = e(y ). Then there exists a unique group homomorphism h : Z[u] G such that e = h β. If, moreover, e is multiplicative then h is a ring homomorphism. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 11 / 21

Theorem (Toshizumi Fukui, Krzysztof Kurdyka, A. P.) Let h : (R n,0) (R n,0) be a semialgebraic homeomorphism germ. Suppose that h and h 1 are arc-analytic. Then h Lipschitz = h 1 Lipschitz Definition f : U R n R m is arc-analytic if for every real analytic arc germ γ : (R,0) (R n,p), p U, f γ is again real analytic. Examples: f (x,y) = x 4 + y 4 and Q(x,y) = x5 y x 6 + y4, Q(0,0) = 0, Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 12 / 21

Definition f : U R n R m is blow-analytic if there is a composition of finitely many blowings-up with smooth nowhere dense centers, σ : M U, such that f σ is real analytic. Always : f blow-analytic = f arc-analytic. Theorem (Bierstone & Milman, A.P.) If the graph of f is semi-algebraic then, f blow-analytic f arc-analytic Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 13 / 21

Theorem Let h : (R n,0) (R n,0) be a semialgebraic homeomorphism germ. Suppose that h and h 1 are arc-analytic. Then h Lipschitz = h 1 Lipschitz Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 14 / 21

Theorem Let h : (R n,0) (R n,0) be a semialgebraic homeomorphism germ. Suppose that h and h 1 are arc-analytic. Then h Lipschitz = h 1 Lipschitz To prove this theorem it suffices to show that Theorem Suppose that h and h 1 are arc-analytic and semialgebraic. If the jacobian determinant det(dh) is bounded from above, then it is bounded from below. C det(dh) C = c 0 < c det(dh) Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 14 / 21

Proof. The jacobian matrix of h 1 equals 1 (cofactor matrix of the jacobian matrix of h) det(dh) Since h is Lipschitz, each coefficient of the jacobian matrix of h 1 is bounded, and hence h 1 is Lipschitz. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 15 / 21

Choice of a good category : { algebraic sets, polynomial maps } { AS- sets (arc-symmetric semialgebraic sets), maps with AS-graphs } It contains, in particular, Nash maps and Nash manifolds (Nash = real analytic + semialgebraic), connected or analytic components of compact real algebraic sets, and differences of such sets. The virtual Poincaré polynomial can be extended to this category. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 16 / 21

Arc space and jet spaces of a Nash manifold M at S M. L(M,S) = {γ : (R,0) (M,p); p S, γ analytic} L k (M,S) = L(M,S)/ k γ 1 (t) k γ 2 (t) iff γ 1 (t) γ 2 (t) mod t k+1. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 17 / 21

Arc space and jet spaces of a Nash manifold M at S M. L(M,S) = {γ : (R,0) (M,p); p S, γ analytic} L k (M,S) = L(M,S)/ k γ 1 (t) k γ 2 (t) iff γ 1 (t) γ 2 (t) mod t k+1. Let σ : (M,E) (R n,0) be a modification. L(M,E) π k σ L(R n,0) π k L k (M,E) σ,k L k (R n,0) Then σ is a bijection (up to sets of infinite codimension). Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 17 / 21

Arc space and jet spaces of a Nash manifold M at S M. L(M,S) = {γ : (R,0) (M,p); p S, γ analytic} L k (M,S) = L(M,S)/ k γ 1 (t) k γ 2 (t) iff γ 1 (t) γ 2 (t) mod t k+1. Let σ : (M,E) (R n,0) be a modification. L(M,E) π k σ L(R n,0) π k L k (M,E) σ,k L k (R n,0) Then σ is a bijection (up to sets of infinite codimension). But σ,k is not! Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 17 / 21

Given a positive integer e. Define e,k = π k ( e ). e = {γ L(M,E);ord t det dσ(γ(t)) = e} Theorem (Kontsevich, Denef & Loeser) If k 2e, e 1, then e,k σ,k ( e,k ) is a piece-wise trivial fibration with fiber R e. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 18 / 21

Given a positive integer e. Define e,k = π k ( e ). e = {γ L(M,E);ord t det dσ(γ(t)) = e} Theorem (Kontsevich, Denef & Loeser) If k 2e, e 1, then e,k σ,k ( e,k ) is a piece-wise trivial fibration with fiber R e. In particular : β( e,k ) = β(σ,k ( e,k ))β(r e ) Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 18 / 21

For simplicity we suppose that we have a diagram (M,E) σ H=Id (M,E) σ (R n,0) h (R n,0) with σ,σ modifications over the origin : E = σ 1 (0) = σ 1 (0) = E i and [det(dσ)] = i I ν i E i, and [det(dσ )] = i I ν i E i. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 19 / 21

For simplicity we suppose that we have a diagram (M,E) σ H=Id (M,E) σ (R n,0) h (R n,0) with σ,σ modifications over the origin : E = σ 1 (0) = σ 1 (0) = E i and [det(dσ)] = i I ν i E i, and [det(dσ )] = i I ν i E i. It suffices to show that i I ν i ν i = i I ν i = ν i. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 19 / 21

For a vector j = (j i ) i I, j i N we denote E j = j i 0 E i \ j l =0 E l j = {γ L(M, E j ) : ord γ(t) E i = j i, i I }. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 20 / 21

For a vector j = (j i ) i I, j i N we denote E j = j i 0 E i \ j l =0 E l j = {γ L(M, E j ) : ord γ(t) E i = j i, i I }. Then j e(j), where e(j) = ν,j = i I ν ij i. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 20 / 21

For a vector j = (j i ) i I, j i N we denote E j = j i 0 E i \ j l =0 E l j = {γ L(M, E j ) : ord γ(t) E i = j i, i I }. Then j e(j), where e(j) = ν,j = i I ν ij i. Denote j,k = π k ( j ), s j = #{i;j i 0}. Lemma. β(σ,k ( j,k )) = β( E j ) (u 1) j u nk s j ν,j, Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 20 / 21

L k (R n,0) = Z k (σ) σ,k ( j,k ). j A k (σ) where A k (σ) = {j : ν,j k/2} and codim Z k (σ) as k. Conclusion. u n(k+1) = β(z k (σ)) + β( E j )(u 1) j u nk s j ν,j j A k (σ) = β(z k (σ )) + β( E j )(u 1) j u nk s j ν,j j A k (σ ) This shows that i I ν i ν i = i I ν i = ν i. Adam Parusiński (Nice) Weight filtration for real algebraic varieties Liverpool, 18-22 June 2012 21 / 21