$\underset{\longrightarrow}{\lim}~{}\mathcal{H}^{k}(U;S)\longrightarrow\underset{\longrightarrow}{\lim}~{}\mathcal{H}^{k}(U-\partial
X;S)$

induced on direct limits by the restriction maps

(17) $\mathcal{H}^{k}(U;S)\longrightarrow\mathcal{H}^{k}(U-\partial X;S).$

Since an arbitrary neighborhood $U$ contains one of the form $U^{\prime}$,
neighborhoods of the latter type are cofinal in the directed set of all open
neighborhoods. So the direct limit can be computed on neighborhoods of type
$U^{\prime}$. When $S=\mathbb{Q}_{X}$ is the constant sheaf, the collar
homeomorphism $\phi$ identifies the restriction map

$\mathcal{H}^{k}(U^{\prime};\mathbb{Q}_{X})\longrightarrow\mathcal{H}^{k}(U^{\prime}-\partial
X;\mathbb{Q}_{X})$

with the restriction map

(18) $\mathcal{H}^{k}([0,\epsilon)\times
V;\mathbb{Q}_{X})\longrightarrow\mathcal{H}^{k}((0,\epsilon)\times
V;\mathbb{Q}_{X}).$

Consider the commutative diagram

$\textstyle{(0,\epsilon)\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[0,\epsilon)\times
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V,}$

where the maps to $V$ are the second factor projections. These projections
have contractible fibers and are homotopy equivalences. It follows that the
horizontal inclusion is a homotopy equivalence. On singular cohomology, which
is homotopy invariant, we get an induced restriction isomorphism

(19) $H^{k}([0,\epsilon)\times
V;\mathbb{Q})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}H^{k}((0,\epsilon)\times
V;\mathbb{Q}).$

For semi-locally contractible topological spaces, there is a natural
isomorphism between singular cohomology $H^{*}(-;\mathbb{Q})$ and sheaf
cohomology $H^{*}(-;\mathbb{Q}_{X})$ (Sella [30]). Locally contractible spaces
are in particular semi-locally contractible. The space $[0,\epsilon)\times V$
is locally contractible, since $V$ is an open subset of the locally
contractible space $\partial X$. The natural isomorphism between singular and
sheaf cohomology thus identifies the maps (19) and (18). Therefore, the latter
is an isomorphism as well. The basic principle on direct limits recalled
earlier implies that the direct limit of the maps (17) is an isomorphism. ∎

We shall use the term _cohomologically constructible_ , a property of
differential graded sheaves, in the sense of Borel [11, p. 69]. We will not
recall the definition here, but point out that this notion does not require a
stratification. According to [11, p. 79, Cor. 3.11 (i)], the constant sheaf on
a stratified pseudomanifold without boundary is cohomologically constructible.
We need to extend this statement to pseudomanifolds with boundary:

######  Lemma 44.

Let $(X,\partial X)$ be a topologically stratified pseudomanifold with
boundary. Then the constant sheaf $\mathbb{Q}_{X}$ on $X$ is cohomologically
constructible.

###### Proof.

Since the interior $X-\partial X$ is a pseudomanifold without boundary,
$\mathbb{Q}_{X}$ is cohomologically constructible at every point of the
interior by [11, p. 79, Cor. 3.11 (i)]. It remains to verify that
$\mathbb{Q}_{X}$ is cohomologically constructible at boundary points
$x\in\partial X$. Using a collar as in the proof of Lemma 43, there exists a
neighborhood basis $\\{U_{i}\\}_{i\in I}$ of $x$ in $X$ consisting of open
sets $U_{i}$ that are homeomorphic to $[0,\epsilon_{i})\times V_{i}$,
$V_{i}\subset\partial X$, $V_{i}$ open and contractible, $\epsilon_{i}>0$. If
$i<j$ so that $U_{i}\supset U_{j}$, then the restriction map

$\mathcal{H}^{k}(U_{i};\mathbb{Q}_{X})\longrightarrow\mathcal{H}^{k}(U_{j};\mathbb{Q}_{X})$

is an isomorphism, since in the commutative diagram

$\textstyle{U_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{pt}}$

the constant maps to the point are homotopy equivalences, and thus the
horizontal inclusion is a homotopy equivalence. (Here, as earlier, we may
identify singular and sheaf cohomology.) Thus, the direct system
$\\{\mathcal{H}^{k}(U_{i};\mathbb{Q}_{X})\\}_{i\in I}$ is constant, hence also
essentially constant in the sense of Borel’s definition [11, p. 68]. Since
$\\{U_{i}\\}_{i\in I}$ is cofinal in the directed set of all open
neighborhoods of $x$, the direct system
$\\{\mathcal{H}^{k}(U;\mathbb{Q}_{X})\\}$ over all neighborhoods of $x$ is
also essentially constant ([11, p. 69]). We observe furthermore that the
direct limit is finitely generated as it is one-dimensional for $k=0$ and $0$
otherwise. This proves condition CC2 of Borel.

We shall next treat the dual case of inverse systems given by compactly
supported cohomology. Thus we consider the inverse system with groups
$\mathcal{H}^{k}_{c}(U_{i};\mathbb{Q}_{X}),$ $i\in I,$ where $\\{U_{i}\\}$ is
the same neighborhood basis of $x\in\partial X$ as before. For $i<j$, so that
$U_{i}\supset U_{j}$, the transition maps are given by extension by zero

$\mathcal{H}^{k}_{c}(U_{j};\mathbb{Q}_{X})\longrightarrow\mathcal{H}^{k}_{c}(U_{i};\mathbb{Q}_{X}).$

We claim that in fact all of these groups vanish. Indeed, we may use the
Künneth formula

$R\Gamma_{c}^{\bullet}(A\times
B;\mathbb{Q})=R\Gamma_{c}^{\bullet}(A;\mathbb{Q})\otimes_{\mathbb{Q}}R\Gamma_{c}^{\bullet}(B;\mathbb{Q})$

valid for finite dimensional locally compact spaces $A,B$, [24, p. 323]. The
spaces $[0,\epsilon_{i})$ and $V_{i}$ satisfy these assumptions and therefore

$\mathcal{H}^{*}_{c}(U_{i};\mathbb{Q}_{X})\cong\mathcal{H}^{*}_{c}([0,\epsilon_{i})\times
V_{i};\mathbb{Q}_{X})\cong\mathcal{H}^{*}_{c}([0,\epsilon_{i});\mathbb{Q})\otimes\mathcal{H}^{*}_{c}(V_{i};\mathbb{Q}).$

Now, the compactly supported cohomology with constant coefficients of a closed
halfspace vanishes, [24, p. 189, 8.4]. So
$\mathcal{H}^{*}_{c}([0,\epsilon_{i});\mathbb{Q})=0$ and hence
$\mathcal{H}^{*}_{c}(U_{i};\mathbb{Q}_{X})=0$ as claimed. The inverse system
$\\{\mathcal{H}^{k}_{c}(U_{i};\mathbb{Q}_{X})\\}_{i\in I}$ is thus constant,
hence also essentially constant. Since $\\{U_{i}\\}_{i\in I}$ is cofinal in
the directed set of all open neighborhoods of $x$, the inverse system
$\\{\mathcal{H}^{k}_{c}(U;\mathbb{Q}_{X})\\}$ over all open neighborhoods of
$x$ is also essentially constant ([11, p. 69]). The inverse limit is zero, so
in particular finitely generated. This proves condition CC1 of Borel.

We have verified Borel’s conditions CC1 and CC2. These imply the remaining
conditions CC3 and CC4, as pointed out by Borel. ∎

Let $(X,\partial X)$ be a rational homology $n$-manifold with boundary. By
definition, $X-\partial X$ is a rational homology $n$-manifold (without
boundary). Thus the dualizing complex $\mathbb{D}_{X-\partial X}[-n]$ is
naturally quasi-isomorphic to the orientation sheaf
$\operatorname{or}_{X-\partial X}$ of $X-\partial X$.

######  Definition 45.

The homology manifold $(X,\partial X)$ with boundary is called orientable if
there exists an isomorphism $\operatorname{or}_{X-\partial
X}\cong\mathbb{Q}_{X-\partial X}$. A choice of such an isomorphism, if it
exists, is called an orientation.

Let $(X,\partial X)$ be an oriented rational homology $n$-manifold with
boundary. The orientation induces a quasi-isomorphism

$\mathbb{D}_{X-\partial X}[-n]\cong\operatorname{or}_{X-\partial
X}\cong\mathbb{Q}_{X-\partial X}.$

Using Verdier’s dualizing functor $\mathcal{D}$, we may express the dualizing
complex as $\mathbb{D}_{X-\partial X}=\mathcal{D}\mathbb{Q}_{X-\partial X}$
([23, p. 90]). The orientation therefore yields a self-duality isomorphism

$(\mathcal{D}\mathbb{Q}_{X-\partial X})[-n]\cong\mathbb{Q}_{X-\partial X}.$

######  Lemma 46.

(Borel [11, p. 137, Prop. 8.8 (3)].) Let $Y$ be a first-countable locally
compact space. Let $i:U\subset Y$ be an open inclusion and let $B$ be a
differential graded sheaf on $U$ such that $Ri_{*}B$ is cohomologically
constructible. Then

$\mathcal{D}Ri_{*}B=i_{!}\mathcal{D}B.$

We apply this lemma to $i:X-\partial X\hookrightarrow X$, observing that $X$
is locally compact and first-countable. We take $B=\mathbb{Q}_{X-\partial X}$.
By Lemma 43,

$Ri_{*}B=Ri_{*}i^{*}\mathbb{Q}_{X}\cong\mathbb{Q}_{X},$

the constant sheaf on $X$, which is cohomologically constructible by Lemma 44.
By Lemma 46,

$\mathcal{D}Ri_{*}\mathbb{Q}_{X-\partial
X}=i_{!}\mathcal{D}\mathbb{Q}_{X-\partial X}.$

Using the above self-duality isomorphism,

$\displaystyle i_{!}i^{*}\mathbb{Q}_{X}$
$\displaystyle=i_{!}\mathbb{Q}_{X-\partial X}\cong
i_{!}\mathcal{D}\mathbb{Q}_{X-\partial X}[-n]$
$\displaystyle=\mathcal{D}Ri_{*}\mathbb{Q}_{X-\partial
X}[-n]=\mathcal{D}Ri_{*}i^{*}\mathbb{Q}_{X}[-n]$
$\displaystyle=\mathcal{D}\mathbb{Q}_{X}[-n].$

On hypercohomology, there is an induced isomorphism

$\mathcal{H}^{k}(X;i_{!}i^{*}\mathbb{Q}_{X})\cong\mathcal{H}^{k-n}(X;\mathcal{D}\mathbb{Q}_{X}).$

The global effect of dualizing a differential graded sheaf is

$\mathcal{H}^{k-n}(X;\mathcal{D}\mathbb{Q}_{X})=\operatorname{Hom}(\mathcal{H}^{n-k}_{c}(X;\mathbb{Q}_{X}),\mathbb{Q}).$

(This holds on any locally compact space which has finite cohomological
dimension over $\mathbb{Q}$.) For compact $X$,
$\mathcal{H}^{n-k}_{c}(X;\mathbb{Q}_{X})=\mathcal{H}^{n-k}(X;\mathbb{Q}_{X}),$
so we obtain an isomorphism

$\mathcal{H}^{k-n}(X;\mathcal{D}\mathbb{Q}_{X})=\operatorname{Hom}(\mathcal{H}^{n-k}(X;\mathbb{Q}_{X}),\mathbb{Q}).$

Composing, we receive a duality isomorphism

$\mathcal{H}^{k}(X;i_{!}i^{*}\mathbb{Q}_{X})\cong\operatorname{Hom}(\mathcal{H}^{n-k}(X;\mathbb{Q}_{X}),\mathbb{Q})$

for compact $X$. The left hand group is nothing but the relative group
$H^{k}(X,\partial X;\mathbb{Q})$ (Iversen [24, p. 249]), while the universal
coefficient theorem identifies the right hand group with the homology group
$H_{n-k}(X;\mathbb{Q})$. The resulting isomorphism

$H^{k}(X,\partial X;\mathbb{Q})\cong H_{n-k}(X;\mathbb{Q})$

is Lefschetz duality for an oriented compact rational homology $n$-manifold
$(X,\partial X)$ with boundary. We have proved:

######  Theorem 47.

Let $(X,\partial X)$ be an oriented compact rational homology $n$-manifold
with boundary. Then $(X,\partial X)$ has a Lefschetz duality isomorphism

$H^{k}(X,\partial X;\mathbb{Q})\cong H_{n-k}(X;\mathbb{Q}).$

######  Corollary 48.

Let $X$ be an $n$-dimensional compact oriented $\mathbb{Q}$-pseudomanifold
with $\mathbb{Q}$-stratification

$\displaystyle X=X_{n}\supset X_{0}.$

Let $(\mathcal{M},\partial\mathcal{M})$ be the pseudomanifold with boundary
obtained by cutting out the 0-dimensional stratum in the
$\mathbb{Q}$-stratification. Then $(\mathcal{M},\partial\mathcal{M})$
satisfies Lefschetz duality with respect to ordinary rational homology.

###### Proof.

The pair $(\mathcal{M},\partial\mathcal{M})$ is a compact oriented rational
homology manifold with boundary. The result follows from Theorem 47. ∎

######  Corollary 49.

Let $X$ be an $n$-dimensional closed oriented $\mathbb{Q}$-pseudomanifold with
$\mathbb{Q}$-stratification $X=X_{n}$. (Such an $X$ is a rational homology
manifold.) Then $X$ satisfies Poincaré duality rationally.

######  Proposition 50.

If $(X,\partial X)$ is a rational homology $n$-manifold with boundary, then
$\partial X$ is a rational homology $(n-1)$-manifold (without boundary).

###### Proof.

Sheaf-theoretically, this is a direct consequence of [3, Chapter 4] and Lemma
43. The subspace $\partial X$ has a stratum preserving collar neighborhood in
$X$. This implies that links in $\partial X$ can be taken to be links in
$X-\partial X$. ∎

######  Example 51.

Let $X$ be a 4-dimensional toric variety. Then $X$ satisfies Poincaré duality
rationally.

######  Example 52.

Let $X$ be a 6-dimensional toric variety. The pseudomanifold obtained by
cutting out $X_{0}$ satisfies Lefschetz duality, rationally.

Concerning intersection homology, we shall use the following standard
notation.

######  Definition 53.

(Goresky-MacPherson.) A perversity

$\displaystyle\bar{p}:\mathbb{Z}_{\geq 2}\longrightarrow\mathbb{Z}$

is a function such that
$\bar{p}(2)=0\;\text{and}\;\bar{p}(k+1)-\bar{p}(k)\in\\{1,0\\}.$ The
complementary perversity $\bar{q}$ of $\bar{p}$ is the one with
$\bar{p}(k)+\bar{q}(k)=k-2$.

We denote the $i$-th intersection homology group of $X$ with coefficients in
$A$ and the perversity $\bar{p}$ with $I^{\bar{p}}H_{i}(X;A).$ Similarly, for
the $i$-th intersection cohomology group, we write $I_{\bar{p}}H^{i}(X;A).$
For an introduction to the theory of intersection homology, the reader may
consult [20] by Friedman, [25] by Kirwan and Woolf, or [4].

##  6\. Intersection Spaces and Isolated Singularities

In this section, first, we briefly recall the theory of intersection spaces
introduced by the first named author in [6]. For toric varieties, we only
consider the duality of Betti numbers for the middle perversity. Throughout
this section, we work with rational coefficients unless otherwise specified.

######  Definition 54.

Let $n$ be a natural number. A CW complex $\mathcal{K}$ is called rationally
$n$-segmented if it contains a sub-complex
$\mathcal{K}_{<n}\subset\mathcal{K}$ such that
$H_{r}(\mathcal{K}_{<n})=0\;\;\text{for}\;r\geq
n\;\text{and}\;i_{\ast}:H_{r}(\mathcal{K}_{<n})\xrightarrow{\;\;\cong\;\;}H_{r}(\mathcal{K})\;\text{for}\;r<n,$
where $i$ is the inclusion of $\mathcal{K}_{<n}$ into $\mathcal{K}$.

Given any CW complex $\mathcal{K}$ and natural number $n$, there exists a
homology $n$-truncation (Moore approximation)
$f:\mathcal{K}_{<n}\to\mathcal{K}$, i.e. a continuous map from a CW complex
$\mathcal{K}_{<n}$ to $\mathcal{K}$ such that
$f_{\ast}:H_{\ast}(\mathcal{K}_{<n};\mathbb{Z})\to
H_{\ast}(\mathcal{K};\mathbb{Z})$ is an isomorphism for $\ast<n$ and
$H_{\ast}(\mathcal{K}_{<n};\mathbb{Z})=0$ for $\ast\geq n$, see [6] (for the
simply connected case) and Wrazidlo [34] (in general). The map $f$ cannot in
general be taken to be a sub-complex inclusion. We shall prove in Proposition
67 that the 5-dimensional links in a 6-dimensional toric variety are
rationally 3-segmented.

Let $X\supset X_{0}$ be an $n$-dimensional compact oriented topological
pseudomanifold with isolated singularities. For $x_{i}\in X_{0}$, let
$\mathcal{L}_{i}$ be an associated link. Although two different links
associated with the same isolated singularity need not be homeomorphic, we
shall justify that the homology of links is well-defined. Let
$\mathcal{L}_{i}$ be as above and $\mathcal{U}$ an open distinguished
neighborhood of $x_{i}$. Then there exists a homeomorphism
$\phi:\mathcal{U}\xrightarrow{\;\;\cong\;\;}{\mathcal{C}}(\mathcal{L})$ which
sends $x_{i}$ to the cone point, $c$. We define
${\mathcal{C}}_{1/2}(\mathcal{L}_{i})\subset{\mathcal{C}}(\mathcal{L}_{i})$ as
the open cone over $\mathcal{L}_{i}$ by considering the interval
$[0,\frac{1}{2})\subset[0,1)$. Let $\mathcal{U}_{1/2}\subset\mathcal{U}$ be
the preimage of ${\mathcal{C}}_{1/2}(\mathcal{L}_{i})$ under the homeomorphism
$\phi$. By employing the excision axiom, we have $H_{\ast}(X,X-x_{i})\cong
H_{\ast}(X-(X-\mathcal{U}_{1/2},(X-x_{i})-(X-\mathcal{U}_{1/2}))=H_{\ast}(\mathcal{U}_{1/2},\mathcal{U}_{1/2}-x_{i}).$
By using the homeomorphism $\phi$ we arrive at $H_{\ast}(X,X-x_{i})\cong
H_{\ast}({\mathcal{C}}_{1/2}(\mathcal{L}_{i}),{\mathcal{C}}_{1/2}(\mathcal{L}_{i})-\\{c\\})$.
As ${\mathcal{C}}_{1/2}(\mathcal{L}_{i})\simeq\ast$ and
${\mathcal{C}}_{1/2}(\mathcal{L}_{i})-\\{c\\}\simeq\mathcal{L}_{i}$, the
connecting homomorphism in the long exact sequence of the relative homology
groups of the pair
$({\mathcal{C}}_{1/2}(\mathcal{L}_{i}),{\mathcal{C}}_{1/2}(\mathcal{L}_{i})-\\{c\\})$
is an isomorphism. If $\mathcal{L}^{\prime}_{i}$ is another link of $x_{i}$,
then the above procedure yields $H_{\ast}(\mathcal{L}_{i})\cong
H_{\ast}(\mathcal{L}^{\prime}_{i})$. Hence the homology of links is well-
defined.

Assume that all links $\mathcal{L}_{i}$ can be equipped with CW structures
such that they are rationally $k$-segmented, where $k=n-1-\bar{p}(n),$ for the
perversity $\bar{p}$. Let $(\mathcal{M},\partial\mathcal{M})$ be the manifold
with boundary obtained by cutting out all isolated singularities of $X$. Let
$(\mathcal{L}_{i})_{<k}$ be a sub-complex of $\mathcal{L}_{i}$ that truncates
the homology. Then, we have
$\partial\mathcal{M}=\bigsqcup_{i}\mathcal{L}_{i}.$ Let
$\mathcal{L}_{<k}=\bigsqcup_{i}(\mathcal{L}_{i})_{<k}$ and define a homotopy
class $g$ by the composition
$g:\mathcal{L}_{<k}\xhookrightarrow{\phantom{-}\phantom{-}}\partial\mathcal{M}\xhookrightarrow{\phantom{-}\operatorname{incl.}\phantom{-}}\mathcal{M}$.
For the purposes of the present paper we adopt the following definition:

######  Definition 55.

The perversity $\bar{p}$ rational intersection space $I^{\bar{p}}X$ of $X$ is
defined to be

$\displaystyle
I^{\bar{p}}X=\operatorname{cone}(g)=\mathcal{M}\bigcup_{g}\mathcal{C}(\mathcal{L}_{<k}).$

Due to the use of rational coefficients, $I^{\bar{p}}X$ as defined above need
not be integrally homotopy equivalent to the construction of [6], but the
rational homology groups are isomorphic.

######  Theorem 56.

Let $X$ be an $n$-dimensional compact oriented topological pseudomanifold with
only isolated singularities. Let $\bar{p}$ and $\bar{q}$ be complementary
perversities. Then, we have the duality isomorphism

$\displaystyle
d:\widetilde{H}_{r}(I^{\bar{p}}X)^{\ast}\xrightarrow{\phantom{-}\cong\phantom{-}}\widetilde{H}_{n-r}(I^{\bar{q}}X),$

where

$\displaystyle\widetilde{H}_{r}(I^{\bar{p}}X)^{\ast}=\operatorname{Hom}(\widetilde{H}_{r}(I^{\bar{p}}X),\mathbb{Q}).$

###### Proof.

The detailed construction of the above duality isomorphism can be found in the
proof of [6, Theorem 2.12] by the first named author. However, as mentioned,
$I^{\bar{p}}X$, as defined above need not be homotopy equivalent to the
construction of [6]. Hence, we need to adjust the previous proof slightly. In
our setting we have the inclusion
$\mathcal{L}_{<k}\hookrightarrow\mathcal{L}=\partial\mathcal{M}$. We consider
the same braid as in the proof in loc. cit. One can then proceed along the
same line as in [6]. ∎

######  Remark 57.

Let $k=n-1-\bar{p}(n)$. Then, we have the following isomorphisms. For $r>k$,

$\displaystyle H_{r}(\mathcal{M})$
$\displaystyle\xrightarrow{\phantom{-}\cong\phantom{-}}\widetilde{H}_{r}(I^{\bar{p}}X)$
$\displaystyle\widetilde{H}_{n-r}(I^{\bar{q}}X)$
$\displaystyle\xrightarrow{\phantom{-}\cong\phantom{-}}H_{n-r}(\mathcal{M},\partial\mathcal{M}).$

For $r<k$,

$\displaystyle\widetilde{H}_{r}(I^{\bar{p}}X)$
$\displaystyle\xrightarrow{\phantom{-}\cong\phantom{-}}H_{r}(\mathcal{M},\partial\mathcal{M})$
$\displaystyle H_{n-r}(\mathcal{M})$
$\displaystyle\xrightarrow{\phantom{-}\cong\phantom{-}}\widetilde{H}_{n-r}(I^{\bar{q}}X).$

######  Corollary 58.

If $n=\dim(X)$ is even, then the difference between the Euler characteristic
of $\widetilde{H}_{\ast}(I^{\bar{p}}X)$ and $I^{\bar{p}}H_{\ast}(X)$ is given
by

$\displaystyle\chi(\widetilde{H}_{\ast}(I^{\bar{p}}X))-\chi(I^{\bar{p}}H_{\ast}(X))=-2\chi_{<n-1-\bar{p}(n)}(\mathcal{L}),$

where $\mathcal{L}$ is the disjoint union of the links of the isolated
singularities of $X$. If $n=\dim(X)$ is odd, then

$\displaystyle\chi(\widetilde{H}_{\ast}(I^{\bar{n}}X))-\chi(I^{\bar{n}}H_{\ast}(X))=(-1)^{\frac{n-1}{2}}b_{\frac{n-1}{2}}(\mathcal{L}),$

where $b_{\frac{n-1}{2}}(\mathcal{L})$ is the middle dimensional Betti number
of $\mathcal{L}$ and $\bar{n}$ is the upper middle perversity. Regardless of
the parity of $n$, the identity

$\displaystyle\operatorname{rk}(\widetilde{H}_{k}(I^{\bar{p}}X))+\operatorname{rk}(I^{\bar{p}}H_{k}(X))=\operatorname{rk}(H_{k}(\mathcal{M}))+\operatorname{rk}(H_{k}(\mathcal{M},\mathcal{L}))$

always holds in degree $k=n-1-\bar{p}(n)$, where $\mathcal{M}$ is the exterior
of the singular set of $X$.

###### Proof.

A proof of the above corollary can be found in [6] by the first named author.
∎

######  Lemma 59.

The rational homology of $I^{\bar{p}}X$ is independent of the choice of
$(\mathcal{L}_{i})_{<k}$.

###### Proof.

From Remark 57 it follows immediately that $H_{r}(I^{\bar{p}}X)$ does not
depend of the choice of $(\mathcal{L}_{i})_{<k}$ for $r<k$ and $r>k$. For the
case $r=k$ the claim follows from the last identity in Corollary 58. ∎

##  7\. Construction of Intersection Spaces for $\mathbb{Q}$-Pseudomanifolds

In this section, we generalize the theory of intersection spaces to
$\mathbb{Q}$-pseudomanifolds with $\mathbb{Q}$-isolated singularities. To do
this, we revisit the proof of Theorem 56 introduced by the first named author
in [6]. The central idea is to indicate that the main ingredients of that
proof remain intact, even for $\mathbb{Q}$-pseudomanifold with isolated
singularities. Before proving our main theorem, we want to compute the
intersection homology of $\mathbb{Q}$-pseudomanifolds with
$\mathbb{Q}$-isolated singularities. But first of all, we need to generalize
the previous definition of intersection spaces to the $\mathbb{Q}$-isolated
setting. Let $X$ be an $n$-dimensional compact oriented
$\mathbb{Q}$-pseudomanifold with $\mathbb{Q}$-isolated singularities, which
means $X$ has a $\mathbb{Q}$-stratification of the form

$\displaystyle X=X_{n}\supset X_{0}.$

For $x_{i}\in X_{0}$, let $\mathcal{L}_{i}$ be an associated link. Assume that
all links are rationally $k$-segmented, where $k=n-1-\bar{p}(n),$ for the
perversity $\bar{p}$. Let $\mathcal{M}$ be the compact $\mathbb{Q}$-manifold
with boundary obtained by cutting out all $\mathbb{Q}$-isolated singularities
of $X$. Then, we have $\partial\mathcal{M}=\bigsqcup_{i}\mathcal{L}_{i}.$ Let
$\mathcal{L}_{<k}=\bigsqcup_{i}(\mathcal{L}_{i})_{<k},$ and define a homotopy
class
$g:\mathcal{L}_{<k}\xhookrightarrow{\phantom{-}\phantom{-}}\partial\mathcal{M}\xhookrightarrow{\phantom{-}\operatorname{incl.}\phantom{-}}\mathcal{M}$.

######  Definition 60.

The perversity $\bar{p}$ generalized intersection space $I^{\bar{p}}X$ of $X$
is defined to be

$\displaystyle
I^{\bar{p}}X=\operatorname{cone}(g)=\mathcal{M}\bigcup_{g}\mathcal{C}(\mathcal{L}_{<k}).$

We shall use the standard cone computation for intersection homology:

######  Proposition 61.

Suppose $Y$ is a compact topological pseudomanifold of dimension $m-1\geq 0$.
Then for a perversity $\bar{p}$,

$\displaystyle
I^{\bar{p}}H_{r}(\mathcal{C}(Y))=\begin{cases}I^{\bar{p}}H_{r}(Y),&\phantom{--}r<m-1-\bar{p}(m)\\\
0,&\phantom{--}\text{otherwise.}\end{cases}$

######  Proposition 62.

Let $X$ be an $n$-dimensional $\mathbb{Q}$-pseudomanifold with
$\mathbb{Q}$-isolated singularities, where $n\geq 1$. Then for perversity
$\bar{p}$, we have

$\displaystyle
I^{\bar{p}}H_{r}(X)\cong\begin{cases}H_{r}(\mathcal{M},\mathcal{L}),&\phantom{--}r>k\\\
H_{r}(\mathcal{M}),&\phantom{--}r<k,\end{cases}$

where $\mathcal{M}$ is again the $\mathbb{Q}$-manifold obtained by cutting out
the $\mathbb{Q}$-isolated singularities and
$\mathcal{L}=\bigsqcup\mathcal{L}_{x_{i}}$, where for each $x_{i}\in X_{0}$,
$\mathcal{L}_{x_{i}}$ is a link associated to $x_{i}$, and $k=n-1-\bar{p}(n)$.

###### Proof.

We cut out cone-like neighborhoods of $\mathbb{Q}$-isolated singularities. The
obtained topological space is a pseudomanifold with boundary,
$(\mathcal{M},\partial\mathcal{M})$. In fact
$(\mathcal{M},\partial\mathcal{M})$ is a $\mathbb{Q}$-manifold with boundary.
Let $\mathcal{C}(\mathcal{L}_{x_{i}})$ be the removed open cone-like
neighborhood of $x_{i}$. Let $\mathcal{V}$ be an open neighborhood of
$\mathcal{M}$ in $X$ that deformation retracts in a stratum preserving way to
$\mathcal{M}$ and set
$\mathcal{U}=\bigsqcup_{i}\mathcal{C}(\mathcal{L}_{x_{i}})$. Thus, we have
$\cap:=\mathcal{U}\cap\mathcal{V}\cong\bigsqcup_{i}\mathcal{L}_{x_{i}}\times(\frac{1}{2},1)$.
The Mayer–Vietoris sequence for intersection homology is a long exact sequence

$\displaystyle\dots\longrightarrow I^{\bar{p}}H_{r}(\cap)\longrightarrow
I^{\bar{p}}H_{r}(\mathcal{U})\oplus
I^{\bar{p}}H_{r}(\mathcal{V})\longrightarrow
I^{\bar{p}}H_{r}(X)\longrightarrow\dots.$

For $r>k$, we have $I^{\bar{p}}H_{r}(\mathcal{U})=0$ by Proposition 61.
Consider the long exact sequence for relative ordinary homology

$\displaystyle\dots\longrightarrow H_{r}(\partial\mathcal{M})\longrightarrow
H_{r}(\mathcal{\mathcal{M}})\longrightarrow
H_{r}(\mathcal{M},\partial\mathcal{M})\longrightarrow\dots.$

We construct a commutative diagram as follows:

(24) $\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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240.11905pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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31.43576pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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99.93898pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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162.44742pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
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240.80481pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
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313.97472pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
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We look at the morphism
$I^{\bar{p}}C_{\ast}(\partial\mathcal{M})\hookrightarrow
C_{\ast}(\partial\mathcal{M})$ of singular ordinary and intersection chain
complexes. Since $\mathcal{M}$ is a $\mathbb{Q}$-manifold, this morphism
induces an isomorphism
$I^{\bar{p}}H_{\ast}(\partial\mathcal{M})\xrightarrow{\cong}H_{\ast}(\partial\mathcal{M})$.
The outer left vertical homomorphism, which turns out to be an isomorphism, is
the composition
$I^{\bar{p}}H_{\ast}(\cap)\xrightarrow{\cong}I^{\bar{p}}H_{\ast}(\partial\mathcal{M})\xrightarrow{\cong}H_{\ast}(\partial\mathcal{M})$.
The stratum preserving deformation retraction
$\mathcal{V}\rightarrow\mathcal{M}$, induces an isomorphism
$I^{\bar{p}}H_{\ast}(\mathcal{V})\xrightarrow{\cong}I^{\bar{p}}H_{\ast}(\mathcal{M})$.
The morphism $I^{\bar{p}}C_{\ast}(\mathcal{M})\hookrightarrow
C_{\ast}(\mathcal{M})$ induces an isomorphism
$I^{\bar{p}}H_{\ast}(\mathcal{M})\xrightarrow{\cong}H_{\ast}(\mathcal{M})$
since $\mathcal{M}$ is a $\mathbb{Q}$-manifold. The second left vertical
homomorphism is the composition
$I^{\bar{p}}H_{\ast}(\mathcal{V})\xrightarrow{\cong}I^{\bar{p}}H_{\ast}(\mathcal{M})\xrightarrow{\cong}H_{\ast}(\mathcal{M})$.
The construction of the two right hand vertical homomorphisms goes along the
same lines. Describing the middle homomorphism requires a bit more effort.
First, note that
$H_{\ast}(\mathcal{M},\partial\mathcal{M})\cong\tilde{H}_{\ast}(\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}})$.
We show that $H_{j}(X)\cong
H_{j}(\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}})$ for $j\geq 2$. We use
the Mayer-Vietories sequence for ordinary homology groups. Let
$A_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}}\cong\mathcal{C}(\bigsqcup_{i}\mathcal{L}_{x_{i}})\subset\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}$
and
$B_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}}\cong\mathcal{M}\subset\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}$
such that $A_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}}\cap
B_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}}\cong\bigsqcup_{i}\mathcal{L}_{x_{i}}\times(0,1)$.
We also set $A_{X}\cong\bigsqcup_{i}\mathcal{C}(\mathcal{L}_{x_{i}})\subset X$
and $B_{X}\cong\mathcal{M}\subset X$ such that $A_{X}\cap
B_{X}\cong\bigsqcup_{i}\mathcal{L}_{x_{i}}\times(0,1)$. For $j\geq 1$, we have
$H_{j}(A_{X})\oplus H_{j}(B_{X})\cong
H_{j}(A_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}})\oplus
H_{j}(B_{\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}})\cong
H_{j}(\mathcal{M})$. The claim follows by using 5-lemma. Note that $k\geq 1$.
Let $I^{\bar{p}}H_{\ast}(X)\rightarrow H_{\ast}(X)$ be the homomorphism
induced by the morphism $I^{\bar{p}}C_{\ast}(X)\hookrightarrow C_{\ast}(X)$.
The middle vertical homomorphism is the composition

$\displaystyle I^{\bar{p}}H_{\ast}(X)\rightarrow
H_{\ast}(X)\xrightarrow{\cong}H_{\ast}(\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}})\xrightarrow{\cong}H_{\ast}(\mathcal{M},\partial\mathcal{M}),\;\ast\geq
2.$

We still need to check the commutativity of the above diagram. Consider the
commutative diagram

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
Each inclusion induces a morphism between intersection chain complexes, and
the induced diagram on intersection chain complexes commutes. Hence, the
induced diagram on intersection homology groups

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commutes. Recall that vertical homomorphisms are isomorphisms. Furthermore,
the diagram

commutes. Note that the vertical morphisms induced by inclusions of singular
intersection chain complexes into ordinary singular chain complexes are
isomorphisms, as mentioned. Hence, the outer left square commutes. Using the
same procedure, one can show that the outer right square commutes. Consider
the commutative diagram

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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31.58334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
36.11806pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
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Each inclusion induces a morphism between intersection chain complexes, and
the induced diagram on intersection chain complexes commutes. Hence, the
induced diagram on intersection homology groups

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.02501pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.02501pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{I^{\bar{p}}H_{\ast}(X)}$}}}}}}}{\hbox{\kern-22.02501pt\raise-41.6044pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{I^{\bar{p}}H_{\ast}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
59.01018pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
65.00139pt\raise-41.6044pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

commutes. The commutativity of

ensures that

commutes. Thus, the second left square commutes. Consider the short exact
sequence

$\displaystyle 0\longrightarrow I^{\bar{p}}C_{i}(\cap)\longrightarrow
I^{\bar{p}}C_{i}(\mathcal{U})\oplus
I^{\bar{p}}C_{i}(\mathcal{V})\longrightarrow
I^{\bar{p}}C_{i}(\mathcal{U})+I^{\bar{p}}C_{i}(\mathcal{V})\longrightarrow 0.$

Note that $I^{\bar{p}}C_{i}(\mathcal{U})+I^{\bar{p}}C_{i}(\mathcal{V})$ is the
sub-complex of $I^{\bar{p}}C_{i}(X)$ generated by allowable chains supported
in $\mathcal{U}$ or in $\mathcal{V}$. The commutativity of the diagram

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{I^{\bar{p}}C_{i}(\cap)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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92.91681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{I^{\bar{p}}C_{i}(\mathcal{U})\oplus
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203.63922pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{I^{\bar{p}}C_{i}(\mathcal{U})+I^{\bar{p}}C_{i}(\mathcal{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
314.36163pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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314.36163pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-40.8022pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.09029pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.09029pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C_{i}(\cap)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
102.09737pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.09737pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C_{i}(\mathcal{U})\oplus
C_{i}(\mathcal{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
212.81978pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
212.81978pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C_{i}(\mathcal{U})+C_{i}(\mathcal{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
314.36163pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
314.36163pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

implies that the diagram

commutes. The commutativity of the right square in the above diagram ensures
that the second right square in Diagram 24 commutes. The desired result
follows from 5-lemma. Thus, we have

$\displaystyle I^{\bar{p}}H_{r}(X)\cong
H_{r}(\mathcal{M},\partial\mathcal{M}),\text{for}\phantom{..}r>k.$

For $r<k$, the proof goes as follows. Let $\mathcal{U}$ and $\mathcal{V}$ be
as above. Employing Proposition 61 yields $I^{\bar{p}}H_{r}(\mathcal{U})\cong
I^{\bar{p}}H_{r}(\mathcal{L})$. Recall that $\mathcal{L}$ is a
$\mathbb{Q}$-manifold. Hence, we have the following diagram.

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.375pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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16.375pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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64.66385pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{I^{\bar{p}}H_{r}(\mathcal{M})\oplus
I^{\bar{p}}H_{r}(\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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158.11655pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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208.56787pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{I^{\bar{p}}H_{r-1}(\cap)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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261.0567pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
362.9094pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-6.375pt\raise-39.83554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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73.32635pt\raise-39.83554pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
212.89912pt\raise-39.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
269.71921pt\raise-39.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H_{r-1}(\mathcal{\mathcal{M}})\oplus
H_{r-1}(\mathcal{\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
362.9094pt\raise-39.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
362.9094pt\raise-39.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The commutativity of the left hand and right hand squares follows from the
commutativity of

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.14377pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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44.14377pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.3022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
44.14377pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{I^{\bar{p}}C_{\ast}(\mathcal{L})\oplus
I^{\bar{p}}C_{\ast}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
88.58403pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
88.58403pt\raise-30.3022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C_{\ast}(\cap)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
53.32433pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.32433pt\raise-40.8022pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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C_{\ast}(\mathcal{M})}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The vertical monomorphisms induce isomorphisms between the intersection and
ordinary homology groups. By [6, Lemma 2.46], we can construct the middle
vertical homomorphism such that the diagram commutes. Finally, by 5-lemma, we
have

$\displaystyle I^{\bar{p}}H_{r}(X)\cong
H_{r}(\mathcal{M}),\text{for}\phantom{..}r<k.$

This concludes the proof. ∎

We are now at the point where we can start with the proof of the main theorem
of this section. In [6], the first named author shows more than just the
duality of Betti numbers of intersection spaces. Although we are only
interested in the duality of the Betti numbers of intersection spaces, we can
generalize [6, Theorem 2.12] to $\mathbb{Q}$-pseudomanifolds. For a brief
introduction to reflective algebra, the reader may consult [6, Section 2.1].

######  Remark 63.

As mentioned earlier, we only consider homology with rational coefficients.
Note also that because the underlying space, $X$, is a
$\mathbb{Q}$-pseudomanifold with $\mathbb{Q}$-isolated singularities,
$I^{\bar{p}}X$ refers to the generalized intersection space of $X$ in the
sense of Definition 60.

######  Theorem 64.

Let $X$ be an $n$-dimensional compact oriented $\mathbb{Q}$-pseudomanifold
with only $\mathbb{Q}$-isolated singularities. Let $\bar{p}$ and $\bar{q}$ be
complementary perversities. Assume that the link of each $\mathbb{Q}$-isolated
singularity is rationally $(n-1-\bar{p}(n))$-segmented. Then

  1. (1)

The pair $(\widetilde{H}_{\ast}(I^{\bar{p}}X),I^{\bar{p}}H_{\ast}(X))$ is
$(n-1-\bar{p}(n))$-reflective across the homology of the links and

  2. (2)

$(\widetilde{H}_{\ast}(I^{\bar{p}}X),I^{\bar{p}}H_{\ast}(X))$ and
$(\widetilde{H}_{\ast}(I^{\bar{q}}X),I^{\bar{q}}H_{\ast}(X))$ are $n$-dual
reflective pairs.

###### Proof.

We mainly mimic the proof of [6, Theorem 2.12]. By the above considerations,
the main ingredients of the proof remain intact for
$\mathbb{Q}$-Pseudomanifolds with $\mathbb{Q}$-isolated singularities. We
start with the homology braid of the triple

$\displaystyle\mathcal{L}_{<k}\xhookrightarrow{\phantom{-f}\phantom{-}}\mathcal{L}\xrightarrow{\phantom{-}j\phantom{-}}\mathcal{M}.$

By Proposition 62, the same argument as in the proof of [6, Theorem 2.12] can
be applied. The rest of the proof relies on Lefschetz duality for
$(\mathcal{M},\partial\mathcal{M})$ and Poincaré duality for
$\partial\mathcal{M}$. For Lefschetz duality, we employ Theorem 47, The
topological space $\partial\mathcal{M}$ is a $\mathbb{Q}$-manifold by
Proposition 50, and therefore it satisfies Poincaré duality rationally, using
Corollary 49. This concludes our proof. ∎

######  Corollary 65.

Let $X$ be an $n$-dimensional, compact, oriented, topological
$\mathbb{Q}$-pseudomanifold with only $\mathbb{Q}$-isolated singularities,
which means there is a $\mathbb{Q}$-stratification of the form $X_{n}\supset
X_{0}.$ Let $\bar{p}$ and $\bar{q}$ be complementary perversities. Then, we
have the duality isomorphism

$\displaystyle
d:\widetilde{H}_{r}(I^{\bar{p}}X)^{\ast}\xrightarrow{\phantom{-}\cong\phantom{-}}\widetilde{H}_{n-r}(I^{\bar{q}}X),$

where

$\displaystyle\widetilde{H}_{r}(I^{\bar{p}}X)^{\ast}=\operatorname{Hom}(\widetilde{H}_{r}(I^{\bar{p}}X),\mathbb{Q}).$

###### Proof.

The construction of the duality isomorphism goes along the same line as in the
proof of [6, Theorem 2.12] and using Corollary 49 and Theorem 47. ∎

Corollary 58 continues to hold in the $\mathbb{Q}$-isolated setting for the
generalized intersection spaces:

######  Corollary 66.

Let $X$ be an $n$-dimensional, compact, oriented $\mathbb{Q}$-pseudomanifold
with only $\mathbb{Q}$-isolated singularities. Let $I^{\bar{p}}X$ be the
associated generalized intersection spaces. If $n=\dim(X)$ is even, then the
difference between the Euler characteristic of
$\widetilde{H}_{\ast}(I^{\bar{p}}X)$ and $I^{\bar{p}}H_{\ast}(X)$ is given by

$\displaystyle\chi(\widetilde{H}_{\ast}(I^{\bar{p}}X))-\chi(I^{\bar{p}}H_{\ast}(X))=-2\chi_{<n-1-\bar{p}(n)}(\mathcal{L}),$

where $\mathcal{L}$ is the disjoint union of the links of the
$\mathbb{Q}$-isolated singularities of $X$. If $n=\dim(X)$ is odd, then

$\displaystyle\chi(\widetilde{H}_{\ast}(I^{\bar{n}}X))-\chi(I^{\bar{n}}H_{\ast}(X))=(-1)^{\frac{n-1}{2}}b_{\frac{n-1}{2}}(\mathcal{L}),$

where $b_{\frac{n-1}{2}}(\mathcal{L})$ is the middle dimensional Betti number
of $\mathcal{L}$ and $\bar{n}$ is the upper middle perversity. Regardless of
the parity of $n$, the identity

$\displaystyle\operatorname{rk}(\widetilde{H}_{k}(I^{\bar{p}}X))+\operatorname{rk}(I^{\bar{p}}H_{k}(X))=\operatorname{rk}(H_{k}(\mathcal{M}))+\operatorname{rk}(H_{k}(\mathcal{M},\mathcal{L}))$

always holds in degree $k=n-1-\bar{p}(n)$, where $\mathcal{M}$ is the
$\mathbb{Q}$-manifold obtained by cutting out the $\mathbb{Q}$-isolated
singularities of $X$.

##  8\. Intersection Spaces of 6-dimensional Toric Varieties

As mentioned in Section 4.3, a 6-dimensional toric variety is a
$\mathbb{Q}$-pseudomanifold with $\mathbb{Q}$-isolated singularities. Such
varieties are therefore amenable to the theory developed in Section 7.
Although we will not compute the homology groups of the intersection spaces
directly via CW structures, we will show that considering the middle
perversity, links of $\mathbb{Q}$-isolated singularities are rationally
3-segmented. Using Lefschetz duality, we then compute the rational homology
groups of generalized intersection spaces of 6-dimensional toric varieties. As
a side remark, we will also show that Corollary 66 yields the well-known Euler
formula for 3-dimensional polytypes. Let $X_{\mathcal{P}}$ be a 6-dimensional
toric variety associated with $\mathcal{P}$, the underlying polytope. We can
endow $X_{\mathcal{P}}$ with a $\mathbb{Q}$-stratification of the form
$X_{\mathcal{P}}=X_{6}\supset X_{0}.$ We have shown in Section 4.3 that links
of $\mathbb{Q}$-isolated singularities are $\mathbb{Q}$-manifolds and satisfy
Poincaré duality rationally.

######  Proposition 67.

Let $X$ be a 6-dimensional toric variety with a $\mathbb{Q}$-stratification of
the form $X_{6}\supset X_{0}$. For each $x_{i}\in X_{0}$, let
$\mathcal{L}_{i}$ be the associated link. Then $\mathcal{L}_{i}$ is rationally
3-segmented.

###### Proof.

We endow each $\mathcal{L}_{i}$ with the CW structure described in Section
4.3. Recall that we have computed $\partial_{3}$ of the $\mathcal{L}_{i}$ in
Equation 15. From each $T^{2}$ with one 2-cell, we remove this 2-cell. From
each $T^{2}$ with two 2-cells, we omit the 2-cell with the negative sign in
the boundary operator. We denote the obtained 3-dimensional CW sub-complex by
$\mathcal{L}_{i}^{\prime}$. Despite the removal of the 2-cells from the tori,
the 2-skeleton of the link has not changed, i.e $\mathcal{L}_{i}$ and
$\mathcal{L}_{i}^{\prime}$ have the same 2-skeleton. Hence, we get

(26) $\displaystyle\partial_{3}^{\mathcal{L}_{i}^{\prime}}$
$\displaystyle=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
e^{1}_{T^{3}_{1}}\times e^{2}_{\mathcal{M}_{x}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle e^{1}_{T^{3}_{2}}\times
e^{2}_{\mathcal{M}_{x}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
e^{1}_{T^{3}_{3}}\times e^{2}_{\mathcal{M}_{x}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle e^{2}_{(T^{2}_{a_{i}})_{1}}\times
e^{1}_{(\mathcal{M}_{x_{a}})_{i}}\\\e^{0}_{T^{3}}\times
e^{2}_{\mathcal{M}_{x}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\overbrace{0}^{\tilde{a}}\\\e^{1}_{(T^{2}_{a_{i}})_{1}}\times
e^{1}_{(\mathcal{M}_{x_{a}})_{i}}\tilde{a}\Big{\\{}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle m_{a_{i}}l_{a_{i}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1\\\e^{1}_{(T^{2}_{a_{i}})_{2}}\times
e^{1}_{(\mathcal{M}_{x_{a}})_{i}}\tilde{a}\Big{\\{}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
n_{a_{i}}l_{a_{i}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\e^{1}_{(T^{2}_{a_{i}})_{3}}\times
e^{1}_{(\mathcal{M}_{x_{a}})_{i}}\tilde{a}\Big{\\{}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle n_{a_{i}}m_{a_{i}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\e^{1}_{(T^{2}_{b_{j}})_{1}}\times
e^{1}_{(\mathcal{M}_{x_{b}})_{j}}\tilde{b}\Big{\\{}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-m_{b_{j}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0\\\e^{1}_{(T^{2}_{b_{j}})_{2}}\times
e^{1}_{(\mathcal{M}_{x_{b}})_{j}}\tilde{b}\Big{\\{}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
n_{b_{j}}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\$\hfil\kern 5.0pt\crcr}}}}\right]$}}.$

We use the same notation as in Section 4.3. Recall that each 1-dimensional
face of $\mathcal{M}_{x}$, is associated to a 2-dimensional face of
$\mathcal{P}$, which is dual to a 1-dimensional cone in $\Sigma$ such that the
dual cones lie in $\mathcal{S}_{x}$. Here $\tilde{a}$ and $\tilde{b}$ are the
numbers of such 1-dimensional cones in $\Sigma$, whose generators have no zero
entry and at least one zero-entry, respectively. Recall also that
$e^{1}_{(\mathcal{M}_{x_{a}})_{i}}$ and $e^{1}_{(\mathcal{M}_{x_{b}})_{j}}$
are those 1-cells of $\mathcal{M}_{x}$, whose associated $T^{2}$ in
$\mathcal{L}_{x}$ has three and two 1-cells, respectively. Clearly, we have

$\displaystyle\operatorname{Im}(\partial_{3}^{\mathcal{L}_{i}^{\prime}})=\operatorname{Im}(\partial_{3}^{\mathcal{L}_{i}})\>\>\text{and}\>\>\operatorname{ker}(\partial_{3}^{\mathcal{L}_{i}^{\prime}})=0,$

and the inclusion
$incl:\mathcal{L}_{i}^{\prime}\xhookrightarrow{\phantom{---}}\mathcal{L}_{i}$
induces

$\displaystyle
incl_{j}:H_{j}(\mathcal{L}_{i}^{\prime})\xrightarrow{\phantom{-}\cong\phantom{-}}H_{j}(\mathcal{L}_{i})\>\>\text{for}\>\>j\leq
2.$

Obviously,

$\displaystyle H_{j}(\mathcal{L}_{i}^{\prime})=0\>\>\text{for}\>\>j>2.$

∎

######  Theorem 68.

Let $X_{\mathcal{P}}$ be an arbitrary compact 6-dimensional toric variety with
$m$ $\mathbb{Q}$-isolated singularities, $m\geq 1$, where $\mathcal{P}$ is the
underlying polytope. Let $\Sigma$ be the dual fan to $\mathcal{P}$. We denote
the number of 1-dimensional and 2-dimensional cones of $\Sigma$ with $f_{1}$
and $f_{2}$, respectively. Then,

$\displaystyle\operatorname{rk}(\widetilde{H}_{6}(I^{\bar{n}}X))$
$\displaystyle=0$
$\displaystyle\operatorname{rk}(\tilde{H}_{5}(I^{\bar{n}}X))$
$\displaystyle=m-1$
$\displaystyle\operatorname{rk}(\tilde{H}_{4}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\tilde{H}_{3}(I^{\bar{n}}X))$
$\displaystyle=2(3f_{1}-f_{2}-b-6)$
$\displaystyle\operatorname{rk}(\tilde{H}_{2}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\tilde{H}_{1}(I^{\bar{n}}X))$
$\displaystyle=m-1$
$\displaystyle\operatorname{rk}(\tilde{H}_{0}(I^{\bar{n}}X))$
$\displaystyle=0,$

where $I^{\bar{n}}X$ is the generalized intersection space associated to the
$\mathbb{Q}$-pseudomanifold $X_{\mathcal{P}}$, and the parameter $b$ is
introduced and has been studied in Section 4.2.1.

###### Proof.

We start by cutting out the $\mathbb{Q}$-isolated singularities of
$X_{\mathcal{P}}$, in the sense of Definition 41. We denote the resulting
$\mathbb{Q}$-manifold with boundary by ($\mathcal{M},\partial\mathcal{M}$). It
satisfies Lefschetz duality rationally by Corollary 48. Now, let
$\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}}$ be the topological
pseudomanifold obtained by coning off the boundary of
$(\mathcal{M},\partial\mathcal{M})$. As shown, we have

$\displaystyle H_{i}(X)\cong
H_{i}(\nicefrac{{\mathcal{M}}}{{\partial\mathcal{M}}})\cong
H_{i}(\mathcal{M},\partial\mathcal{M})\>\>\text{for}\>i\geq 2.$

Hence by Proposition 27, we get

$\displaystyle\operatorname{rk}(H_{6}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{0}(\mathcal{M}))$
$\displaystyle=1$
$\displaystyle\operatorname{rk}(H_{5}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{1}(\mathcal{M}))$
$\displaystyle=0$
$\displaystyle\operatorname{rk}(H_{4}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{2}(\mathcal{M}))$
$\displaystyle=f_{1}-3$
$\displaystyle\operatorname{rk}(H_{3}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{3}(\mathcal{M}))$
$\displaystyle=3f_{1}-f_{2}-b-6$
$\displaystyle\operatorname{rk}(H_{2}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{4}(\mathcal{M}))$
$\displaystyle=f_{1}-3-b.$

The group $H_{6}(\mathcal{M}))$ vanishes since by Lefschetz duality
$\operatorname{rk}(H_{6}(\mathcal{M}))=\operatorname{rk}(H_{0}(\mathcal{M},\partial\mathcal{M}))=0$
(Note that $\partial\mathcal{M}$ is not empty as $m\geq 1$.) Consequently, by
considering the long exact sequence of the pair
$(\mathcal{M},\partial\mathcal{M})$, we arrive at the short exact sequence

$\displaystyle 0\longrightarrow
H_{6}(\mathcal{M},\partial\mathcal{M})\longrightarrow
H_{5}(\partial\mathcal{M})\longrightarrow H_{5}(\mathcal{M})\longrightarrow
0.$

Since $\partial\mathcal{M}=\bigsqcup_{i=1}^{m}\mathcal{L}_{i}$ and each
$\mathcal{L}_{i}$ is orientable, connected and 5-dimensional,
$\operatorname{rk}(H_{5}(\partial\mathcal{M}))=m$. This implies

$\displaystyle\operatorname{rk}(H_{1}(\mathcal{M},\partial\mathcal{M}))=\operatorname{rk}(H_{5}(\mathcal{M}))$
$\displaystyle=m-1.$

Before dealing with the homology groups of the generalized intersection space,
we need to compute the intersection homology groups of $X_{\mathcal{P}}$. In
[32], Stanley shows that the odd degree intersection homology groups of toric
varieties vanish. We merely need this result for the middle degree. The
relevant cut off degree is $k=6-1-\bar{n}(6)=3$. Using Proposition 62, we
finally get

$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{6}(X))$ $\displaystyle=1$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{5}(X))$ $\displaystyle=0$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{4}(X))$ $\displaystyle=f_{1}-3$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{3}(X))$ $\displaystyle=0$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{2}(X))$ $\displaystyle=f_{1}-3$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{1}(X))$ $\displaystyle=0$
$\displaystyle\operatorname{rk}(I^{\bar{n}}H_{0}(X))$ $\displaystyle=1.$

Proposition 67 ensures the existence of the associated generalized
intersection space of $X_{\mathcal{P}}$. By using the duality isomorphisms of
Theorem 64, we can read off the homology groups of $I^{\bar{n}}X$ above and
below the middle degree:

$\displaystyle
H_{r}(\mathcal{M})\xrightarrow{\phantom{--}\cong\phantom{--}}\widetilde{H}_{r}(I^{\bar{n}}X)\>\>\text{for}\>\>r>k=3$
$\displaystyle
H_{r}(\mathcal{M},\partial\mathcal{M})\xrightarrow{\phantom{--}\cong\phantom{--}}\widetilde{H}_{r}(I^{\bar{n}}X)\>\>\text{for}\>\>r<k=3.$

Finally, we use Proposition 33 and the last identity in Corollary 66 and
compute $\operatorname{rk}(\widetilde{H}_{3}(I^{\bar{n}}X))$. ∎

######  Corollary 69.

Let $X_{\mathcal{P}}$ be an arbitrary compact 6-dimensional toric variety,
where $\mathcal{P}$ is the underlying polytope. Let $\Sigma$ be the fan dual
to $\mathcal{P}$. We denote the 1-dimensional, 2-dimensional, and
3-dimensional cones of $\Sigma$ with $f_{1}$, $f_{2}$, and $f_{3}$,
respectively. We further assume that all vertices in $\mathcal{P}$ are
$\mathbb{Q}$-isolated singularities. Then,

$\displaystyle\operatorname{rk}(\widetilde{H}_{6}(I^{\bar{n}}X))$
$\displaystyle=0$
$\displaystyle\operatorname{rk}(\tilde{H}_{5}(I^{\bar{n}}X))$
$\displaystyle=f_{3}-1$
$\displaystyle\operatorname{rk}(\tilde{H}_{4}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\tilde{H}_{3}(I^{\bar{n}}X))$
$\displaystyle=2(3f_{1}-f_{2}-b-6)$
$\displaystyle\operatorname{rk}(\tilde{H}_{2}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\tilde{H}_{1}(I^{\bar{n}}X))$
$\displaystyle=f_{3}-1$
$\displaystyle\operatorname{rk}(\tilde{H}_{0}(I^{\bar{n}}X))$
$\displaystyle=0.$

###### Proof.

The number of vertices in $\mathcal{P}$ is equal to $f_{3}$, which equals $m$
by the assumption. ∎

######  Remark 70.

For $X_{\mathcal{P}}$ an arbitrary compact 6-dimensional toric variety with
$m$ $\mathbb{Q}$-isolated singularities, $m\geq 1$, we employ the first
identity on the Euler characteristic in Corollary 66 to obtain

$\displaystyle\big{(}0-(m-1)+(f_{1}-3-b)-2(3f_{1}-f_{2}-b-6)+(f_{1}-3-b)$
$\displaystyle-(m-1)+0\big{)}-\big{(}1+(f_{1}-3)+(f_{1}-3)+1\big{)}$
$\displaystyle=-2\big{(}\sum_{i=1}^{m}1-0+\sum_{i=1}^{m}(f_{1}^{i}-3)\big{)},$

where $f_{1}^{i}$ denotes the number of 1-dimensional neighboring faces of the
$i$-th vertex of $\mathcal{P}$. We can also interpret $f_{1}^{i}$ as the
number of 1-dimensional faces of $\mathcal{M}_{i}$, where $\mathcal{M}_{i}$ is
the underlying 2-dimensional polygon of the link of the $i$-th vertex. Note
that for a $\mathbb{Q}$-isolated non-singular point $x_{j}$, we have
$f_{1}^{j}=3$. This implies
$\sum_{i=1}^{m}(f_{1}^{i}-3)=\sum_{i=1}^{f_{3}}(f_{1}^{i}-3)$. Counting the
1-dimensional faces of $\mathcal{M}_{i}$ for each vertex $x_{i}$ equals
counting the neighboring 1-dimensional faces of each vertex in the underlying
polytope $\mathcal{P}$. Since each 1-dimensional face of the polytype has
precisely two vertices in its topological boundary, we have
$\sum_{i=1}^{f_{3}}f_{1}^{i}=2f_{2}$. Thus, we get

$\displaystyle\sum_{i=1}^{f_{3}}(f_{1}^{i}-3)=2f_{2}-3f_{3}.$

Then, we have
$\big{(}-2m-4f_{1}+2f_{2}+8\big{)}-\big{(}2f_{1}-4\big{)}=-2\big{(}m+2f_{2}-3f_{3}\big{)}.$
Finally, we get

$\displaystyle f_{3}-f_{2}+f_{1}=2,$

which is the well-known Euler formula for 3-dimensional polytopes.

######  Corollary 71.

We assume that all vertices in $\mathcal{P}$ are $\mathbb{Q}$-isolated
singularities. Considering the above remark, we arrive at

$\displaystyle\operatorname{rk}(\widetilde{H}_{6}(I^{\bar{n}}X))$
$\displaystyle=0$
$\displaystyle\operatorname{rk}(\widetilde{H}_{5}(I^{\bar{n}}X))$
$\displaystyle=f_{2}-f_{1}+1$
$\displaystyle\operatorname{rk}(\widetilde{H}_{4}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\widetilde{H}_{3}(I^{\bar{n}}X))$
$\displaystyle=2(3f_{1}-f_{2}-b-6)$
$\displaystyle\operatorname{rk}(\widetilde{H}_{2}(I^{\bar{n}}X))$
$\displaystyle=f_{1}-3-b$
$\displaystyle\operatorname{rk}(\widetilde{H}_{1}(I^{\bar{n}}X))$
$\displaystyle=f_{2}-f_{1}+1$
$\displaystyle\operatorname{rk}(\widetilde{H}_{0}(I^{\bar{n}}X))$
$\displaystyle=0.$

######  Remark 72.

In Theorem 68, we have

$\displaystyle\chi(\tilde{H}_{\ast}(I^{\bar{n}}X))=2(-2f_{1}+f_{2}-m+4),$

which is even. This observation is consistent with the following more general
principle. Let $X$ be a $(4n+2)$-dimensional $\mathbb{Q}$-pseudomanifold with
$\mathbb{Q}$-isolated singularities. By using the duality on the intersection
homology groups, we find

$\displaystyle\chi(I^{\bar{n}}H_{\ast}(X))=\sum_{i=1}^{4n+2}(-1)^{i}\operatorname{rk}(I^{\bar{n}}H_{i}(X))=2\sum_{i=1}^{2n}(-1)^{i}\operatorname{rk}(I^{\bar{n}}H_{i}(X))-\operatorname{rk}(I^{\bar{n}}H_{2n+1}(X)).$

The intersection form on $I^{\bar{n}}H_{2n+1}(X)$ is skew-symmetric and non-
degenerate. Hence, there is a basis with respect to which the intersection
form has the matrix representation

$\displaystyle\left(\begin{matrix}\begin{matrix}0&1\\\ -1&0\\\
\end{matrix}&0&\cdots&0\\\ 0&\begin{matrix}0&1\\\ -1&0\\\
\end{matrix}&&\vdots\\\ \vdots&&\ddots&0\\\ 0&\cdots&0&\begin{matrix}0&1\\\
-1&0\\\ \end{matrix}\end{matrix}\right).$

Consequently, we have $\operatorname{rk}(I^{\bar{n}}H_{2n+1}(X))\equiv 0(2)$
and thus $\chi(I^{\bar{n}}H_{\ast}(X))\equiv 0(2).$ For an intersection space
$I^{\bar{n}}X$ of $X$, it follows from Lefschetz duality that
$\operatorname{rk}(H_{2n+1}(I^{\bar{n}}X))=-\operatorname{rk}(I^{\bar{n}}H_{2n+1}(X))+2\operatorname{rk}(H_{2n+1}(\mathcal{M}))$.
Therefore,

$\displaystyle\chi(H_{\ast}(I^{\bar{n}}X))\equiv 0(2).$

######  Remark 73.

In the context of Corollary 71, the homology groups of $I^{\bar{n}}X$
determine the ordinary homology groups of $X_{\mathcal{P}}$ up to and
including the middle degree. In contrast, the intersection homology groups do
not determine the ordinary homology below the middle degree. In this sense,
the homology of intersection spaces of toric varieties sees more structure
than the intersection homology.

######  Remark 74.

We can nicely fit all of the above data into a Mayer-Vietoris sequence. Let
$I^{\bar{n}}X$ be the associated generalized intersection space of a
6-dimensional toric variety $X$. Then we take $\mathcal{U}\simeq\mathcal{M}$,
where $\mathcal{M}$ is the $\mathbb{Q}$-manifold obtained by cutting out the
$\mathbb{Q}$-isolated singularities of $X$. We also take
$\mathcal{V}\cong\mathcal{C}(\displaystyle{\sqcup}\mathcal{L}_{<3})$. The
intersection $\cap:=\mathcal{U}\cap\mathcal{V}$ is homotopy equivalent to
$\mathcal{L}_{<3}$. Thus, we have

$\displaystyle 0$
$\displaystyle\xrightarrow{\phantom{-\cong-}}\overbrace{H_{6}(\cap)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{6}(\mathcal{U})}^{=0}\xrightarrow{\phantom{-}\cong\phantom{-}}\overbrace{H_{6}(I^{\bar{n}}X)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{5}(\cap)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{5}(\mathcal{U})}^{\cong\mathbb{Q}^{m-1}}\xrightarrow{\phantom{-}\cong\phantom{-}}\overbrace{H_{5}(I^{\bar{n}}X)}^{\cong\mathbb{Q}^{m-1}}$
$\displaystyle\xrightarrow{\phantom{-\cong-}}\overbrace{H_{4}(\cap)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{4}(\mathcal{U})}^{\cong\mathbb{Q}^{f_{1}-3-b}}\xrightarrow{\phantom{-}\cong\phantom{-}}\overbrace{H_{4}(I^{\bar{n}}X)}^{\cong\mathbb{Q}^{f_{1}-3-b}}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{3}(\cap)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{3}(\mathcal{U})}^{\cong\mathbb{Q}^{3f_{1}-f_{2}-b-6}}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{3}(I^{\bar{n}}X)}^{\cong\mathbb{Q}^{2(3f_{1}-f_{2}-b-6)}}$
$\displaystyle\xrightarrow{\phantom{-\cong-}}\overbrace{H_{2}(\cap)}^{\cong\mathbb{Q}^{r}}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{2}(\mathcal{U})}^{\cong\mathbb{Q}^{f_{1}-3}}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{2}(I^{\bar{n}}X)}^{\cong\mathbb{Q}^{f_{1}-3-b}}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{1}(\cap)}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{1}(\mathcal{U})}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{H_{1}(I^{\bar{n}}X)}^{\cong\mathbb{Q}^{m-1}}$
$\displaystyle\xrightarrow{\phantom{-}\cong\phantom{-}}\overbrace{\widetilde{H}_{0}(\cap)}^{\cong\mathbb{Q}^{m-1}}\xrightarrow{\phantom{-\cong-}}\overbrace{\widetilde{H}_{0}(\mathcal{U})}^{=0}\oplus\overbrace{\widetilde{H}_{0}(\mathcal{V})}^{=0}\xrightarrow{\phantom{-\cong-}}\overbrace{\widetilde{H}_{0}(I^{\bar{n}}X)}^{=0}\xrightarrow{\phantom{-\cong-}}0,$

where $r:=\sum_{i=1}^{m}(f_{1}^{i}-3)=2f_{2}-3f_{3}$.

######  Corollary 75.

For each 6-dimensional toric variety $X$ with $H_{3}(X)=0$, the equality $b=r$
holds and $b$ is combinatorially invariant.

###### Proof.

The assumption implies that $H_{3}(\mathcal{U})=0$ and hence
$H_{3}(I^{\bar{n}}X)=0$. By using the Mayer-Vietoris sequence of Remark 74, we
get $b=r=2f_{2}-3f_{3}.$ Reformulating the assumption yields
$b=3f_{1}-f_{2}-6.$ Setting both expressions equal to each other will result
in the Euler formula. ∎

##  9\. Conclusion

We summarize the results obtained in this work. We started by considering
toric varieties as topological pseudomanifolds. We introduced the construction
of a link for an arbitrary stratum in a given toric variety. We also endowed
links of strata with real co-dimensions 4 and 6 with CW structures and
computed their homology groups. We showed that the link of each connected
component of the real 4-co-dimensional stratum is always a rational homology
3-sphere. Thus, for an arbitrary $2n$-dimensional toric variety $X$, we get a
$\mathbb{Q}$-stratification whose depth 1 stratum is of real co-dimension 6.
We generalized the theory of intersection spaces to rational homology
stratified pseudomanifolds with rationally isolated singularities. Our main
objects of study are 6-dimensional toric varieties. For these, we compare the
Betti numbers of the associated generalized intersection spaces $IX$ and
intersection homology in the following table.

$b_{\ast}$ Theory | $H_{\ast}(IX)$ | $IH_{\ast}(X)$
---|---|---
$b_{6}$ | $0$ | $1$
$b_{5}$ | $m-1$ | $0$
$b_{4}$ | $f_{1}-3-b$ | $f_{1}-3$
$b_{3}$ | $2(3f_{1}-f_{2}-b-6)$ | $0$
$b_{2}$ | $f_{1}-3-b$ | $f_{1}-3$
$b_{1}$ | $m-1$ | $0$
$b_{0}$ | $1$ | $1$
Table 1. The Betti numbers associated to $HI_{\ast}$ and $IH_{\ast}$.

The left hand column of the table shows that the $b_{\ast}^{IX}$ are not
combinatorially invariant. Indeed, we can determine the ordinary Betti numbers
of $X$ up to and including the middle degree by using $b_{\ast}^{IX}$.
However, the right hand column, as already observed by Fieseler in [18], shows
that the associated Betti numbers of intersection homology neither fix the
combinatorial data of the fan nor the ordinary homology below the middle
degree.

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they are Cartier divisors on $X$. If we drop the coefficients of $D_{1}$ and
$D_{2}$ in $D$ to some $c\in(0,\frac{1}{2})$ and denote the new boundary
divisor by $D^{\prime}$, then $K_{X_{1}}+D^{\prime}|_{X_{1}}+\Gamma_{1}$
remains unchanged, while $K_{X_{2}}+D^{\prime}|_{X_{2}}+\Gamma_{2}$ has
negative intersection with $E$ (we have
$(K_{X_{2}}+D|_{X_{2}}+\Gamma_{2})\cdot E=D_{1}\cdot E=D_{2}\cdot E=1$, hence
$(K_{X_{2}}+D^{\prime}|_{X_{2}}+\Gamma_{2})\cdot E=2c-1$), thus the MMP
$\pi\colon X_{2}\to X^{\prime}_{2}$ on $X_{2}$ contracts the curve $E$ and we
get $X^{\prime}_{2}=\mathbb{P}^{1}\times\mathbb{P}^{1}$. However, the pair
$(X^{\prime}_{2},\pi_{*}(D^{\prime}|_{X_{2}}))$ no longer glues with
$(X_{1},D^{\prime}|_{X_{1}})$, since
$(\pi_{*}(D^{\prime}|_{X_{2}})\cdot\pi_{*}\Gamma_{2})=2+2c<3=(D^{\prime}|_{X_{1}}\cdot\Gamma_{1})$,
i.e. the differents along the conductors do not match.

###  7.2. Reduction morphism does not factor

By modifying an example in [ABIP23]*Section 8.3, we show that in Theorem 6.2
the reduction morphism $B\to\mathcal{N}_{\mathbf{b}}$ does not factor through
$\mathcal{N}_{\mathbf{a}}$ if we do not assume that the generic fiber of
$(X,\mathbf{a}D)\to B$ is stable (in other words, if $K_{X/B}+\mathbf{a}D$ is
only relatively big but not ample along the generic fiber). Our examples have
klt generic fibers, so they also illustrate that the ampleness assumption in
the main results of [ABIP23] is necessary (more specifically, the only place
in _loc. cit._ that relies on this assumption is their Theorem 7.6).

######  Example 7.2.

Consider a line $L_{0}$ in $\mathbb{P}^{2}$ and a smooth plane curve
$C_{0}\subseteq\mathbb{P}^{2}$ of degree $d\gg 0$ that has transversal
intersection with $L_{0}$. Let $p\in C_{0}\cap L_{0}$, let $\pi\colon
X\to\mathbb{P}^{2}$ be the blow up at $p$, let $C$ (resp. $L$) be the strict
transform of $C_{0}$ (resp. $L_{0}$), and let $E$ be the $\pi$-exceptional
curve. Choose some $a\in(\frac{1}{2},1)$. We have

$K_{X}+a(C+L)+(2a-1)E=\pi^{*}(K_{\mathbb{P}^{2}}+a(C_{0}+L_{0})).$

Thus if $b\geq 2a-1$, the ample model of $(X,a(C+L)+bE)$ is
$(\mathbb{P}^{2},a(C_{0}+L_{0}))$, while for $b\in(2a-1-\varepsilon,2a-1)$
(where $0<\varepsilon\ll 1$), the ample model of $(X,a(C+L)+bE)$ is itself.
Now consider the moduli space $U$ of such (marked) family $(X;C,L,E)$ as
$C_{0},L_{0}$ and $p\in C_{0}\cap L_{0}$ vary. Note that for a fixed choice of
$C_{0}$ and $L_{0}$ there are exactly $d$ possible ways to choose $p$.
Moreover, since $E$ is the only $(-1)$-curve on $X$, any isomorphism
$(X,C+L)\cong(X^{\prime},C^{\prime}+L^{\prime})$ necessarily comes from an
isomorphism
$(\mathbb{P}^{2},C_{0}+L_{0})\cong(\mathbb{P}^{2},C^{\prime}_{0}+L^{\prime}_{0})$
sending $p$ to $p^{\prime}$. For general choice of $C_{0}$ and $L_{0}$, the
pair $(\mathbb{P}^{2},C_{0}+L_{0})$ does not have any non-trivial
automorphisms, hence different choices of the intersection point $p$ give non-
isomorphic $(X,C+L)$. Putting the above discussions together, we see that the
reduction morphism $U\to\mathcal{N}_{a,b}$ is birational when
$2a-1-\varepsilon<b<2a-1$, and is generically finite of degree $d>1$ when
$b\geq 2a-1$. It follows that the natural (wall-crossing) map
$\mathcal{N}_{a,2a-1-\varepsilon}\to\mathcal{N}_{a,2a-1}$ is also generically
finite of degree $d>1$ and the reduction morphism
$U\to\mathcal{N}_{a,2a-1-\varepsilon}$ does not factor through
$\mathcal{N}_{a,2a-1}$.

###  7.3. Walls for $\mathcal{K}_{\mathbf{a}}$ when coefficients vary

In general, the moduli

$\mathcal{K}_{\mathbf{a},\leq
v}:=\cup_{v_{0}\in[0,v]}\mathcal{K}_{\mathbf{a},v_{0}}$

of stable pairs with coefficient vector $\mathbf{a}$ and volume at most $v$
may have infinitely many connected components. A priori, we could still
investigate whether there is some finite wall-and-chamber structure in this
setting. We declare that a coefficient vector $\mathbf{a}$ sits on a wall if
there exist a pair $(X,D)$ and some $\mathbf{a}_{i}$ $(i=1,2,\dots)$
converging to $\mathbf{a}$ such that $(X,\mathbf{a}_{i}D)$ are stable pairs of
volume at most $v$ for all $i$ but $(X,\mathbf{a}D)$ is not. The next series
of examples show that such walls are often badly behaved.

######  Example 7.3.

Let $C_{d}\subseteq\mathbb{P}^{2}$ be a smooth plane curve of degree $d\geq
4$, then $(\mathbb{P}^{2},aC_{d})$ is stable if and only if $a>\frac{3}{d}$.
Thus we hit a wall at $a=\frac{3}{d}$. As $d$ varies, we get infinitely many
walls near $0$.

######  Example 7.4.

Fix an integer $d\geq 2$. For each $m\in\mathbb{N}^{*}$, let $C_{m}$ be a
general member of the base point free linear system $|\mathcal{O}(dm)|$ on
$\mathbb{P}(1,1,m)$. Then $(\mathbb{P}(1,1,m),aC_{m})$ is stable if and only
if $a>\frac{1}{d}+\frac{2}{dm}$. Thus there are walls at
$\frac{1}{d}+\frac{2}{dm}$ for each $m$, and we get infinitely many walls near
$\frac{1}{d}$.

In these two examples, the walls accumulate from above. One may ask whether
the walls also accumulate from below (equivalently, whether the walls satisfy
the ascending chain condition, or ACC). The next example shows that this can
also happen.

######  Example 7.5.

If $X$ is a klt surface with ample canonical divisor, and there exists a
divisor $E$ over $X$ such that $a(E,X,0)<0$, then there is a wall at
$-a(E,X,0)$. This is because there exists a birational morphism $\pi\colon
Y\to X$ that extracts $E$ as the unique exceptional divisor, and
$K_{Y}+(-a(E,X,0)-\varepsilon)E=\pi^{*}K_{X}-\varepsilon E$ is ample when
$0<\varepsilon\ll 1$. By [Kol08-topo-BMY-ineq]*43, there is an unbounded set
of klt surfaces with ample canonical divisor and volume $\leq 2$. Moreover,
the minimal log discrepancies of these surfaces can be arbitrarily close to
zero (this follows from the general boundedness result in [HMX14-ACC], but can
also be deduced through a direct computation). By our previous discussion,
this implies that there are walls accumulating near $1$ (the accumulation is
necessarily from below).

## References