Faithful tropicalization of hypertoric varieties

Transcription

1 University of Oregon STAGS June 3, 2016

2 Setup Fix an algebraically closed field K, complete with respect to a non-archimedean valuation ν : K T := R { } (possibly trivial). Let X be a K-variety. An embedding of X into a toric variety (e.g. A n, P n ) yields a tropicalization of X, which is the image of a proper continuous map X an Trop(X). Here, X an is the Berkovich analytic space associated to X. Think of Trop(X) as a combinatorial shadow of the Berkovich space.

3 Analytification If X = Spec A, then X an = {val: A T val is a ring valuation extending ν}. Each f A defines an evaluation function ev f : X an T, val val(f). Give X an the coarsest topology such that ev f continuous for all f. T = (0, 1].

4 Analytification For general X, X an = {Spec L X L is a valued extension of K}/. Note that X(K) X an. Foster-Gross-Payne: If X has an embedding into a toric variety, then X an lim Trop(X) is a homeomorphism. Goal: Find a particular toric embedding of X so that the corresponding Trop(X) reflects enough of the topology of X an.

5 Faithful tropicalization A tropicalization is faithful if there is a continuous section to the tropicalization map X an Trop(X). Such a section identifies Trop(X) with a closed subset of X an.

6 Faithful tropicalization Brief history: BPR (2011): Case when X is a curve CHW (2014): Gr(2, n) faithfully tropicalized by Plücker embedding GRW (2014): In torus, multiplicity one is sufficient GRW (2015): Sufficient conditions for general case K (2016): Hypertoric varieties are faithfully tropicalized

7 Hypertoric varieties Hypertoric varieties are hyperkähler analogues of toric varieties. Every hypertoric variety is constructed from an arrangement A of affine hyperplanes. There is a natural embedding of each hypertoric variety into its corresponding Lawrence toric variety.

8 Notation Let M be a lattice of rank d, N = Hom(M, Z) the dual lattice. Set M R = M Z R and N R = N Z R = Hom(M, R). Let T = Spec K[M] be the torus with character lattice M.

9 Toric varieties Rational polyhedral cones in N R Affine T -toric varieties σ U σ := Spec K[σ M] Rational polyhedral fans in N R T -toric varieties Y := Torus orbits in Y are indexed by cones in : O(σ) := Spec K[σ M]. σ U σ

10 Compactifications of N R Rational polyhedral cone σ in N R N σ R := Hom(σ M, T) Rational polyhedral fan in N R N R := Analogous to the torus orbit O(σ), we have N R (σ) := N R / σ. σ N σ R

11 Tropicalization Clear analogy between Y = U σ = O(σ) σ σ and N R = N σ R = N R (σ). σ σ Tropicalization relates these two spaces.

12 Tropicalization On the torus T = Spec K[M], the tropicalization map is trop: T an N R = Hom(M, R), val val M. This extends to U σ = Spec K[σ M] in the obvious way: trop: U an σ N σ R = Hom(σ M, T), val val σ M. Glue to obtain trop: Y an N R Restriction to O(σ) an is the usual tropicalization map for a torus.

13 Tropicalization Now, if X Y is a closed subvariety, then X an Y an. Let Trop(X) = trop(x an ). Trop(X T ) is the support of a balanced finite polyhedral complex, of pure dimension dim X. Trop(X) is a (partial) compactification of Trop(X T ) by lower-dimensional finite polyhedral complexes.

14 Faithful tropicalization Gubler-Rabinoff-Werner give the following conditions on X Y, which imply X an Trop(X) has a unique continuous section. Trop(X) has multiplicity one everywhere X T is dense in X X O(σ) is either empty or equidimensional for all σ There is a polyhedral structure on Trop(X) such that for each maximal polyhedron P in Trop(X T ), the closure P is a union of polyhedra, each of which is maximal in its respective stratum

15 Hyperplane arrangements A hyperplane arrangement A in M R is given by: A finite set E A tuple a N E of primitive elements r Z E. The arrangement A consists of the multiset of affine hyperplanes H e = {m M R m, a e + r e = 0}, e E. Usually require {a e e E} to span N.

16 Hyperplane arrangements Each H e is cooriented by its normal vector a e, giving positive and negative closed halfspaces H + e, H e. The arrangement A is central if r = 0. In general, obtain the centralization of A, denoted A 0, by setting r = 0.

17 Matroids The centralization A 0 defines a matroid on E, via rank function rk S = dim span{a e e S}. A flat is a set which is maximal for its rank. A flag of flats F is a chain where each F i is a flat. = F 0 F 1 F k 1 F k = E

18 Matroids This matroid is the matroid of the linear space L A0 A E = Spec K[x e e E], defined by the linear forms c e x e e E for each relation c e a e = 0 in N. e E Every linear subspace of A E not contained in a coordinate hyperplane arises from an arrangement in this way.

19 Localization and restriction Two ways to obtain new arrangements from A: For any subset S E, the localization of A at S, denoted A S, is the arrangement of hyperplanes {H e e S}. For a flat F E, the restriction of A 0 to F, denoted A F 0, is the arrangement of hyperplanes {(H e ) 0 H F e / F } in H F = e F H F. Flats of A F 0 are flats of A containing F.

20 The Lawrence toric variety of an arrangement Let A be an arrangement in M R. This defines a surjection G E m T via Z E N, δ e a e. We get a map of short exact sequences ι 0 Λ Z E N 0 a id 0 Λ Z E Z E Ñ 0 where is the antidiagonal embedding.

21 The Lawrence toric variety of an arrangement The lattice Ñ has rank d + E and generators ρ± e. Let M be its dual lattice, and T = Spec K[ M]. Definition The Lawrence toric variety B A defined by A is the T -toric variety T A E α G, where G = ker(g E m T ) and α = ι (r).

22 The Lawrence toric variety of an arrangement Given a subset S E, and a face R of the localization A S, we let σ S,R be the cone in ÑR with rays generated by the vectors {ρ + e R H + e } {ρ f R H f }. Proposition (Hausel-Sturmfels 04, K 16) The collection of cones A = {σ S,R S E and R is a face of A S } forms a rational polyhedral fan in ÑR, with face relations σ S,R σ S,R if and only if S S and R R. We have Y A = B A.

23 The hypertoric variety of an arrangement Recall ι 0 Λ Z E N 0 a id 0 Λ Z E Z E Ñ 0 The cokernel of the inclusion N Ñ is Ñ ZE, ρ ± e δ e. Tensor with R to get linear ÑR R E. Maps σ S,R onto R S 0.

24 The hypertoric variety of an arrangement Thus obtain surjection Φ: B A A E. Definition The Hypertoric variety M A defined by A is Φ 1 (L A0 ) B A. Note: M A can also be defined as the GIT quotient of a closed subset of T A E by G, without reference to B A. Examples: A 2n, T P n, T (P n P m ),...

25 The hypertoric variety of an arrangement The key observation for us is that M A is built out of linear spaces: Proposition Let S E and let R be a face of A S. The intersection M A O(σ S,R ) is nonempty if and only if S is a flat of A 0, in which case it is a linear subvariety of O(σ S,R ) of dimension 2d rk S codim R. Note: The expected dimension of M A O(σ S,R ) is dim M A dim σ S,R = 2d S codim R.

26 The tropicalization of M A It follows that Trop(M A O(σ F,R )) is the support of a fan with cones C (F,R) F indexed by flags F of flats of A F 0. The fan is balanced, with all multiplicities equal to one.

27 The tropicalization of M A Theorem (K 16) The tropicalization Trop(M A ) has the combinatorial structure of a finite polyhedral complex, under the closure relation C (F,R ) F C (F,R) F if and only if the following conditions hold: F F ; R R; F is a flat in F, and trunc F (F) is a refinement of F. Moreover, dim C (F,R) F = d + l(f) codim R.

28 Faithful tropicalization Recall the criteria of GRW: X Y is faithfully tropicalized if Trop(X) has multiplicity one everywhere X T is dense in X X O(σ) is either empty or equidimensional for all σ There is a polyhedral structure on Trop(X) such that for each maximal polyhedron P in Trop(X T ), the closure P is a union of polyhedra, each of which is maximal in its respective stratum

29 Faithful tropicalization Theorem There is a unique continuous section s: Trop(M A ) M an A tropicalization map. of the

30 References M. Kutler, Preprint, arxiv: