Torus actions, Morse homology, and the Hilbert scheme of points on affine space
aa r X i v : . [ m a t h . AG ] S e p Torus actions, Morse homology, and the Hilbert schemeof points on affine space
Burt Totaro
We formulate a conjecture on actions of the multiplicative group G m in algebraicgeometry. (Over the complex numbers, this group may be called C ∗ .) In short, if G m acts on a quasi-projective scheme U which is attracted as t → G m toa closed subset Y in U , then the inclusion Y → U should be an A -homotopyequivalence (Conjecture 1.1). This is not obvious, in that the action of G m on U usually does not extend to a morphism A × U → U ; compare Figure 1. Weshow that the inclusion Y → U over the complex numbers is at least a homotopyequivalence in the classical topology (Theorem 1.2). This extends work of Hauseland Rodriguez-Villegas on the case where U is smooth [13, Corollary 1.3.6]. Weprove several other results in the direction of the conjecture, including a homotopyequivalence on real points (Theorem 4.1) and, when U is smooth, an A -homotopyequivalence after a suitable suspension (Theorem 7.1). The proofs use the ideas ofMorse homology, translated into algebraic geometry (Proposition 2.1).We apply these results to the Hilbert scheme of points on affine space. TheHilbert scheme of points on an algebraic surface is smooth, and its Betti num-bers were computed by G¨ottsche [11]. The Hilbert scheme of points on a higher-dimensional variety, even affine space A n , is more mysterious. It has many irre-ducible components [16], and for n ≥
16 its singularities satisfy Murphy’s law upto retraction [17]. Nonetheless, progress was recently made toward understandingthe homotopy type (and even the A -homotopy type) of Hilb d ( A n ) for n large com-pared to d . In particular, in the limit where n goes to infinity, Hilb d ( A ∞ ) has the A -homotopy type of the infinite Grassmannian Gr d − ( A ∞ ) ≃ BGL ( d −
1) [15].There are also corresponding stability theorems. In particular, over the complexYUFigure 1: Example of a T -action on U , with Y = P shown as the horizontal line.The arrows point in the direction t →
0. The fixed point set Y T consists of twopoints. 1umbers, the resulting homomorphism on integral cohomology, H ∗ ( BGL ( d − , C ) , Z ) = Z [ c , . . . , c d − ] → H ∗ (Hilb d ( A n ) , Z ) , is an isomorphism in degrees at most 2 n − d + 2 [15].This paper considers another homotopical property of the Hilbert scheme Hilb d ( A n )for finite n . Namely, over the complex numbers, we show that Hilb d ( A n ) (in theclassical topology) has the homotopy type of Hilb d ( A n , G m -actions. For example, it follows that the weightfiltration on the rational cohomology H i (Hilb d ( A n ) , Q ) is concentrated in weights ≤ i , since that holds for proper schemes over C [9].It remains open whether Hilb d ( A n ,
0) is A -homotopy equivalent to Hilb d ( A n ),over the complex numbers or any other field. This would follow from our generalConjecture 1.1. We can say something about the unstable A -homotopy type ofthese spaces, namely that Hilb d ( A n ,
0) and Hilb d ( A n ) are A -connected (Theorems5.1 and 5.3).As a tool, we extend one of Bachmann’s conservativity theorems, relating themotivic stable homotopy category to the derived category of motives along withreal realizations (Theorem 6.1).This work was supported by NSF grant DMS-1701237. I spoke about it inthe Algebraic Geometry and Moduli Zoominar at ETH Zurich. Thanks to TomBachmann, David Hemminger, Marc Hoyois, Joachim Jelisiejew, Denis Nardin, andMaria Yakerson for their suggestions. G m -actions in mo-tivic homotopy theory We formulate here a general conjecture about actions of the multiplicative group T = G m in motivic homotopy theory. (For motivic homotopy theory as defined byMorel and Voevodsky, a reference is [22] and an introduction is [1].) Namely, if T acts on a quasi-projective scheme U which is attracted as t → T to a closedsubset Y in U , then the inclusion Y → U should be an A -homotopy equivalence(Conjecture 1.1). We show that (over the complex numbers) the inclusion Y → U is at least a homotopy equivalence in the classical topology (Theorem 1.2, provedin section 3).Let Hilb d ( A n ) be the quasi-projective scheme of zero-dimensional degree- d closedsubschemes of affine space A n over a field k . When k is the complex numbers, wededuce from Theorem 1.2 that Hilb d ( A n ) has the homotopy type of Hilb d ( A n , Conjecture 1.1.
Let X be a projective scheme over a field k with an action of T = G m . Suppose that there is a T -equivariant ample line bundle on X . Let Y be a T -invariant closed subscheme of X such that every point x in X withlim t →∞ ( tx ) ∈ Y is in Y . Suppose that the fixed point set Y T is open in X T . Let U be the subset of points x in X such that lim t → ( tx ) is in Y ; then U is open in X ,and Y is contained in U . 2hen the inclusion Y → U is an A -homotopy equivalence (that is, an isomor-phism in the A -homotopy category H ( k )).The assumption that X has a T -equivariant ample line bundle is automatic if X is normal [27, Theorem 1.6].The conjecture would be useful for motivic homotopy theory, since G m -actionsoccur everywhere. When U is smooth, both Y and U are unions of affine bundlesover the connected components of Y T , by Bia lynicki-Birula [6]. But even then, weonly know how to prove that Y → U is an A -homotopy equivalence after a suitablesuspension (Theorem 7.1). Regardless of whether U is smooth, the conjecture wouldbe clear if the T -action on U extended to a morphism A × U → U (since 0 × U would map into Y ), but in general there is no such morphism. (Even A × Y → Y need not be continuous; consider the case where Y is P with thestandard action of G m .)As evidence for the conjecture in the singular case, we prove the following weakerstatement in section 3. Theorem 1.2 was proved in the case where U is smooth byHausel and Rodriguez-Villegas [13, Corollary 1.3.6]. Theorem 1.2.
Under the assumptions of Conjecture 1.1 with base field C , theinclusion Y → U is a homotopy equivalence (in the classical topology). Corollary 1.3.
Over the complex numbers, the inclusion from
Hilb d ( A n , to Hilb d ( A n ) (in the classical topology) is a homotopy equivalence.Proof. (Corollary 1.3) Let X = Hilb d ( P n ), Y = Hilb d ( A n ,
0) and U = Hilb d ( A n ).The idea is to use the action of the multiplicative group T (that is, C ∗ ) on Hilb d ( A n ),coming from the action of T by scaling on A n . Here X has a GL ( n + 1)-equivariantample line bundle by construction. (Namely, Grothendieck constructed the Hilbertscheme as a closed subscheme of the Grassmannian of subspaces of the vector spaceof homogeneous polynomials of sufficiently high degree, sending a closed subscheme S ⊂ P n to the linear subspace of polynomials that vanish on S [19, section I.1]. Thestandard ample line bundle O (1) on the Grassmannian is GL ( n + 1)-equivariant.)In particular, X has a T -equivariant ample line bundle.In the limit as t ∈ T approaches 0, every subscheme of degree d in A n is movedto a subscheme supported at the origin, as we want. The action does not extend toa morphism A × U → U (or even A × Y → Y ). Nonetheless, the desired homotopyequivalence follows from Theorem 1.2. G m -actions and broken trajectories We show here that for an action of the multiplicative group T on a projective scheme,every limit of T -orbits is a broken trajectory , meaning a chain of T -orbits that con-nect a finite sequence of T -fixed points. This is analogous to fundamental results inMorse homology. Namely, given a smooth function on a closed Riemannian man-ifold satisfying some mild conditions, every limit of gradient flow lines is a brokentrajectory, meaning a chain of gradient flow lines that connect a finite sequence of3ritical points [5, Theorem 4.9, Definition 4.10]. For a T -action on a smooth com-plex projective variety, one can in fact deduce the results here from those in Morsehomology, applied to a Hamiltonian function for the T -action. Instead, we give adirect proof over any field. It turns out that smoothness is irrelevant. Proposition 2.1.
Let X be a projective scheme over a field k with an action of T = G m . Suppose that there is a T -equivariant ample line bundle on X . Then everylimit of T -orbit closures in X (in the Chow variety of effective 1-cycles on X ) isa broken trajectory, that is, a chain of T -orbits (with some positive multiplicities)connecting some T -fixed points.In more detail: let C be a smooth curve over k with a morphism f : C → X .Composing f with the action of T gives a morphism T × C → X , which extendsto a morphism P × ( C − Z ) → X for some 0-dimensional closed subset Z in C .This gives a morphism g from C − Z to the Chow variety of 1-cycles on X , whichextends to all of C by properness of Chow varieties. Then for each k -point c in C (possibly in Z ), g ( c ) is a broken trajectory over k , meaning the sum of T -orbits (withsome positive multiplicities) of points y , . . . , y n in X ( k ) that connect T -fixed points x , . . . , x n in X ( k ) . More precisely, lim t → t ( y i ) = x i − and lim t →∞ t ( y i ) = x i foreach ≤ i ≤ n .Proof. There is a T -equivariant embedding of X into the projective space P ( V ) forsome representation V of T . Given that, we can assume that X = P ( V ); this greatlysimplifies the situation. Then T acts on X = P r by t ([ z , . . . , z r ]) = [ t a z , . . . , t a r z r ]for some integers a i . We can assume that a ≤ · · · ≤ a r .The completed local ring of C at c is isomorphic to the power series ring k [[ u ]].So the curve f : C → X near c is given by some power series [ z ( u ) , . . . , z r ( u )] with z i ( u ) ∈ k [[ u ]], not all zero. A reference for the Chow variety of effective 1-cycles is[19, section 1.3]. The limit of the T -orbits of points in C approaching c in the Chowvariety (ignoring the multiplicities of the limit 1-cycle), viewed as a subset of X ( k ),is the set of all k -points in X that can be written as p = lim u → [ g ( u ) a z ( u ) , . . . , g ( u ) a r z r ( u )]for some g in the algebraic closure k (( u )), g = 0.This limit point depends mainly on the rational number b := ord u ( g ). Thesituation is described by the Newton polygon of the pairs ( a i , ord u ( z i )) in Z × ( Z ∪∞ ),as in Figure 2. (Here ord u ( z i ) = ∞ if z i ( u ) is identically zero.)Namely, let I be the set of numbers i ∈ { , . . . , r } such that ba i +ord u ( z i ) reachesits minimum value. (These correspond to the points where a line of slope − b meetsthe boundary of the Newton polygon.) Then we compute that the limit point p defined above has all coordinates zero except the i th coordinate for elements i ∈ I .Replacing g by another function with the same value of b (that is, multiplying g bya unit h ( u )) just replaces p by h (0)( p ), another point in the same T -orbit as p .For all but finitely many rational numbers b , the limit point p above belongsto a set { x , x , . . . , x n } of T -fixed points, these being indexed by the vertices ofthe Newton polygon. (For these values of b , all nonzero coordinates i in the set I above have the same weight a i , which means that p is a T -fixed point.) Forthe remaining n values of b , corresponding to the non-vertical edges of the Newtonpolygon, the limit point can be anywhere in the T -orbit of a certain point y i in X i ord u ( z i )Figure 2: Newton polygon of the pairs ( a i , ord u ( z i ))with lim t → t ( y i ) = x i − and lim t →∞ t ( y i ) = x i . Here the points y , . . . , y n (andhence the points x , x , . . . , x n ) can be taken to be k -points of X , by choosing thefunction g ∈ k (( u )) with a given value of ord u ( g ) ∈ Q to lie in a totally ramifiedextension of k (( u )), for example in k (( u /e )) for a positive integer e . Proof.
To recall the assumptions: we have a projective scheme X over C with anaction of T = G m , and there is a T -equivariant ample line bundle on X . Wehave a T -invariant closed subscheme Y of X such that every point x in X withlim t →∞ ( tx ) ∈ Y is in Y , and the fixed point set Y T is open in X T . Let U be thesubset of points x in X such that lim t → ( tx ) is in Y ; then U is Zariski open in X ,and Y is contained in U . We want to show that the inclusion Y → U is a homotopyequivalence in the classical topology. Lemma 3.1.
Let q , q , . . . be a sequence of complex points in X that converge toa T -fixed point w . Let t , t , . . . be a sequence in C ∗ that converges to zero in C .Then any limit point of the sequence t i ( q i ) in X lies in a broken trajectory “below w .” That is, such a limit point belongs to the union of the T -orbits of some points y , . . . , y n in X and some T -fixed points x , . . . , x n = w such that lim t → t ( y i ) = x i − and lim t →∞ t ( y i ) = x i for each ≤ i ≤ n .Proof. We largely follow the proof of Proposition 2.1. Choose a T -equivariant em-bedding of X into P ( V ) for some representation V of T . We can write the actionof T on P r = P ( V ) by t ([ z , . . . , z r ]) = [ t a z , . . . , t a r z r ] with a ≤ · · · ≤ a r . Afterpassing to a subsequence, we can assume that the points q , q , . . . all have the samelowest weight a j of a nonzero coefficient. On the locally closed subset K in X ofpoints with this lowest weight, the G m -action G m × K → K extends to a morphism f : A × K → K , by inspection. Here K denotes the closure of K in X .By assumption, the points ( t i , q i ) in A × K converge to the point (0 , w ) in A × K . The rational map f : A × K K becomes a morphism after some blow-up M → A × K that is an isomorphism over the complement of 0 × ( K − K ). Soany limit point of the sequence t i ( q i ) in X is equal to f ( m ) for some point m in M over (0 , w ) ∈ A × K . In particular, we can choose a smooth algebraic curve witha morphism to M that goes through m and meets the open set G m × K .5hus, by considering the completion of this curve at the point that maps to m ,we have power series g ( u ) = 0 ∈ C [[ u ]] and z ( u ) ∈ X ( C (( q ))) such that g (0) = 0,lim u → z ( u ) = w , and lim u → ( g ( u ))( z ( u )) is the given limit point in X . The proofof Proposition 2.1 showed that the limit of the T -orbits of z ( u ) as u approaches 0is a broken trajectory in X , which clearly contains w as one of the T -fixed points x , . . . , x n , say w = x j . Moreover, since g (0) = 0 (so that b := ord u ( g ) > u → ( g ( u ))( z ( u )) in P r shows that this limit point is “below w ”, that is, in the union of x , . . . , x j = w and the T -orbits that connect them.We continue the proof of Theorem 1.2. By the triangulation of real semialgebraicsets, there is a triangulation of X with Y as a subcomplex [14, section 1]. Therefore, Y has arbitrarily small simplicial regular neighborhoods N in X , and for these theinclusion Y → N is a homotopy equivalence [25, chapter 3]. Let N be a (compact)regular neighborhood of Y contained in U .Consider the submonoid (0 ,
1] of T = C ∗ . It would be convenient to have(0 , · N ⊂ N , but it is not obvious that we can arrange that. Instead, we argueas follows. I claim that each point w ∈ Y T has a neighborhood N in U suchthat t ( N ) ⊂ N for all t ∈ (0 , q i in U converging to w such that for each positive integer j , (0 , · q j is not contained in N . So there is a sequence t i ∈ (0 ,
1] such that t i ( q i ) is not in N . The sequence t i must converge to zero; otherwise, a subsequence of t i ( q i ) would converge to the T -fixed point w in Y (and hence infinitely many of those points would be in N ).After passing to subsequences, we can assume that t i ( q i ) converges to a point v in X − int( N ), hence not in Y . By Lemma 3.1, v belongs to the union of somefinite chain of T -orbits going “down” from w , meaning the T -orbits of some points y , . . . , y n in X and some T -fixed points x , . . . , x n = w such that lim t → t ( y i ) = x i − and lim t →∞ t ( y i ) = x i . By our assumption on Y , it follows that v is in Y , acontradiction. Thus we have proved the claim that each point w ∈ Y T has aneighborhood N in U such that t ( N ) ⊂ N for all t ∈ (0 , x ∈ U (not just in Y T ), there is a real number a ∈ (0 ,
1] and a neighborhood N in U such that t ( N ) ⊂ N for all t ∈ (0 , a ]. Thatfollows from the previous statement applied to the point y = lim t → t ( x ) ∈ Y T .Therefore, for every compact subset K of U , there is a real number a ∈ (0 ,
1] suchthat t ( K ) ⊂ N for all t ∈ (0 , a ]. Equivalently, K ⊂ a − ( N ). In particular, there isa real number c > N of Y is contained inthe interior of c ( N ). It also follows that U is the union of the subsets c j ( N ) overall j ≥ Y → N is a homotopy equivalence, so is the inclusion Y → c j ( N ) for each integer j . Therefore, each of the inclusions c j ( N ) → c j +1 ( N ) is alsoa homotopy equivalence. Since c j ( N ) is a closed subset contained in the interior of c j +1 ( N ), the union of these subsets (namely, U ) has the colimit topology. Since thisis a filtered colimit, the colimit U is equivalent to the homotopy colimit, and so theinclusion N → U is a homotopy equivalence. Since the inclusion Y → N is also ahomotopy equivalence, we conclude that Y → U is a homotopy equivalence.6 The real case
Theorem 4.1.
Under the assumptions of Conjecture 1.1 with base field R , theinclusion Y ( R ) → U ( R ) is a homotopy equivalence.Proof. This is similar to the complex case (Theorem 1.2). In particular, Lemma 3.1holds by the same proof over R in place of C , using that Proposition 2.1 expressesany limit of T -orbits of R -points as the union of a finite chain of T -orbits of R -points. Given that, the proof of Theorem 1.2 applies verbatim (using a regularneighborhood of Y ( R ) inside U ( R )) to show that the inclusion Y ( R ) → U ( R ) is ahomotopy equivalence. A -connectedness of the Hilbert scheme Hartshorne showed that the Hilbert scheme of projective space over a field k (of sub-schemes with a given Hilbert polynomial) is connected [12]. In particular, Hilb d ( P n )is connected for every n ≥ d ≥
0. The argument was sharpened by Reevesand Pardue [24, 23]. Reeves and Pardue showed that for an infinite field k , any two k -points of Hilb d ( P n ) can be connected by a chain of affine lines over k . By Morel’sresults (Lemma 5.2 below), it follows that Hilb d ( P nk ) is A -connected for k infinite.We now show that Hilb d ( A n ) and Hilb d ( A n ,
0) are A -connected over an infinitefield k . This seems to be harder for Hilb d ( A n , d >
1) this spacecontains no smooth subschemes of A n . When n ≥ d , the A -connectedness of theseHilbert schemes can be proved using the ideas of [15], but here we want the resultsfor all n and d . Theorem 5.1.
Let k be an infinite field, n ≥ , d ≥ . Then Hilb d ( A n ) is A -connected over k .Proof. We use the following result of Morel’s:
Lemma 5.2.
Let X be a separated scheme of finite type over a field k such that X has a k -point. Suppose that for every separable finitely generated field extension F of k , any two F -points of X can be connected by a chain of affine lines A → X over F . Then X is A -connected.Proof. For m ≥
0, Morel showed that an A -local pointed simplicial Nisnevichsheaf X over k is m -connected if and only if the fiber X ( F ) is m -connected forevery separable finitely generated field extension F of k [21, Lemma 6.1.3]. Also,for a simplicial sheaf X , π ( X ) → π A ( X ) is a surjection of Nisnevich sheaves [22,section 2, Corollary 3.22]. In particular, for a separated scheme X of finite typeover k , X ( F ) → π A ( X )( F ) is surjective for every separable finitely generated fieldextension F over k . This implies the lemma.By Lemma 5.2, it suffices to show that for an infinite field k , any two k -pointsof U := Hilb d ( A n ) can be connected by a chain of affine lines A → U over k . So let S be any k -point of U . That is, S is a closed subscheme of A n over k of dimensionzero and degree d . We use a “Gr¨obner degeneration,” as follows. Let c be a largepositive integer, and consider the action of T := G m on A n by t ( x , . . . , x n ) = ( t c x , t c x , . . . , t c n x n ) . S := lim t → t ( S ) exists in U . It is a closed subscheme supported at theorigin in A n , and it is fixed by this T -action. That is, the defining ideal I of S is homogeneous with respect to the weights ( c, c , . . . , c n ) on x , . . . , x n . Taking c big enough compared to d and n , it follows that I is generated by monomials. Byconstruction, we can connect S to S by an affine line over k .Since S has dimension 0 and is defined by monomials, it is smoothable, us-ing Hartshorne’s proof by distraction ; a specific reference is [8, Proposition 4.15].We need the more precise information given by the proof, as follows. Let I =( x M , . . . , x M r ) in multi-index notation, so x M i = Q nj =1 x M ij j . Consider the follow-ing flat family of ideals in k [ x , . . . , x n ] parametrized by affine space A d : for a point( a , . . . , a d − ) in A d , take the ideal J a in k [ x , . . . , x n ] generated by the elements f i := n Y j =1 ( x j − a )( x j − a ) · · · ( x j − a M ij − ) . The initial ideal of J a (with respect to any monomial order compatible with thegrading, say the graded reverse lexicographic order) is I ; so we have a flat family.This defines a morphism A d → Hilb d ( A n ) over k , with the origin mapping to thegiven monomial scheme S . When a , . . . , a d − are distinct elements of k , the sub-scheme Z a of A n defined by J a contains d distinct k -points: namely, for each of the d monomials x L not in I , Z a contains the k -point ( a L , . . . , a L n ). Since the scheme Z a has degree d , it must be smooth over k , equal to those d k -points in A n .Since k is infinite, it follows that we can connect S by an affine line in Hilb d ( A n )to a scheme S which consists of d distinct k -points in A n . If n ≥
2, since thecondition for two points to be equal in A n has codimension at least 2, it is easy toconnect S by a chain of affine lines over k to a fixed arrangement S of d distinct k -points in A n . Thus Hilb d ( A n ) is A -connected when n ≥
2. It is also A -connectedwhen n = 1, since Hilb d ( A ) ∼ = A d . Theorem 5.3.
Let k be an infinite field, n ≥ , d ≥ . Then Hilb d ( A n , is A -connected over k .Proof. By Lemma 5.2, it suffices to show that for every infinite field k , any two k -points of Y := Hilb d ( A n ,
0) can be connected by a chain of affine lines over k . Forlack of a direct proof, we will reduce this to Theorem 5.1.Let X = Hilb d ( P n ), U = Hilb d ( A n ), and T = G m . Consider the action of T on X coming from the action of T by scaling on A n . Then Y is a T -invariant closedsubset of U , and lim t → t ( x ) exists in Y for each point x in U . Clearly we canconnect any k -point x in Y to this limit point by an affine line in Y , and the limitpoint is fixed by T . So it suffices to show that any two k -points p, q in Y T can beconnected by a chain of affine lines in Y .We know by the proof of Theorem 5.1 that p and q can be connected by a chainof affine lines in U . So it suffices to show that for any morphism f : A → U over k , we can connect lim t → t ( f (0)) to lim t → t ( f (1)) by a chain of affine lines in Y .Composing f with the action of T on U gives a morphism G m × A → U over k , which can be viewed as a rational map P × P X over k . Since X is properover k , this map becomes a morphism after blowing up the domain finitely manytimes at closed points. It follows that g ( s ) := lim t → t ( f ( s )) defines a morphism8 : A − Z → Y for some 0-dimensional closed subset Z of A . Since Y is properover k , g extends to a morphism g : A → Y . As a result, for any two k -points s , s in A − Z , lim t → t ( f ( s )) and lim t → t ( f ( s )) can be connected by an affine line in Y . There remains the case where 0 or 1 is in Z . It suffices to show that for any k -point s in Z (which will be 0 or 1 for us), the point y := lim t → t ( f ( s )) can beconnected by a chain of affine lines in Y to g ( s ).By Proposition 2.1, the T -orbits of the points f ( s ) (for s ∈ A − Z ) converge as s approaches s to a “broken trajectory” containing f ( s ). This means the unionof T -orbits of points y , . . . , y n in X ( k ) that connect T -fixed points x , . . . , x n in X ( k ), in the sense that lim t → t ( y i ) = x i − and lim t →∞ t ( y i ) = x i .Both the k -point z := lim t → t ( s ) and the k -point g ( s ) lie in this union of T -orbit closures in X , and both are in the closed subset Y . We know that everypoint x in X with lim t →∞ ( tx ) ∈ Y is in Y . Therefore, all the orbit closures thatconnect z to g ( s ) are in Y . So these two points can be connected by affine linesover k in Y , as we want. Extending one of Tom Bachmann’s results, we prove the following conservativitytheorem, relating the motivic stable homotopy category with the derived categoryof motives along with real realizations. Thanks for Bachmann for his suggestions.This result will be used in the proof of Theorem 7.1.
Theorem 6.1.
Let k be a finitely generated field of characteristic zero. Let A be acompact object in SH ( k ) such that M ( A ) = 0 in DM ( k ) and for every embeddingof k into R , H ∗ ( A ( R ) , Z [1 / . Then A = 0 .Proof. Bachmann showed (in particular) that if A is a compact object in SH ( k )such that M ( A ) = 0 in DM ( k ) and for every σ in the space Sper( k ) of orderings of k , M σ [1 / A ) = 0 in D ( Z [1 / A = 0 [2, Theorem 33]. When σ comes froman embedding of k into R , M σ [1 / A ) is the complex that computes the singularhomology of the corresponding real realization of A , H ∗ ( A ( R ) , Z [1 / k .We use the following property of the space Sper( k ) of orderings of k [10, Lemma1.9]. The topology on Sper( k ) is defined by taking the sets { σ : a > σ } for a ∈ k asa sub-basis for the topology. This makes Sper( k ) into a compact Hausdorff totallydisconnected space. Lemma 6.2.
Let k be a finitely generated field of characteristic zero. Then the setof archimedean orderings of k is dense in the topological space Sper( k ) of orderingsof k , and every archimedean ordering comes from an embedding of k into R . Given that, we are done if we can show that the support in X := Sper( k ) of acompact object in SH ( k ) is open as well as closed. This is related to the generalfact that for a tensor triangulated category K , the support of an object of K in9he Balmer spectrum Spc( K ) is closed and its complement is quasi-compact [4,Proposition 2.14]. However, we will argue more directly.We use that the functors M σ come from a functor from SH ( k ) to the derivedcategory of sheaves D ( X, Z [1 / SH ( k ) to Witt motives DM W ( k, Z [1 / DM W ( k, Z [1 / D ( X, Z [1 / D ( X, Z [1 / Z [1 / X is compact, Hausdorff, and totally disconnected, every open subsetof X is a union of clopen subsets (or equivalently, quasi-compact open subsets). Itfollows that every compact object in D ( X, Z [1 / D ( X, Z [1 / j ! ( Z [1 / U ), with j : U ֒ → X the inclusion of a quasi-compact open subset [26,Lemma 094C]. Clearly such a summand is a perfect complex of Z [1 / X .)Because sections of the sheaf Z [1 /
2] on X = Sper( k ) are locally constant, thesupport of a perfect complex on X is open as well as closed. G m -actions on smooth varieties and motivic homo-topy theory We now consider 1.1 in the special case where U is smooth. (One example wherethis applies is the inclusion from Hilb d ( A ,
0) to Hilb d ( A ).) For U smooth, we showthat the inclusion Y → U becomes an A -homotopy equivalence after suspendingby S , = S ∧ G m . It follows that Y and U have many invariants in common,such as motivic homology and cohomology, l -adic cohomology, and so on. On theother hand, it remains open whether the Nisnevich sheaf π A is the same for Y and U , and likewise for π A . At least for π A , one might hope to imitate the proof ofTheorem 5.3. Theorem 7.1.
Under the assumptions of Conjecture 1.1 with base field k of char-acteristic zero, and assuming that U is smooth over k , the inclusion Y → U becomesan A -homotopy equivalence after suspending by S , = S ∧ G m .Proof. By equivariant resolution of singularities (using that k has characteristiczero), we can assume that X (as well as U ) is smooth over k , while still having a T -action [20, Proposition 3.9.1].We first show that the inclusion Y → U induces an isomorphism in the derivedcategory of motives DM ( k ), M ( Y ) → M ( U ). Namely, since X is smooth over k ,we have the Bia lynicki-Birula decomposition, as follows. The fixed point set X T issmooth over k . Write Z , . . . , Z m for the connected components of X T . For each i , let Z + i = { x ∈ X : lim t → tx ∈ Z i } and Z − i = { x ∈ X : lim t →∞ tx ∈ Z i } bethe stable and unstable manifolds of Z i . Then the action of T gives morphisms Z + i → Z i and Z − i → Z i which are affine-space bundles [6].10arpenko showed that this geometric decomposition gives a direct-sum decom-position of Chow motives over k [18, Theorem 6.5], [7, Theorem 3.5]: M ( X ) ∼ = ⊕ mi =1 M ( Z i ) { a i } , where a i := dim( Z + i ) − dim( Z i ). (Here Z { } denotes the Lefschetz motive, with M ( P ) = Z { }⊕ Z { } .) This implies another decomposition M ( X ) ∼ = ⊕ mi =1 M ( Z i ) { b i } ,where b i := dim( Z − i ) − dim( Z i ), by inverting the T -action on X . Here a i + dim( Z i ) + b i = n, by considering the action on T on the tangent space to X at a point of Z i .The category of Chow motives is a full subcategory of the derived category ofmotives, DM ( k ): the thick subcategory generated by smooth projective schemesover k tensored with Z { a } = Z ( a )[2 a ] for integers a [28]. Every scheme X offinite type over k has a motive M ( X ) and a compactly supported motive M c ( X ) in DM ( k ). We can assume that Z , . . . , Z m are ordered in such a way that the closureof Z − i is contained in X i := ∪ j ≤ i Z − j . Karpenko’s argument shows that the exacttriangle M c ( X i − ) → M c ( X i ) → M c ( Z − i ) ∼ = M ( Z i ) { b i } in DM ( k ) is split [18, Theorem 6.5, part (a)]. (Indeed, his splitting on Chowgroups is defined by an element of CH dim( Z i )+ b i ( Z i × X i ), and that is preciselyHom( M ( Z i ) { b i } , M c ( X i )) since Z i is smooth and proper over k .) In particular, itfollows that M ( X j ) ∼ = ⊕ ji =1 M ( Z i ) { b i } for each 1 ≤ j ≤ m ., and its open complement X − X j satisfies M c ( X − X j ) ∼ = ⊕ mi = j +1 M ( Z i ) { b i } (Here X j need not be smooth, but it is proper over k , and so its motive M ( X j ) isthe same as its compactly supported motive M c ( X j ).)In the notation of Conjecture 1.1, we can assume that the closed subset Y of X is equal to X r for some r ≤ m . So M ( Y ) ∼ = ⊕ ri =1 M ( Z i ) { b i } . Likewise, the opensubset U is the union of the subsets Z + i with i ≤ r . By the splitting in DM ( k )above, applied to the inverse action of T on X , we have M c ( U ) ∼ = ⊕ ri =1 M ( Z i ) { a i } . Since U is smooth of dimension n over k , it follows that M ( U ) ∼ = M c ( U ) ∗ { n }∼ = ⊕ ri =1 M ( Z i ) ∗ { n − a i }∼ = ⊕ ri =1 M ( Z i ) { n − a i − dim( Z i ) }∼ = ⊕ ri =1 M ( Z i ) { b i } . Thus M ( Y ) is isomorphic to M ( U ) in DM ( k ). More precisely, the inclusion Y → U induces an isomorphism M ( Y ) → M ( U ). To see this, one checks fromKarpenko’s construction of the splittings that for i, j ∈ { , . . . , r } , the composition11 ( Z i ) { b i } → M ( Y ) → M ( U ) → M ( Z j ) { b j } is the identity for i = j and zero if i < j .The schemes Y, U, X with T -action are defined over some finitely generatedsubfield of k . So we can assume that the field k is finitely generated over Q . ApplyTheorem 6.1 to the cofiber A = Σ ∞ ( U/Y ) in SH ( k ). We showed above thatthe motive of A in DM ( k ) is zero. Also, for every real embedding of k , the realrealization of A is zero in the stable homotopy category, by Theorem 4.1. Therefore, A = 0 in SH ( k ). That is, the inclusion Y → U induces an isomorphism in SH ( k ).Again using that k has characteristic zero, Bachmann showed that the P -suspension functor Q = Σ ∞ P : H ( k ) ∗ → SH ( k ) is conservative on A -simply con-nected spaces which can be written as homotopy colimits of spaces X + ∧ G m with X ∈ Sm k [3, Theorem 1.3].The S -suspension of every space in H ( k ) ∗ is A -simply connected. Therefore, S ∧ G m ∧ Y + → S ∧ G m ∧ U + is a pointed A -homotopy equivalence.Theorem 7.1 can be slightly strengthened if in addition Y and U are A -connected. In that case, their S -suspensions are A -simply connected, and so P ∧ Y + → P ∧ U + is a pointed A -homotopy equivalence, using that P = S , = S ∧ G m . References [1] B. Antieau and E. Elmanto. A primer for unstable motivic homotopy theory.
Surveys on recent developments in algebraic geometry , Proc. Sympos. PureMath. 95, 305–370. Amer. Math. Soc. (2017). 2[2] T. Bachmann. On the conservativity of the functor assigning to a motivicspectrum its motive.
Duke Math. J. (2018), 1525–1571. 9, 10[3] T. Bachmann. The zeroth P -stable homotopy sheaf of a motivic space. arXiv:2003.12021 J.Reine Angew. Math. (2005), 149–168. 10[5] A. Banyaga and D. Hurtubise. Morse-Bott homology.
Trans. Amer. Math. Soc. (2010), 3997–4043. 4[6] A. Bia lynicki-Birula. Some theorems on actions of algebraic groups.
Annals ofMath. (1973), 480–497. 3, 10[7] P. Brosnan. On motivic decompositions arising from the method of Bia lynicki-Birula. Invent. Math. (2005), 91–111. 11[8] D. Cartwright, D. Erman, M. Velasco, and B. Viray. Hilbert schemes of 8points.
Algebra Number Theory (2009), 763–795. 8[9] P. Deligne. Poids dans la cohomologie des vari´et´es alg´ebriques. Proceedingsof the International Congress of Mathematicians (Vancouver, 1974), 79–85.Canad. Math. Congress, Montreal (1975). 21210] M. Fried, D. Haran, and H. V¨olklein. Real Hilbertianity and the field of to-tally real numbers.
Arithmetic geometry (Arizona State University, 1993), 1-34.Amer. Math. Soc. (1994). 9[11] L. G¨ottsche. Hilbert schemes of points on surfaces.
Proceedings of the Inter-national Congress of Mathematicians (Beijing, 2002), Vol. II, 483–494. HigherEd. Press, Beijing (2002). 1[12] R. Hartshorne. Connectedness of the Hilbert scheme.
Publ. Math. IHES (1966), 5–48. 7[13] T. Hausel and F. Rodriguez Villegas. Cohomology of large semiprojective hy-perk¨ahler varieties. De la g´eometrie alg´ebrique aux formes automorphes (II) ,113–156. Ast´erisque 370 (2015). 1, 3[14] H. Hironaka. Triangulations of algebraic sets.
Algebraic geometry (Arcata,1974), Proc. Sympos. Pure Math., Vol. 29, 165–185. Amer. Math. Soc. (1975).6[15] M. Hoyois, J. Jelisiejew, D. Nardin, B. Totaro, M. Yakerson. The Hilbertscheme of infinite affine space. arXiv:2002.11439
1, 2, 7[16] A. Iarrobino. Reducibility of the families of 0-dimensional schemes on a variety.
Invent. Math. (1972), 72–77. 1[17] J. Jelisiejew. Pathologies on the Hilbert scheme of points. Invent. Math. (2020), 581–610. 1[18] N. Karpenko. Cohomology of relative cellular spaces and of isotropic flag vari-eties.
Algebra i Analiz (2000), 3–69. 11[19] J. Koll´ar. Rational curves on algebraic varieties.
Springer (1996). 3, 4[20] J. Koll´ar.
Lectures on resolution of singularities.
Princeton (2007). 10[21] F. Morel. The stable A -connectivity theorems. K-Theory (2005), 1–68. 7[22] F. Morel and V. Voevodsky. A -homotopy theory of schemes. Publ. Math. IHES (1999), 45–143. 2, 7[23] K. Pardue. Deformation classes of graded modules and maximal Betti numbers. Ill. J. Math. (1996), 564–585. 7[24] A. Reeves. The radius of the Hilbert scheme. J. Alg. Geom. (1995), 639–657.7[25] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology.
Springer (1972). 6[26] The Stacks Project Authors.
Stacks Project (2018). http://stacks.math.columbia.edu
J. Math. Kyoto Univ. (1975), 573–605. 3 1328] V. Voevodsky. Triangulated categories of motives over a field. Cycles, transfers,and motivic homology theories , 188–238. Princeton (2000). 11