{'fraction_non_alphanumeric': 0.067071233398996, 'fraction_numerical': 0.014011565101353499, 'mean_word_length': 3.5728606712189452, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
ON INTERSECTION COHOMOLOGY WITH TORUS ACTIONS OF COMPLEXITY ONE

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus T, one of our result determines the intersection cohomology Betti numbers of any normal projective T-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.


Introduction
In this article, we are interested in the rational intersection cohomology for complex normal algebraic varieties endowed with an action of an algebraic torus T.
Let us recall that a T-variety is a normal variety endowed with an effective T-action. The complexity of a T-variety X is the non-negative number dim(X) − dim(T); it corresponds geometrically to the codimension of a general T-orbit. Moreover, we say that a T-variety X is contraction-free (see [IV13,Definition 1.8]) if the rational quotient map given by the T-action on X is a morphism. The best concrete examples of T-varieties are toric varieties (corresponding to the complexity zero) and affine G m -surfaces. Both admit a combinatorial description. The first one can be defined by a fan of strongly convex polyhedral cones in the rational vector space N Q associated with the lattice N of one-parameter subgroups of the torus T, see for instance [CLS11]. The second one can be described by a Q-divisor or a pair of Q-divisors on a smooth algebraic curve (see [FZ03] for details). These two combinatorial descriptions admit a generalization to the setting of T-varieties (see [AH06,AHS08,Tim08,AIPSV12,Lan14]). The description in [AH06] of an affine T-variety is in term of a divisor on a normal variety with polyhedral coefficients in N Q which is called a polyhedral divisor. More generally, the combinatorial object introduced in [AHS08] describing a T-variety is a divisorial fan which corresponds to a finite set of polyhedral divisors.
One of the results of this article is an explicit description (in terms of divisorial fans) of the intersection cohomology Betti numbers of rational projective contraction-free T-varieties of complexity one (see Theorem 3.7). As an intermediate step, we compute the classical Betti numbers for every smooth projective T-variety of complexity one (see Propositions 2.4 and 2.7). These results can be related to classical one in the field of intersection cohomology with a torus action. See [Kir88] for a general description in the projective case using the Bialynicki-Birula decomposition. See also [Sta87,DL91,Fie91] for the toric projective case which is related to h-polynomials and [FK86,FK93] for the case of affine G m -surfaces.
To motivate our result, let us make some comments on the description in [Sta87] for the toric projective case. In the sequel, for every algebraic variety V we will denote by P V (t) ∈ Z[t] the Poincaré polynomial of V which is the generating function of the intersection cohomology Betti numbers of V (see 1.8). Let X be a projective toric variety (for the torus T) with defining fan Σ X . Then, since X is projective, the fan Σ X is the normal fan of a rational polytope Q. In particular, the set of faces of dimension i of Q is in bijection with the set of cones of codimension i. In [Sta87] a polynomial h(Λ; t) (called h-polynomial) depending on each polytope Λ is introduced, so that we have the equality P X (t) = h(Q; t 2 ) (see [Sta87,Theorem 3.1]). In the smooth case, the polynomial h(Q; t 2 ) can be defined by the relation where d i (Q) is the number of faces of dimension i. However, in the non-smooth case, the polynomial above in the right-hand side has generally negative coefficients and therefore the definition of an h-polynomial is different (see [Sta87, Section 2] and also the remainder in 1.14). Our main result is an adaptation of the description of the h-polynomials explained above (see [Sta87,Theorem 3.1]) to the setting of torus actions of complexity one. Let us introduce some notation in order to explain our result. Let E be a divisorial fan on the projective line P 1 corresponding to a singular rational projective contraction-free T-variety X(E ) of complexity one. We recall the definition of the divisorial fan E and the construction of the variety X(E ) in 1.4. Note that the curve P 1 can be seen as a quotient for the T-action on X(E ). Let us denote by supp(E ) the support of E which corresponds to points y ∈ P 1 where the polyhedral subdivisions given by the coefficients at y of each element of E are non-trivial (see 1.6 for a precise definition). Then, similarly to the toric case, one can attach rational polytopes Q(E ) and Q y (E ) for every y ∈ supp(E ), see 2.1 for the construction of these objects. Our result can be stated as follows.
Theorem 0.1. Let r be the cardinal of the finite set supp(E ). Then we have the equality In particular, one can see that the odd rational intersection cohomology groups of X(E ) vanish as in the toric projective case. To obtain this result we adapt in our setting a version of the decomposition theorem in [CMM14] given for toric fibrations (see Theorem 3.3). This allows us to show the result by induction on the dimension of X(E ) starting with a projective desingularization of X(E ) given by a subdivision of divisorial fans (see 3.2 for the definition of a subdivision of divisorial fans). Let us outline the structure of this article. In Section 1 we introduce some notation from the toric geometry related to divisorial fans and from the intersection cohomology theory. In Section 2 we describe the classical Betti numbers for every smooth projective T-varieties of complexity one. Finally, Section 3 is devoted to the proof of Theorem 0.1.
Notation 0.2. By a variety we mean an integral separated scheme of finite type over the field of complex numbers C. An algebraic torus (of dimension n) is an algebraic group isomorphic to G n m , where G m is the multiplicative group of the field C. If X is a variety, then C[X] denotes the coordinate ring of X and C(X) denotes the field of rational functions on X. We will identify X with its set of C-points X(C) and we will also see X as a complex manifold. Hence the set X is endowed with two topologies: the Zariski topology and the Euclidian topology.
1.1. Let T be an algebraic torus with lattice of characters M and lattice of one-parameter subgroups N . Then the duality between M and N extends naturally to a duality According to the Sumihiro theorem (see [Sum74, Section 3, Corollary 2]), every T-variety is covered by affine T-stable Zariski open subsets. Thus we recall first how to describe an affine T-variety by combinatorial objects in the setting of the complexity one.

1.2.
A polyhedral cone σ ⊂ N Q is said to be strongly convex if it contains no lines; this condition is equivalent to the dual cone being full dimensional. Let us fix a strongly convex polyhedral cone σ ⊂ N Q . A σ-polyhedron of N Q is a Minkowski sum Q + σ, where Q ⊂ N Q is a polytope (i.e., Q is the convex hull of a non-empty finite subset of N Q ). Let Y be a smooth curve. A σ-polyhedral divisor D on the curve Y is a formal sum where every D y is a σ-polyhedron and D y = σ for all but many finite y ∈ Y . For every m ∈ σ ∨ we define a Q-divisor by letting The curve Y (respectively the cone σ) is usually called the locus (respectively the tail) of D.
The σ-polyhedral divisor D is called proper if Y is affine; or Y is projective and D verifies the additional properties: (i) The degree deg(D) := y∈Y D y is strictly contained in σ.
(ii) For every m ∈ σ ∨ such that min v∈deg(D) m, v = 0, the divisor D(dm) is principal for some d ∈ Z >0 . We denote by PPDiv Q (Y, σ) the set of proper σ-polyhedral divisors on Y .
It is known that for an affine variety X there is a one-to-one correspondence between Tactions on X and M -gradings on C[X]. The next result (see [AH06, Theorems 3.1, 3.4]) gives a combinatorial description of M -graded algebras corresponding to affine T-varieties of complexity one. Theorem 1.3.
(i) Let σ ⊂ N Q be a strongly convex polyhedral cone and let Y be a smooth curve. If D ∈ PPDiv Q (Y, σ), then the M -graded subalgebra where χ m is the Laurent monomial corresponding to m ∈ M , defines an affine T-variety X(D) = X(Y, D) of complexity one with rational quotient Y .
(ii) Conversely, if X is an affine T-variety of complexity one, then there exist a strongly convex polyhedral cone σ ⊂ N Q , a smooth curve Y , and D ∈ PPDiv Q (Y, σ) such that the T-variety X is T-isomorphic to X(D).
We refer to [AH06, Section 8] and [Lan14, Section 4] for the functorial properties and the uniqueness problem of this decomposition. In the next paragraph, we explain the combinatorial description of [AHS08] for (non-necessarily affine) T-varieties specified to the case of the complexity one.
Zariski open dense subset and σ i ⊂ N Q is a strongly convex polyhedral cone such that: (i) For all i, j ∈ I we have For all i, j ∈ I and for every y ∈ Y ij , the polyhedron D i y ∩ D j y is a common face of D i y and D j y . (iii) We have Y = i∈I Y i . By the preceding conditions, the set {σ i | i ∈ I} generates a fan denoted Σ ⋆ (E ). If E is a divisorial fan on (Y, N ), then the natural morphisms (cf [AHS08, Section 5]) are T-invariant Zariski open immersions. The collection of T-varieties X(D i ) can be glued in a T-variety X(E ) in which the Zariski open subsets X(D i ∩D j ) are identified with the intersections X(D i )∩ X(D j ) (see [AHS08,Remark 7.4 (ii)] for the fact that the C-scheme X(E ) is separated). We refer the reader to [IS11, Section 1] for other equivalent conditions to the notion of divisorial fans.
Conversely, if X is a T-variety of complexity one, then there exist a smooth curve Y and a divisorial fan E on (Y, N ) such that the T-variety X is T-isomorphic to X(E ) [AHS08, Theorem 5.6]. Note that the proof of this latter fact uses the Sumihiro theorem. Let us fix a divisorial fan E on (Y, N ) such that X(E ) is contraction-free. In the next paragraph, we recall the description of the orbits of X(E ) in terms of the combinatorial object 1.6. Denote by π : X(E ) → Y the quotient morphism which the restriction on each open subset Given y ∈ Y , let us explain how to describe the orbits of the reduced part π −1 (y) red . For a vertex v ∈ D i y consider the cone where v runs the set of vertices of D i y . Denote by face(E ) y the set of faces of the polyhedra D i y . We have a bijection F → O(y, F ) between the set face(E ) y and the set of T-orbits of π −1 (y) red ; the orbit O(y, F ) is of dimension codim(F ) = rank(N ) − dim(F ) and is seen geometrically as a common part of the component where X(Σ ⋆ (E )) is the toric variety for the torus T associated with the fan Σ ⋆ (E ).
We recall this in the next paragraph.
1.7. We call hyperspace (with the respect to (Y, N )) the set N Q obtained as the quotient of Y × N Q × Q ≥0 by the equivalence relation ∼ defined by The equivalence class of (y, v, l) will be denoted by [y, v, l]. Let D be a σ-polyhedral divisor on a Zariski open dense subset Y 0 ⊂ Y . The associated hypercone of D, denoted by C (D), is the subset where F ⊂ C (D) y is a usual face. There exists a bijection between the set of faces of C (D) and the set of germs of X(D) given as follows. With the same notation as above, let (v, l) be in the relative interior of F .
The germ associated is given by the ideal Let E be a divisorial fan on (Y, N ) with X(E ) contraction-free. Then we denote by H(E ) the set of faces of every element of the form C (D) ⊂ N Q where D runs E . Similarly, the set H(E ) is in bijection with the set of germs of X(E ). In the sequel, we will write by V (τ ) the germ associated with τ ∈ H(E ). Note that in this case every germ of X(E ) is normal (see [Tim00, Theorem 7]).
1.2. Preliminaries on intersection cohomology. In this section, we fix the notation concerning the intersection cohomology theory that we will use in the sequel (see [GM80,GM83] for details concerning the intersection homology). We note that the cohomology sheaves are particularly sheaves of Q-vector spaces with the respect to the Euclidian topology. The perversity for the intersection cohomology groups is the middle perversity.
1.8. Following the construction in [HTT08, Section 8.2], for a variety X we denote by IC X its intersection cohomology complex and by H j (IC X ) the j-th cohomology sheaf of IC X so that the Q-vector space of the global sections of H j−dim(X) (IC X ) is the j-th rational intersection cohomology group IH j (X; Q). The vector spaces IH ⋆ (X; Q) are topological invariants [GM83, Section 4], satisfy the Poincaré duality and they are finitely generated (see [GM80, the Poincaré polynomial of X. 1.9. By a local system on X we mean a locally constant sheaf of Q-vector spaces (for the Euclidian topology) having finite dimensional stalks. A local system is simple if has no nontrivial local subsystems (i.e., simple in the category of local systems) and semisimple if it is direct sum of simple local systems. Let L be a local system defined on a Euclidian open dense subset of the regular locus of X. We denote by IC X (L ) the twisted intersection cohomology complex, so that the Q-vector space of the global sections of H j−dim(X) (IC X (L )) is the twisted j-th rational intersection cohomology group IH j (X; L ).
In the particular case where X is smooth, the rational intersection cohomology coincide with the classical rational cohomology. The following paragraph explains a method to compute the rational cohomology groups of every smooth projective variety (see 1.10 (iii)).
1.10. The Hodge-Deligne polynomial of a variety X is defined by the relation where d = dim(X) and h p,q (H j c (X; C)) is the dimension of the (p, q)-type Hodge component in the j-th cohomology group H j c (X; C) with compact support. The polynomial E( ⋆ ; u, v) satisfies the following properties: (ii) If X 1 and X 2 are two varieties, then (iii) If X is smooth and projective, then P X (t) = E(X; −t, −t).
For instance, if Y is a smooth projective curve of genus g, then Hence E(P 1 ; u, v) = uv + 1, E(A 1 ; u, v) = uv, E(G m ; u, v) = uv − 1 and so E(G r m ; u, v) = (uv − 1) r for every r ∈ Z ≥0 . In the next paragraph, we enunciate the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber which allows us to describe the topology of singular proper algebraic maps (see [BBP82,Theorem 6.25] and [CM09] for a survey article on this topic).
Theorem 1.11. Let f : X → Z be a proper (algebraic) morphism between varieties X and Z. Then there exists a finite family (Z α , L α , d α ) where for every index α, Z α ⊆ Z is a smooth irreducible (algebraic) subvariety which is locally closed (for the Zariski topology), L α is a semisimple local system on Z α and d α ∈ Z such that whereZ α is the Zariski closure of Z α in Z. More concretely, this implies that for every Euclidian open subset U ⊆ Z and for every j ∈ Z, we have an isomorphism of Q-vector spaces In [CMM14] the preceding theorem made explicit in the case where X and Z are toric varieties and f is a proper toric map. By convenience we recall this result.
Theorem 1.12. Let X, Z be toric varieties for the torus T. Let Σ Z be the fan defining Z. For every τ ∈ Σ Z , denote by V (τ ) the corresponding orbit closure. If f : X → Z is a birational proper toric map given by a subdivision of fans, then we have an isomorphism More concretely, this implies that for every Euclidian open subset U ⊆ Z and every j ∈ Z, we have an isomorphism of Q-vector spaces where l τ,b = b+dim(X)−dim(V (τ )). The sequence s τ,b of natural numbers satisfies the following conditions: Note that assertion (iv) in Theorem 1.12 is a consequence of [CMM14, Remark 7.1, Theorems 7.2,7.4]. The following terminology will be useful for the sequel.
Definition 1.13. With the same notation as in 1.12, let us denote by Σ X the fan defining the toric variety X. The sequence of coefficients s τ,b (τ and b run respectively Σ Z and Z) involving in Theorem 1.12 will be called an s-sequence of the subdivision Σ Z of Σ X .
Let us end this section by recalling the notion of h-vectors which describes the intersection cohomology Betti numbers of projective toric varieties (see [Sta87,Theorem 3.1] and [DL91,Fie91]).
1.14. We will consider here the empty set ∅ as a common face of every polytope. Let us define two polynomials h( ⋆ , t), g( ⋆ , t) ∈ Z[t] by induction [Sta87, Section 2] which the first entry ⋆ depends on a polytope. Let g(∅, t) = 1 and dim(∅) = −1. Assume that g( ⋆ , t) is defined for every polytope of dimension < d. Then for a polytope Q of dimension d, we let where the sum runs the set of proper faces of Q (included the empty set ∅). Let us denote by h k (Q) the k-th coefficient of h(Q, t). One defines g(Q, t) by the relation The polynomial h(Q, t) is called the h-polynomial of Q and (h 0 (Q), . . . , h d (Q)) is the h-vector of Q. More generally, we will let h j (Q) = 0 for any j ∈ Z <0 . We refer to [Fie91] for a geometric meaning of h( ⋆ , t) and g( ⋆ , t).
Theorem 1.15. Let X be a projective toric variety defined by a fan Σ X . Let us fix a polytope Q ⊂ M Q such that Σ X is the normal fan of Q. Then we have the equality P X (t) = h(Q, t 2 ).

Betti numbers of complexity-one smooth projective T-varieties
The purpose of this section is to give an explicit description of the classical Betti numbers of every smooth projective T-variety of complexity one. Before stating our results, let us introduce some notation.
2.1. Let E be a divisorial fan on (Y, N ) such that X(E ) is contraction-free. Let D ∈ E and let y ∈ Y be a point belonging to the locus of D. We denote by C − (D) the cone generated by the subset where σ is the tail of D. We denote by E + y resp. E y the fan generated by {C (D) y | D ∈ E with y in the locus of D} resp. {C − (D), C (D) y | D ∈ E with y in the locus of D}. If X(E ) is projective, then E y is a complete fan which is a normal fan of a polytope Q y (E ) (compare with [PS11, Corollary 3.28]). The same holds for the fan Σ ⋆ (E ); we denote by Q(E ) a polytope such that Σ ⋆ (E ) is its normal fan.
Let us illustrate the preceding paragraph by an example.

resp. Spec(C[t]) we have the equality
, for every i ∈ I, where the right-hand side is the semigroup algebra of Consequently, X(E ) is the toric variety for the torus T × G m associated with the fan E 0 .
The description of the toric variety of Example 2.2 allows us to establish relations between h-polynomials of polytopes: Lemma 2.3. Let E be a divisorial fan on (P 1 , N ) as in Example 2.2 and assume that X(E ) is smooth projective. Then we have the equality Proof. First of all, by Theorem 1.15 we know that h(Q 0 (E ); t 2 ) = P X(E ) (t). Let us compute P X(E ) (t) by using the method of E-polynomials (see 1.10). Removing the special fiber at the origin of the quotient map π : X(E ) → P 1 gives Using the orbit decomposition for π −1 (0) red (see 1.6) we obtain that

The result follows from the substitution
The next result gives particularly an explicit formula of the Poincaré polynomial of each smooth projective contraction-free T-variety of complexity one. We consider this result in the general setting of quotient singularities since it will be needed for the proof of Theorem 3.7.
Proposition 2.4. Let X(E ) be a projective contraction-free T-variety of complexity one corresponding to a divisorial fan E on (Y, N ). Denote by g the genus of the curve Y and by r the cardinal of the finite set supp(E ). Assume that every cone of E y is generated by a subset of a basis of (N ⊕ Z) Q for all y ∈ Y . Then we have the equality Proof. First of all, let us compute the Poincaré polynomial P X(E ) (t) with the assumption that X(E ) is smooth. Using again the method with E-polynomials we have By substituting u = v = −t, it follows that According to [LS13, Section 5] the smoothness assumption on X(E ) implies that the toric variety X(E y ) associated with the fan E y is smooth for every y ∈ Y . We conclude by using Lemma 2.3 on the right-hand side (E 0 of Example 2.2 is replaced by E y ) to obtain the required formula.
In the general case, one can realize X(E ) as the quotient X(Ê )/G, where X(Ê ) is smooth and G is a finite abelian group by changing the ambient lattice N . Hence X(E ) is rationally smooth [Bri99, Proposition A1 iii)] and so H j (X(E ); Q) = IH j (X(E ); Q) for all j ∈ Z.
To obtain the last equalities, we can adapt [HTT08, Proposition 8.2.21] for the intersection cohomology with rational coefficients. Since the quotient map by G induces isomorphisms between rational cohomology groups of X(E ) and X(Ê ) we conclude that P X(E ) (t) = P X(Ê ) (t). Finally, we can make the choice to take Q(E ) and Q y (E ) so that Q(E ) = Q(Ê ) and Q y (E ) = Q y (Ê ) for every y ∈ Y . This finishes the proof of the proposition.
2.5. Actually, the h-vectors of simplicial polytopes are simple to describe (compare Theorem 1.15 with [CMM14, Corollary 3.9]) and the formula of Proposition 2.4 can be expressed by the relation: where d = dim(X(E )) and for any simplicial polytope Q, we denote by d i (Q) the number of faces of dimension i.
In the next paragraphs, we complete the computation of Betti numbers of smooth projective T-varieties of complexity one. Note that according to [LS13, Section 5] a smooth non-contractionfree T-variety of complexity one is automatically rational. Hence we can restrict ourselves to divisorial fans on the projective line.
2.6. Let E be a divisorial fan on (P 1 , N ) such that X(E ) is complete. Let us introduce specific notation. We denote by Σ ⋆ big (E ) the subfan generated by the elements τ ∈ Σ ⋆ (E ) such that there exists D ∈ E satisfying the conditions: (1) The cone τ is a face of the tail cone σ of the polyhedral divisor D. For every F ∈ face(E ) y and D ∈ E (with tail σ) such that F is a face of D y , we let We will also consider the finite set where R is the equivalence relation defined by (y, F ) R (y ′ , F ′ ) if and only if: • y = y ′ and F = F ′ , or • there exists D ∈ E with P 1 as locus, the tail σ of D is a maximal cone of Σ ⋆ (E ) (we say that σ is marked following the terminology of [IS11, Section 1]), and F, F ′ are respectively faces of D y and D y ′ such that λ(D, F, y) = λ(D, F ′ , y ′ ). Furthermore, we require that λ(D, F, y) is not contained in a dual face of a cone of Σ ⋆ big (E ).
The equivalence class of (y, F ) will be denoted by [y, F ]. Finally, let us remark that the map is well defined and so for every s ∈ Z ≥0 we can let Proposition 2.7. Let E be a divisorial fan on (P 1 , N ) such that X(E ) is smooth and projective. Let r be the cardinal of supp(E ). With the same notation as in 2.6, we have the equality Proof. Let us consider the equivariant map q :X → X(E ) obtained by gluing the natural maps where D ∈ E and Y 0 resp. σ is the locus resp. the tail of D. The map q is proper and birational (see [AH06, Theorem 3.1]). In particular, X(E ) = q(X). Let Then we have the decomposition 3. Poincaré polynomials in the singular case 3.1. A description of the decomposition theorem. In this section, we investigate the decomposition theorem for equivariant proper maps between contraction-free T-varieties of complexity one induced by subdivisions of divisorial fans (see Theorem 3.3). Let us introduce some notation.
3.1. Let E be a divisorial fan on (Y, N ) of a contraction-free T-variety X(E ). Recall that π : X(E ) → Y stands for the quotient map. We denote by SH(E ) the finite subset of H(E ) (see notation in 1.7) which consists of elements of the form: ( for some D ∈ E and some y ∈ supp(E ). The set of elements of type (1) resp. (2) will be denoted by SH(E ) hor resp. SH(E ) ver . Note that the germ V (τ ) associated with τ ∈ SH(E ) hor is the germ obtained as the Zariski closure of (Y \ supp(E )) ×Ṽ (τ ) in X(E ), whereṼ (τ ) is the orbit closure of X(Σ ⋆ (E )) corresponding to τ and (Y \ supp(E )) × X(Σ ⋆ (E )) is identified with the complement of π −1 (supp(E )) red in X(E ). To every τ ∈ SH(E ) hor one can attach a divisorial fan E (τ ) describing the germ V (τ ) as follows. We consider the natural projection and we choose E (τ ) such that H(E (τ )) is generated by the subset with the condition X(E (τ )) to be contraction-free (i.e., the locus of every element of E (τ ) is affine).
Moreover, every germ V (τ ) with τ ∈ SH(E ) ver is contained in π −1 (supp(E )) red . If X(E ) is projective, then V (τ ) corresponds to a projective toric variety; we denote by Q τ a polytope such that the fan defining V (τ ) is the normal fan of Q τ . The notation E (τ ) resp. Q τ will apply in Section 3.2.
The next paragraph introduces notions of subdivision of divisorial fans and s-sequence similarly to the toric case.
Definition 3.2. With the same notation as in 3.1 for the divisorial fan E , we say that a divisorial fan E ′ on (Y, N ) is a subdivision of E if it verifies: (ii) For all D ∈ E there exist D 1 , . . . , D m ∈ E ′ with the same loci as D such that C (D) = i∈{1,...,m} C (D i ). Clearly a subdivision E ′ of E naturally induces an equivariant birational morphism which is proper (see [Tim11,Theorem 12.13]). Note that X(E ′ ) is also contraction-free and that (E ′ ) + y is a subdivision fan of E + y for every y ∈ Y . Similarly, Σ ⋆ (E ′ ) is a subdivision of Σ ⋆ (E ). An s-sequence of the subdivision E ′ of E is a sequence s τ,b of natural numbers where τ ∈ SH(E ) and b ∈ Z such that: (a) The sequence s b,τ (for τ ∈ SH(E ) hor = Σ ⋆ (E )) is a usual s-sequence (see Definition 1.13) of the fan subdivision Σ ⋆ (E ′ ) of Σ ⋆ (E ). (b) For every y ∈ supp(E ), the sequence s b,τ (for τ ∈ E + y ) is a usual s-sequence of the fan subdivision (E ′ ) + y of E + y . The following is the main result of this section.
Theorem 3.3. Let E be a divisorial fan on (P 1 , N ) corresponding to a rational contraction-free T-variety X(E ) and let E ′ be a subdivision of the divisorial fan E . If is the birational proper equivariant map given by the subdivision E ′ , then we have an isomorphism for all j ∈ Z, where s τ,b is an s-sequence of the subdivision E ′ . More concretely, this implies that for every Euclidian open subset U ⊆ X(E ) and every j ∈ Z, we have an isomorphism of Q-vector spaces Proof. Let D ∈ E . Let U z ⊆ P 1 and V 0 ⊆ A 1 be two Zariski open affine neighborhoods of z ∈ P 1 and 0 ∈ A 1 , respectively. Let us consider an isomorphism ϕ : U z → V 0 sending z to 0. Let D z be the polyhedral divisor over A 1 defined by the conditions D z 0 = D z and D z y = σ for all y = 0, where σ is the tail of D.
Shrinking U z and V 0 if necessary, we may suppose that D y = σ for all y ∈ U z \ {z}. Let Then the map ϕ induces an isomorphism X(D |Uz ) → X(D z |V 0 ). We also remark that X(D z |V 0 ) is seen as a Zariski open subset of the affine toric variety X(C (D) z ) for the torus T×G m associated with the cone C (D) z . Note that X(D z |V 0 ) intersects all the T × G m -orbits closure of X(C (D) z ). Let E ′ (D |Uz ) resp. E ′ (D z |V 0 ) be the divisorial fan of polyhedral divisors D ′ |Uz resp. (D ′ ) z |V 0 , where D ′ runs E ′ , such that C (D ′ ) ⊂ C (D) and D ′ has the same locus as D. Then ϕ induces a commutative diagram: The horizontal maps above are given by subdivisions of divisorial fans and the vertical are isomorphisms. Furthermore, X(E ′ (D z |V 0 )) is seen as a Zariski open subset of a toric variety and the second horizontal map is the restriction of a toric map given by a subdivision of fans. Via this isomorphism, the intersections of T × G m -orbits closure of the toric variety X(C (D) z ) with X(D z |V 0 ) correspond to intersections X(D |Uz ) ∩ V (τ ) where τ runs SH(E ). Applying the decomposition theorem for the commutative diagram above, and Theorem 1.12, we obtain isomorphisms where s τ,b is an s-sequence of the divisorial fan subdivision E ′ . Since X(E ) is contraction free we can cover X(E ) by open subsets and the open dense subset U z i ⊂ P 1 sastifying U z i ∩ supp(D) ⊂ {z i }. By the construction above we also have the isomorphisms Using the uniqueness of the decomposition into simple perverse sheaves, [Dim04, Remark 2.5.3] and (1), we conclude that where s τ,b is an s-sequence of the divisorial fan subdivision E ′ . Note that the local systems in the right-hand side of the previous relation are trivial since by (1)  Lemma 3.4. Let X(E ) be a projective contraction-free T-variety of complexity one corresponding to a divisorial fan E on (Y, N ). Then there exists a subdivision E ′ of the divisorial fan E such that for all y ∈ Y each cone of E ′ y is generated by a subset of a basis of (N ⊕ Z) Q and the natural morphism X(E ′ ) → X(E ) is projective.
Proof. First of all, we consider a star subdivision (see [CLS11,Section 11.1]) for every onedimensional cone of Σ ⋆ (E ). This induces a subdivision E ′′ of E . Secondly, to each y ∈ supp(E ′′ ) we consider a star subdivision of (E ′′ ) + y for every one-dimensional cone of (E ′′ ) + y \ Σ ⋆ (E ′′ ). This last process induces again a subdivision E ′ of E . We conclude by [PS11, Corollary 3.28] and the proof of [CLS11, Proposition 11.1.7] that the subdivision E ′ is the desired one.
As in Section 2, the next result establishes relations between h-polynomial of polytopes starting with Example 2.2.
Lemma 3.5. Let E , E ′ be divisorial fans on (P 1 , N ) as in Example 2.2. Assume that E ′ 0 is a subdivision of E 0 such that X(E ′ ) is projective and each cone of E ′ 0 is generated by a subset of a basis of (N ⊕ Z) Q . Let d = dim(X(E )) and d τ = dim(V (τ )) for τ ∈ H(E ). Then we have the formula for every j ∈ Z ≥0 , where s τ,b is any s-sequence of the subdivision E ′ (see the notation in 3.1).
Proof. Applying Theorem 1.12 for the (toric) proper map X(E ′ ) → X(E ) induced by the sub- for every j ∈ Z. We conclude by using Theorem 1.15 and the fact that s 0,0 = 1 and s 0,b = 0 for all b ∈ Z \ {0}.
The reader may remark that the formula of the next result depends only of the tail fans Σ ⋆ (E ) and Σ ⋆ (E ′ ) of E and E ′ . Furthermore, the proof of this result is a straightforward consequence of Theorem 1.12 and so we omitted it.
Lemma 3.6. Let E be a divisorial fan on (Y, N ) such that X(E ) is contraction-free and projective. Let E ′ be a subdivision of E as in Lemma 3.4. We recall that Q(E ) denotes a polytope such that Σ ⋆ (E ) is a normal fan of Q(E ). Then we have the equality h j+d (Q(E )) = h j+d (Q(E ′ )) − τ ∈SH(E ) hor , τ =0 b∈Z s τ,b · h j+dτ −b (Q(E (τ ))) for every j ∈ Z, where s τ,b is any s-sequence of the subdivision E ′ .
The following is the main result of this section.
Theorem 3.7. Let X(E ) be a singular rational projective contraction-free T-variety of complexity one corresponding to a divisorial fan E on (P 1 , N ). Denote by r the cardinal of the set supp(E ). Then we have the equality P X(E ) (t) = ((1 − r)t 2 + 1 − r)h(Q(E ); t 2 ) + y∈supp(E ) h(Q y (E ); t 2 ).
We apply the induction process for the germs V (τ ) = X(E (τ )) X(E ) so that b j+d (X(E )) = We observe that since supp(E (τ )) ⊂ supp(E ) for all τ ∈ SH(E ) (see 3.1) we can substitute the sets supp(E (τ )) by the set supp(E ) in the second and third sums of the right-hand side of the preceding equality. Using Lemmas 3.5 and 3.6 we have This finishes the proof of the theorem.