Katzarkov-Kontsevich-Pantev conjecture for minimal adjoint orbits

TL;DR

This paper proves that Landau-Ginzburg models for minimal semisimple adjoint orbits fulfill the Katzarkov-Kontsevich-Pantev conjecture regarding new Hodge theoretical invariants.

Contribution

It establishes the conjecture for a specific class of LG models related to minimal semisimple adjoint orbits, advancing understanding of their Hodge invariants.

Findings

LG models for minimal semisimple adjoint orbits satisfy the conjecture

Verification of Hodge theoretical invariants in this context

Progress in understanding the conjecture's scope

Abstract

We prove that LG models for minimal semisimple adjoint orbits satisfy the Katzarkov-Kontsevich-Pantev conjecture about new Hodge theoretical invariants.

Equations

$$f^{p,q}(Y,w),h^{p,q}(Y,w),i^{p,q}(Y,w)$$

$$h^m(Y,Y_b\text{mathchar}24635\mathbb{C})=\sum _{p+q=m}{i}^{p,q}(Y,w)=\sum _{p+q=m}{h}^{p,q}(Y,w)=\sum _{p+q=m}{f}^{p,q}(Y,w)$$

$$h^{p,q}(Y,w)=f^{p,q}(Y,w)=i^{p,q}(Y,w).$$

$$\mathcal{O}(H_0):=\{\mathrm{Ad}(g)H_0,g\in G\}\text{.}$$

$$\begin{array}[]{rll}f_{H}\colon\mathcal{O}(H_{0})&\rightarrow&\mathbb{C}\cr X&\mapsto&\langle H,X\rangle\end{array}$$

$$\frac{d{z}_1}{z_1},\dots ,\frac{d{z}_k}{z_k},d{z}_{k+1},\dots ,d{z}_n$$

$${\Omega }_Z^a(log{D}_Z,f)=\{\alpha \in {\Omega }_Z^a(log{D}_Z)|df\wedge \alpha \in {\Omega }_Z^{a+1}(log{D}_Z)\},$$

$$f^{p,q}(Y,w)=\dim H^p(Z,{\Omega }_Z^q(log{D}_Z,f)).$$

$$0\subset W_0(N,m)\subset W_1(N,m)\subset \dots \subset W_{2m-1}(N,m)\subset W_{2m}(N,m)=V$$

$$T:H^{\bullet }(Y_b)\to H^{\bullet }(Y_b)$$

$$T:H^{\bullet }(Y,Y_b)\to H^{\bullet }(Y,Y_b).$$

$$\dots \to H^m(Y,Y_b)\to H^m(Y)\to H^m(Y_b)\to H^{m+1}(Y,Y_b)\to \dots$$

$$h^{p,n-q}(Y,w)=\dim g{r}_{2(n-p)}^{W,n-a}{H}^{n+p-q}(Y,Y_b)\text{if a=p−q≥0},$$

$$h^{p,n-q}(Y,w)=\dim g{r}_{2(n-q)}^{W,n+a}{H}^{n+p-q}(Y,Y_b)\text{if a=p−q<0}.$$

$$i^{p,q}(Y,w)=\sum _{\lambda \in \mathbb{C}}\sum _k{i}^{p,q+k}{\mathbb{H}}^{p+q-1}(Y_{\lambda },{\varphi }_{w-\lambda }{\mathbb{C}}_Y).$$

$$i^{p,q}(Y,w)=\sum _{\lambda \in \mathbb{C}}\sum _k{i}^{p,q+k}{\mathbb{H}}^{p+q-1}(Y_{\lambda },{\varphi }_{w-\lambda }\mathbf{R}{j}_{\ast }{\mathbb{C}}_Y).$$

$${\mathfrak{n}}_{H_{\Theta }}^{+}=\sum _{\alpha \left(H_{\Theta }\right)>0}{\mathfrak{g}}_{\alpha },{\mathfrak{n}}_{H_{\Theta }}^{-}=\sum _{\alpha \left(H_{\Theta }\right)<0}{\mathfrak{g}}_{\alpha }$$

$${\mathfrak{p}}_{\Theta }=\sum _{\lambda \ge 0}{\mathfrak{g}}_{\lambda }={\mathfrak{z}}_{\Theta }\oplus {\mathfrak{n}}_{H_{\Theta }}^{+}$$

$${\Theta }^{\ast }:=-w_0\left(\Theta \right)\subset \Sigma$$

$${\mathfrak{q}}_{{\Theta }^{\ast }}=\sum _{\lambda \le 0}{\mathfrak{g}}_{\lambda }={\mathfrak{z}}_{\Theta }\oplus {\mathfrak{n}}_{H_{\Theta }}^{-},$$

$$\left(\begin{array}[]{cc}\ast_{1\times 1}&\ast\\ 0&\ast_{n\times n}\end{array}\right).$$

$$\left(\begin{array}[]{cc}\ast_{1\times 1}&0\\ 0&\ast_{n\times n}\end{array}\right)$$

$$R_H:{\mathbb{P}}^n\times Gr(n,n+1)\to {\mathbb{P}}^1,$$

$$R_H([v],[\epsilon ])=\frac{\mathrm{tr}((v\otimes \epsilon )\rho (H))}{\mathrm{tr}(v\otimes \epsilon )}=\frac{{\sum }_{i=1}^{n+1}{\lambda }_i{a}_{i1}(\mathrm{adj}g{)}_{1i}}{{\sum }_{i=1}^{n+1}{a}_{i1}(\mathrm{adj}g{)}_{1i}}.$$

$$R_H([x_1,\dots ,x_{n+1}],[y_1,\dots ,y_{n+1}])=\left[\sum _{i=1}^{n+1}{\lambda }_i{x}_i{y}_i,\sum _{i=1}^{n+1}{x}_i{y}_i\right].$$

$$\mathcal{I}:=\mathbb{F}(1,n)\cap \{{\lambda }_1{x}_1{y}_1+\cdots +{\lambda }_{n+1}{x}_{n+1}{y}_{n+1}=0\}.$$

$$J=\left(\begin{array}[]{cccccc}\lambda_{1}y_{1}&\dots&\lambda_{n+1}y_{n+1}&\lambda_{1}x_{1}&\dots&\lambda_{n+1}x_{n+1}\\ y_{1}&\dots&y_{n+1}&x_{1}&\dots&x_{n+1}\end{array}\right)$$

$$w:Z\subset {\mathbb{P}}^n\times {\mathbb{P}}^n\times {\mathbb{P}}^1$$

$$(x_1,\dots ,x_{n+1},y_1,\dots ,y_{n+1},t,s)$$

$$[t,s]=\left[\frac{t}{s},1\right]=\left[\frac{f_H}{g},1\right]=\left[f_H,g\right]$$

$$U_{12}=\{x_1\ne 0,y_2\ne 0\}\subset {\mathbb{P}}^n\times {\mathbb{P}}^n$$

Full text

MSC]58A14, 14J33; 17B08, 32M10

1 KKP conjecture and our result

The Katzarkov–Kontsevich–Pantev (KKP) conjecture is a numerical prediction expected to follow from the Homological Mirror Symmetry conjecture of M. Kontsevich, a master conjecture predicting a wide range of categorical equivalences which so far have been established in only a few cases.

In KKP three types of new Hodge theoretical invariants were defined:

[TABLE]

for tamely compactifiable Landau–Ginzburg (LG) models $w\colon Y\to\mathbb{C}$; we recall their definitions in section 2. Katzarkov–Kontsevich–Pantev proved in KKP that these numbers satisfy the identities

[TABLE]

where $Y_{b}$ is a smooth fibre of $w$, and conjectured the equality of the 3 invariants.

KKP Conjecture**.**

KKP ; LP * Assume that $(Y,w)$ is tamely compactified Landau–Ginzburg. Then for every $p,q$ there are equalities*

[TABLE]

For $Y$ a specific rational surface with a map $w\colon Y\to\mathbb{C}$ such that the generic fibre is an elliptic curve, Lunts and Przyjalkowski LP proved the equality $f^{p,q}(Y,w)=h^{p,q}(Y,w)$ and gave an example where $i^{p,q}(Y,w)\neq h^{p,q}(Y,w)$. Shamoto Sh gave sufficient conditions for a tamely compactifiable LG models to satisfy $f^{p,q}(Y,w)=h^{p,q}(Y,w)$. There remains open the question of what LG models satisfy this KKP conjecture. The goal of this paper is to provide a family of examples of LG models coming from Lie theory that do satisfy the KKP conjecture in the sense that the 3 invariants coincide on weakly tame compactifications.

We note that Katzarkov–Kontsevich–Pantev posed a second conjecture in KKP , namely they predicted that if $(Y,w)$ is the LG model mirror to $X$, then $f^{p,q}(Y,w)=h^{p,n-q}(X)$ where $n=\dim X=\dim Y$. This conjecture was previously discussed by Gross–Katzarkov–Ruddat GKR in the context of mirror symmetry for varieties of general type. Harder Ha described how to compute the Hodge numbers $f^{p,q}(X)$ via classical mixed Hodge theory, proving $h^{p,q}(X)=f^{3-q,p}(Y,w)$ for a crepant resolution of a Gorenstein Fano threefold $X$ and its LG mirror $(Y,w)$. Cheltsov and Przyjalkowski CP showed that this second conjecture holds for the Hodge numbers of Fano threefolds and their mirror LG models.

In this work we consider the KKP conjecture about the equality of the 3 invariants defined by Katzarkov–Kontsevich–Pantev, providing examples where the 3 numbers coincide for a family of Landau–Ginzburg models $(Y,w)$ admitting weakly tame compactifications $((Z,f),D_{Z})$. We call our compactifications weakly tame because, even though the the divisor $D_{Z}$ is the sum of the fibre at infinity $D^{v}$ and the horizontal divisor at infinity $D^{h}$, only multiples of these divisors give the anti-canonical class, failing the strict requirement of multiplicity 1 in Definition 2.2. The appearance of these multiplicities is unavoidable here, because our compactifications $Z$ are Fano.

We consider the symplectic Lefschetz fibrations constructed in GGSM1 , defined over semisimple adjoint orbits via the height function from Lie theory. The novelties of the cases that we present are: these adjoint orbits are affine, non-toric, Calabi–Yau varieties, which occur in arbitrarily high dimensions. Indeed, the minimal semisimple orbit of $\mathfrak{sl}(n+1)$ has dimension $2n$. Thus, we present the first explicit family of arbitrarily large Calabi–Yau varieties where the KKP conjecture holds true in the sense that all 3 invariants coincide.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Lie group $G$, and $\mathfrak{h}$ the Cartan subalgebra. Consider the adjoint orbit $\mathcal{O}(H_{0})$ of an element $H_{0}\in\mathfrak{h}$, that is,

[TABLE]

Let $H\in\mathfrak{h}_{\mathbb{R}}$ be a regular element, and $\langle\cdot,\cdot\rangle$ the Cartan–Killing form. Then (GGSM1, , Thm. 2.2) shows that the height function

[TABLE]

gives the orbit the structure of a symplectic Lefschetz fibration.

Here we consider the case of $G=\mathrm{SL}(n+1,\mathbb{C})$ and focus on the adjoint orbit passing through $H_{0}=\operatorname{Diag}(n,-1,\ldots,-1)$. The diffeomorphism type is then $\mathcal{O}(H_{0})\simeq T^{*}\mathbb{P}^{n}$. Among all choices of elements $H_{0}\in\mathfrak{h}\subset{\mathfrak{s}l}(n+1,\mathbb{C})$, the choice $H_{0}=\operatorname{Diag}(n,-1,\ldots,-1)$ produces the homogeneous manifold of smallest dimension, for this reason we set the following terminology:

Definition 1.1**.**

Let $\mathcal{O}_{n}$ denote the adjoint orbit of $H_{0}=\operatorname{Diag}(n,-1,\ldots,-1)$ in ${\mathfrak{s}l}(n+1,\mathbb{C})$, we call it the minimal orbit.

For every $n\geq 2$ and for every choice of regular element $H\in\mathfrak{h}$, we prove:

Theorem**.**

6.1* The LG model $(\mathcal{O}_{n},f_{H})$ admits a weakly tame compactification and satisfies the KKP conjecture.*

Since the 3 Hodge theoretical invariants $f^{p,q}(Y,w),h^{p,q}(Y,w),i^{p,q}(Y,w),$ of our LG models coincide, we may depict diamonds containing their common values, that we name KKP diamonds. For our examples, these turn out to have a unique nonzero entry. Thus, our Lie theoretical constructions give rise to LG models where computations of Hodge theoretical invariants turn out rather simple, which is arguably the best feature of our choice of LG models. It is at this moment unknown how KKP invariants behave for LG models on more general adjoint orbits.

2 Landau–Ginzburg Hodge numbers

This section is just a summary of parts of the nicely written text of LP .

Definition 2.1**.**

A Landau–Ginzburg model* is a pair $(Y,w)$, where*

$Y$* is a smooth complex quasi-projective variety with trivial canonical bundle $K_{Y}$;* 2. 2.

$w\colon Y\to\mathbb{C}$* is a morphism with a compact critical locus $crit(w)\subset Y$.*

Definition 2.2**.**

(LP*, *, Def. 3)** A tame compactified Landau–Ginzburg model is the data $((Z,f),D_{Z})$, where

$Z$* is a smooth projective variety and $f\colon Z\to\mathbb{P}^{1}$ is a flat morphism.* 2. 2.

$D_{Z}=(\cup_{i}D^{h}_{i})\cup(\cup_{j}D_{j}^{v})$* is a reduced normal crossings divisor such that*

• (i)

$D^{v}=\cup_{j}D^{v}_{j}$* is a scheme theoretical pole divisor of $f$, i.e. $f^{-1}(\infty)=D^{v}$. In particular $ord_{D^{v}_{j}}(f)=-1$ for all $j$;*

• (ii)

each component $D_{i}^{h}$ of $D^{h}=\cup_{i}D^{h}_{i}$ is smooth and horizontal for $f$, i.e. $f|_{D^{h}_{i}}$ is a flat morphism;

• (iii)

The critical locus $crit(f)\subset Z$ does not intersect $D^{h}$. 3. 3.

$D_{Z}$* is an anticanonical divisor on $Z$.*

One says that $((Z,f),D_{Z})$ is a compactification of the Landau–Ginzburg model $(Y,w)$ if in addition the following holds: 4. 4.

$Y=Z\setminus D_{Z}$, $f|_{Y}=w$.

Assuming that we are given a Landau–Ginzburg model $(Y,w)$ with a tame compactification $((Z,f),D_{Z})$ as above, in KKP the authors define the numbers $i^{p,q}(Y,w)$, $h^{p,q}(Y,w)$, $f^{p,q}(Y,w)$. We recall their definitions, and we will use them in our more general situation of weakly tame compatification. We denote by $n=\dim Y=\dim Z$ the (complex) dimension of $Y$ and $Z$. Choose a point $b\in\mathbb{C}$ which is near $\infty$ and such that the fibre $Y_{b}=w^{-1}(b)\subset Y$ is smooth.

2.1 $f^{p,q}(Y,w)$

Recall the logarithmic de Rham complex $\Omega^{\bullet}_{Z}(log\,D_{Z})$. Namely, $\Omega^{s}_{Z}(log\,D_{Z})=\wedge^{s}\Omega^{1}_{Z}(log\,D_{Z})$ and $\Omega^{1}_{Z}(log\,D_{Z})$ is a locally free $\mathcal{O}_{Z}$-module generated locally by

[TABLE]

if $z_{1}\cdot\ldots\cdot z_{k}=0$ is a local equation of the divisor $D_{Z}$. Hence in particular $\Omega^{0}_{Z}(log\,D_{Z})=\mathcal{O}_{Z}$.

The numbers $f^{p,q}(Y,w)$ are defined using the subcomplex $\Omega^{\bullet}_{Z}(log\,D_{Z},f)\subset\Omega^{\bullet}_{Z}(log\,D_{Z})$ of $f$-adapted forms, which we recall next.

Definition 2.3**.**

For each $a\geq 0$ define a sheaf $\Omega^{a}_{Z}(log\,D_{Z},f)$ of $f$-adapted logarithmic forms as a subsheaf of $\Omega^{a}_{Z}(log\,D_{Z})$ consisting of forms which stay logarithmic after multiplication by $df$. Thus

[TABLE]

where one considers $f$ as a meromorphic function on $Z$ and $df$ is viewed as a meromorphic 1-form.

Definition 2.4**.**

(KKP*, *, Def. 3.1)** The Landau–Ginzburg Hodge numbers $f^{p,q}(Y,w)$ are defined as follows:

[TABLE]

2.2 $h^{p,q}(Y,w)$

Let $N\colon V\to V$ be a nilpotent operator on a finite dimensional vector space $V$ such that $N^{m+1}=0$. Such data defines a canonical weight filtration centered at $m$, $W=W_{\bullet}(N,m)$ of $V$

[TABLE]

with the properties

$N(W_{i})\subset W_{i-2}$, 2. 2.

the map $N^{l}\colon gr^{W,m}_{m+l}V\to gr^{W,m}_{m-l}V$ is an isomorphism for all $l\geq 0$.

Let $S^{1}\simeq C\subset\mathbb{P}^{1}$ be a loop passing through the point $b$ that goes once around $\infty$ in the counter clockwise direction in such a way that there are no singular points of $w$ on or inside $C$. It gives the monodromy transformation

[TABLE]

with corresponding monodromy transformation on the relative cohomology

[TABLE]

in such a way that the sequence

[TABLE]

is $T$-equivariant, where $T$ acts trivially on $\operatorname{\mathrm{H}}^{\bullet}(Y)$.

Since we assume that the infinite fibre $f^{-1}(\infty)\subset Z$ is a reduced divisor with normal crossings, by Griffiths–Landman–Grothendieck Theorem (see Ka ) the operator $T\colon\operatorname{\mathrm{H}}^{m}(Y_{b})\to\operatorname{\mathrm{H}}^{m}(Y_{b})$ is unipotent and $\left(T-\mathrm{id}\right)^{m+1}=0$. It follows that the transformation (3) is also unipotent.

Definition 2.5**.**

(KKP*, *, Def. 3.2)** $h^{p,q}(Y,w)=\dim gr_{p}^{W,p+q}\operatorname{\mathrm{H}}^{p+q}(Y,Y_{b}).$

Remark 2.1**.**

*Definition  2.5 differs from (LP, , Def. 8) by the indices of the grading. Denoting by $N$ the logarithm of the transformation (3), which is assumed to be a nilpotent operator on $\operatorname{\mathrm{H}}^{\bullet}(Y,Y_{b})$ with $N^{m+1}=0$, LP call the Landau–Ginzburg model $(Y,w)$ of Fano type if the operator $N$ on the relative cohomology $\operatorname{\mathrm{H}}^{n+a}(Y,Y_{b})$ has the properties: $N^{n-|a|}\neq 0,\quad\text{and}\quad N^{n-|a|+1}=0,$ and define: *

[TABLE]

[TABLE]

We shall not worry about details of the grading here; it is trivial in our examples, since we have no critical points on the fibre at infinity. Such a possibility is explicitly mentioned in (KKP, , Def. 3.2(iii)).

2.3 $i^{p,q}(Y,w)$

For each $\lambda\in\mathbb{C}$ one has the corresponding sheaf $\phi_{w-\lambda}\mathbb{C}_{Y}$ of vanishing cycles for the fibre $Y_{\lambda}$. The sheaf $\phi_{w-\lambda}\mathbb{C}_{Y}$ is supported on the fibre $Y_{\lambda}$ and is equal to zero if $\lambda$ is not a critical value of $w$. From the works of Schmid, Steenbrink, and Saito it is classically known that the constructible complex $\phi_{w-\lambda}\mathbb{C}_{Y}$ carries a structure of a mixed Hodge module and so its hypercohomology inherits a mixed Hodge structure. For a mixed Hodge module $S$ we will denote by $i^{p,q}S$ the $(p,q)$ Hodge numbers of the $p+q$ weight graded piece $gr^{W}_{p+q}S$.

Definition 2.6**.**

(KKP*, *, Def. 3.4)**

Assume that the horizontal divisor $D^{h}\subset Z$ is empty, i.e. assume that the map $w\colon Y\to\mathbb{C}$ is proper. Then the Landau–Ginzburg Hodge numbers $i^{p,q}(Y,w)$ are defined as follows:

[TABLE] 2. 2.

In the general case denote by $j\colon Y\hookrightarrow Z$ the open embedding and define similarly

[TABLE]

3 Lie theoretical compactification

Let $\mathfrak{g}$ be a noncompact semisimple Lie algebra (not necessarily complex) with group $G$. A compactification of $\mathcal{O}(H_{0})$ to a product of flags $F_{\Theta}\times F_{\Theta^{\ast}}$ is described in (GGSM2, , Sec. 3). We now describe the orbits of the diagonal action of $G$ in this product. For the case considered here, the one of minimal orbits, we will have $F_{\Theta}=\mathbb{P}^{n}\simeq Gr(n,n+1)=F_{\Theta^{\ast}}$, see example 3.1.

Let $\Sigma$ be a system of simple roots of $\left(\mathfrak{g},\mathfrak{a}\right)$ (where $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}^{+}$ is an Iwasawa decomposition) and $\Theta\subset\Sigma$ a subset of roots, cf. example 3.2. Choose $H_{\Theta}$ defined by $\Theta=\{\alpha\in\Sigma:\alpha\left(H_{\Theta}\right)=0\}$, then set

[TABLE]

and take the parabolic subalgebra

[TABLE]

where $\lambda$ varies over the eigenvalues of $\mathrm{ad}\left(H_{\Theta}\right)$ and $\mathfrak{z}_{\Theta}$ is the centralizer of $H_{\Theta}$. The dual of $\Theta$ is by definition

[TABLE]

where $w_{0}$ is the main involution of the Weyl group $\mathcal{W}$. Set

[TABLE]

the parabolic subalgebra of $\mathfrak{g}$ conjugate to $\mathfrak{p}_{\Theta^{\ast}}$. In fact, $\mathfrak{q}_{\Theta^{\ast}}=\mathrm{Ad}\left(\overline{w}_{0}\right)\left(\mathfrak{p}_{\Theta^{\ast}}\right)$ where $\overline{w}_{0}$ is a representative of the main involution $w_{0}$ in $\mathrm{Norm}_{G}\left(\mathfrak{a}\right)$, and this is precisely the reason to consider here the dual flag $F_{\Theta^{\ast}}$.

The parabolic subgroups $P_{\Theta}$ and $P_{\Theta^{\ast}}$ are the normalizers of $\mathfrak{p}_{\Theta}$ and $\mathfrak{p}_{\Theta^{\ast}}$ respectively. Their flags are $F_{\Theta}=G/P_{\Theta}$ and $F_{\Theta^{\ast}}=G/P_{\Theta^{\ast}}$.

Denote by $b_{\Theta}=1\cdot P_{\Theta}$ the origin of $F_{\Theta}=G/P_{\Theta}$ and by $b_{\Theta^{\ast}}=1\cdot P_{\Theta^{\ast}}$ the origin of $F_{\Theta^{\ast}}=G/P_{\Theta^{\ast}}$. If $w\in\mathcal{W}$ then $wb_{\Theta}$ (respectively $wb_{\Theta^{\ast}}$) denotes the image of $b_{\Theta}$ by $w$ (actually, the image $\overline{w}b_{\Theta}$ of any representative $\overline{w}\in\mathrm{Norm}_{G}\left(\mathfrak{a}\right)$ of $w$).

The diagonal action is given by $g\left(x,y\right)=\left(gx,gy\right)$, $g\in G$, $x\in F_{\Theta}$ and $y\in F_{\Theta^{\ast}}$. The following statements describe the diagonal action and its properties.

Orbits of the diagonal action have the form $G\cdot\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ with $w\in\mathcal{W}$.

In fact, given $\left(x,y\right)\in F_{\Theta}\times F_{\Theta^{\ast}}$ there exists $g\in G$ such that $x=gb_{\Theta}$. Therefore, $\left(x,y\right)$ is in the orbit of $\left(b_{\Theta},z\right)$ for some $z\in F_{\Theta^{\ast}}$.

On the other hand $F_{\Theta^{\ast}}$ is the union of orbits $N^{+}\cdot wb_{\Theta^{\ast}}$, $w\in\mathcal{W}$. Thus, $z\in N^{+}\cdot wb_{\Theta^{\ast}}\subset P_{\Theta}\cdot wb_{\Theta^{\ast}}$ for some $w\in\mathcal{W}$. This shows that any $\left(x,y\right)$ belongs to an orbit $G\cdot\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ for some $w\in\mathcal{W}$.

Note that for different $w\in\mathcal{W}$ it might happen that the orbits $G\cdot\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ coincide. 2. 2.

Dualizing, it follows that orbits of the diagonal action are of the form $G\cdot\left(wb_{\Theta},b_{\Theta^{\ast}}\right)$ with $w\in\mathcal{W}$. The two descriptions are equivalent, since $\left(wb_{\Theta},b_{\Theta^{\ast}}\right)$ and $\left(b_{\Theta},w^{-1}b_{\Theta^{\ast}}\right)$ belong to the same orbit. 3. 3.

The two previous items show that the diagonal action has only a finite number of orbits.

The orbits $G\cdot\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ are in bijection with the orbits $P_{\Theta}\cdot wb_{\Theta^{\ast}}$, which are all the orbits of $P_{\Theta}$ in $F_{\Theta^{\ast}}$. They are also in bijection with the orbits $P_{\Theta^{\ast}}\cdot wb_{\Theta}$ which are the orbits of $P_{\Theta^{\ast}}$ in $F_{\Theta}$. In fact, if $\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ and $\left(b_{\Theta},w_{1}b_{\Theta^{\ast}}\right)$ belong to the same orbit, then there exists $g\in G$ such that $g\left(b_{\Theta},wb_{\Theta^{\ast}}\right)=\left(b_{\Theta},w_{1}b_{\Theta^{\ast}}\right)$. This means that $gb_{\Theta}=b_{\Theta}$, that is, $g\in P_{\Theta}$. Consequently $wb_{\Theta^{\ast}}=gw_{1}b_{\Theta^{\ast}}$ with $g\in P_{\Theta}$, so $wb_{\Theta^{\ast}}$ and $w_{1}b_{\Theta^{\ast}}$ belong to the same orbit of $P_{\Theta}$. Reciprocally, if $wb_{\Theta^{\ast}}$ and $w_{1}b_{\Theta^{\ast}}$ are in the same orbit of $P_{\Theta}$ then $\left(b_{\Theta},wb_{\Theta^{\ast}}\right)$ and $\left(b_{\Theta},w_{1}b_{\Theta^{\ast}}\right)$ belong to the the same orbit of $G$.

Example 3.1**.**

If $F_{\Theta}$ is a projective space (real or complex) $\mathbb{P}^{n}$, then $F_{\Theta^{\ast}}$ is the Grassmannian $\mathrm{Gr}\left(n,n+1\right)$. In the language of roots, $\Theta$ is the complement of $\{\alpha_{12}\}$ and $\Theta^{\ast}$ is the complement of $\{\alpha_{n,n+1}\}$.

Taking the basis $\{e_{1},\ldots,e_{n+1}\}$, in the canonical realization, $b_{\Theta}=\left[e_{1}\right]$ whereas $b_{\Theta^{\ast}}=\left[e_{1},\ldots,e_{n}\right]$. An element $w\in\mathcal{W}$ is a permutation, so that $wb_{\Theta}$ (respectively $wb_{\Theta^{\ast}}$) is obtained from $b_{\Theta}$ (respectively $b_{\Theta^{\ast}}$) by permutation of the indices. For instance, $w_{0}\left[e_{1}\right]=\left[e_{n+1}\right]$ and $w_{0}b_{\Theta^{\ast}}=\left[e_{2},\ldots,e_{n+1}\right]$ since $w_{0}$ inverts the order of the indices.

In this case $P_{\Theta}$ is the group of $(n+1)\times(n+1)$ matrices of type

[TABLE]

It has two orbits in $\mathrm{Gr}\left(n,n+1\right)$. They are:

The hyperplanes containing $\left[e_{1}\right]$, that is, the orbit of $b_{\Theta^{\ast}}=\left[e_{1},\ldots,e_{n}\right]$. In fact, such a hyperplane is determined by its intersection with $\left[e_{2},\ldots,e_{n+1}\right]$ and the subgroup of matrices

[TABLE]

is transitive already in the Grassmannian of subspaces of $\dim=n-1$ in $\left[e_{2},\ldots,e_{n+1}\right]$. 2. 2.

The hyperplanes transversal to $\left[e_{1}\right]$, that is, the orbit of $w_{0}b_{\Theta^{\ast}}=\left[e_{2},\ldots,e_{n+1}\right]$. In fact, if $V$ is a hyperplane transversal to $\left[e_{1}\right]$ then the matrix $g\in P_{\Theta}$ whose columns from $2$ to $n+1$ are the coordinates of a basis $\{v_{2},\ldots,v_{n+1}\}$ of $V$ satisfies $g\left[e_{2},\ldots,e_{n+1}\right]=V$.

In conclusion, the diagonal action of $\mathrm{SL}\left(n+1,\ast\right)$ in $\mathbb{P}^{n}\times\mathrm{Gr}\left(n,n+1\right)$ has two orbits, an open one and a closed one. The open orbit is isomorphic to the adjoint orbit $\mathrm{Ad}\left(G\right)H_{\Theta}$ with $H_{\Theta}=\operatorname{Diag}(n,-1,\ldots,-1)$ and is formed by the pairs of transversal elements in $\mathbb{P}^{n}\times\mathrm{Gr}\left(n,n+1\right)$. On the other hand, the closed orbit is isomorphic to the flag $F_{\Theta\cap\Theta^{\ast}}$. Since $\Theta\cap\Theta^{\ast}$ is the complement of $\{\alpha_{12},\alpha_{n,n+1}\}$ it follows that $F_{\Theta\cap\Theta^{\ast}}=F\left(1,n\right)$.

Example 3.2**.**

Consider the Grassmanians $F_{\Theta}=\mathrm{Gr}\left(k,n+1\right)$ and $F_{\Theta^{\ast}}=\mathrm{Gr}\left(n+1-k,n+1\right)$, real or complex (it is preferable to assume $k<(n+1)/2$). Here, in terms of roots, $\Theta$ is the complement of $\{\alpha_{12},\ldots,\alpha_{k-1,k}\}$ while $\Theta^{\ast}$ is the complement of $\{\alpha_{k,k+1},\ldots,\alpha_{n,n+1}\}$. In $\mathrm{Gr}\left(k,n+1\right)\times\mathrm{Gr}\left(n+1-k,n+1\right)$ there exist $k+1$ orbits of the diagonal action determined by the pairs $\left(V,W\right)\in\mathrm{Gr}\left(k,n+1\right)\times\mathrm{Gr}\left(n+1-k,n+1\right)$ such that $\dim\left(V\cap W\right)=0,1,\ldots,k$. The orbit determined by $\dim\left(V\cap W\right)=0$ is the open orbit (transversal pairs) whereas the closed orbit is given by $\dim\left(V\cap W\right)=k$, that is $V\subset W$. This closed orbit is the flag $F\left(k,n+1-k\right)$ given by $F_{\Theta}$ with $\Theta$ the complement of $\{\alpha_{k,k+1},\alpha_{n+1-k,n+1-k+1}\}$.

Example 3.3**.**

The choice $F_{\Theta}=F$ gives the maximal flag, which is self-dual (for any group). The orbits of $P_{\Theta}=P$ are the same as the orbit of $N^{+}$ which give the Bruhat decomposition. In this case the closed orbit is $F$ itself.

4 Partial extension of the potential

Now we wish to describe how to extend the potential of our LG-models to their compactifications. Let $H=\mathrm{Diag}(\lambda_{1},\dots,\lambda_{n+1})\in\mathfrak{sl}(n+1,\mathbb{C})$ be a regular element so that $\lambda_{i}$ are all distinct. We consider the Landau–Ginzburg models $(\mathcal{O}_{n},f_{H})$ where $\mathcal{O}_{n}$ is the adjoint orbit of $H_{0}=\operatorname{Diag}(n,-1,\ldots,-1)$ as in definition 1.1 and $f_{H}$ is the height function on the orbit as described in (GGSM1, , Thm. 2.2), obtained by pairing with $H$ via the Cartan–Killing form.

Notation 4.1**.**

We denote by $\mathrm{LG}_{n}$ the Landau–Ginzburg model $(\mathcal{O}_{n},f_{H})$.

We are looking for a tame compactification $\overline{\mathrm{LG}}_{n}=(Z,w)$ such that $Z\setminus D=\mathcal{O}_{n}$ for some divisor $D$ together with a holomorphic extension $w$ of the potential $f_{H}$. In this section, we accomplish an intermediate step of the construction, namely, that of describing an extension of $f_{H}$ to a rational map $R_{H}$ defined in codimension 2 on the compactification $\mathbb{P}^{n}\times\mathbb{P}^{n}$ as in example 3.1. We also verify that the critical points of $R_{H}$ coincide with those of $f_{H}$ outside of the indeterminacy locus $\mathcal{I}$ (definition 4.2), once this is done we can then obtain a holomorphic extension after blowing up $\mathcal{I}$, which we will do in section 5.

In the case of $\mathrm{LG}_{n}$ our rational map is described in (BGGSM, , Sec. 7) as

[TABLE]

[TABLE]

On $\mathbb{P}^{n}\times\mathbb{P}^{n}$ with bihomogeneous coordinates $x_{1},\dots,x_{n+1},y_{1},\dots,y_{n+1}$:

[TABLE]

For the general case, we have the following definitions:

Definition 4.1**.**

The flag variety $\mathbb{F}(1,n)$ is the homogeneous manifold consisting of pairs $(l,\pi)$ of lines contained in an $n$-dimensional subspaces of $\mathbb{C}^{n+1}$. $\mathbb{F}(1,n)$ can be obtained as the subvariety of $\mathbb{P}^{n}\times\mathbb{P}^{n}$ defined by: $x_{1}y_{1}+\cdots+x_{n+1}y_{n+1}=0.$

Notation 4.2**.**

The indeterminacy locus $\mathcal{I}$ of the rational map $R_{H}$ is given by:

[TABLE]

Lemma 4.1**.**

$\mathcal{I}$* is smooth and $R_{H}$ has no critical points on $\mathbb{F}(1,n)\setminus\mathcal{I}$.*

**Proof: **The Jacobian matrix of $R_{H}$

[TABLE]

has rank 2 on $\mathbb{F}(1,n)\setminus\mathcal{I}$. In fact, since all $\lambda_{i}$ are distinct and $x_{i}$ and $y_{i}$ are coordinates in projective spaces, the only points when the matrix has rank 1 are ones when $x_{i}=y_{j}=0$ unless $i=j=k$ for some $k$, but these do not belong to $\mathbb{F}$ and nor to $\mathcal{I}$ either, showing in particular that $\mathcal{I}$ is smooth.

Furthermore, $R_{H}$ has critical points on the coordinate points $(e_{i},e_{i})$, that is, those points $[x_{1},\dots,x_{n+1}],[y_{1},\dots,y_{n+1}]$ with only 2 nonzero coordinates $x_{i},y_{j}$ with $i=j$, these lie outside $\mathbb{F}(1,n)$ and correspond to the critical points of $f_{H}$. Lemma 4.1 does not verify points of $\mathcal{I}$ where the rational map $R_{H}$ remains undefined (we resolve this issue in the following section, where such points are blown-up).

5 Holomorphic extension of the potential

To extend the potential as a holomorphic map we need to blow up $\mathbb{P}^{n}\times\mathbb{P}^{n}$ along $\mathcal{I}$. We denote the result of the blowing up by $Z$, obtained as follows:

Take $\mathbb{P}^{1}$ with homogeneous coordinates $[t,s]$. The pencil $\{tg+sf_{H}\}_{t,s\in\mathbb{C}}$ induces a rational map $\mathbb{P}^{n}\times\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{1}$ with $\mathcal{I}$ as its indeterminacy locus. Call $Z$ the closure of the graph of this map.

So, we get a map

[TABLE]

Note that if $s\neq 0$ then

[TABLE]

where the middle equality holds since $tg=sf_{H}$, and $g=1$ over the orbit $\mathcal{O}_{n}$ because in fact $g=\det(A_{ij})$ with $A\in\operatorname{\mathrm{SL}}(n+1,\mathbb{C})$. Thus, we obtain the desired holomorphic extension:

Lemma 5.1**.**

The map $w\colon Z\rightarrow\mathbb{P}^{1}$ is a holomorphic extension of $f_{H}$. The critical points of $w$ coincide with the critical points of $f_{H}$.

**Proof: **We know by lemma 4.1 that the indeterminacy locus $\mathcal{I}$ of $R_{H}$ is smooth. We want to show that $w$ has no critical points over the exceptional divisor $E$. Since surjectivity of the derivative is a local question, it is enough to analyse an open neighbourhood of the point in question (the analytic topology is sufficient, though the Zariski topology will work as well).

For instance if we take a point $p$ in $E$ mapping to the coordinate point $P=([1,0,\ldots,0],[0,1,\ldots,0])\in\mathcal{I}$ by the blow down map, hence $p=([1,0,\ldots,0],[0,1,\ldots,0],[t_{0},s_{0}])$. We take the open neighbourhood

[TABLE]

of the point $P$. In this neighbourhood, the defining equations for $S=\mathcal{I}\cap U_{12}$ become

[TABLE]

where $X_{i}=x_{i}/x_{1},Y_{i}=y_{i}/y_{2}$, hence the local expressions for $f$ and $g$ are nonsingular. Since $S$ is a smooth submanifold of $U_{12}\simeq\mathbb{C}^{2n}$ of codimension 2, we can change coordinates to $\Delta=\mathbb{C}[Z_{1},\dots,Z_{2n}]$ so that

[TABLE]

and therefore $S$ is cut out inside $\Delta$ as the linear submanifold $Z_{2n-1}=Z_{2n}=0$. Now, following (GH, , p. 603), we know that an extension of $w$ is defined over $\tilde{\Delta}$ by $(w(z),l^{\prime})$ where

[TABLE]

Now consider the chart on the target $\mathbb{P}^{1}$ where $s\neq 0$, then on $\tilde{\Delta}\setminus E$ we have $w=f/g=t/s$, with $s\neq 0$ hence $g\neq 0$ thus

[TABLE]

So, that $w|_{S}$ can be extended over $E$ to

[TABLE]

without critical points. We conclude that $P$ is not a critical point of $w$.

Calculations at other coordinate points work similarly, and since every point on $E$ belongs to an open neighbourhood of some coordinate point, extension of the result to more general points of $E$ goes through automatically. In all such cases we conclude surjectivity of the derivative, hence $w$ has no critical points on $E$.

Remark 5.1**.**

Here $[t,s]$ are the coordinates of $\mathbb{P}^{1}$ and the fibre over infinity $Z_{\infty}=w^{-1}([1,0])$ happens when $s=0$, giving the equation

[TABLE]

The set $V$ of solutions of (7) inside $\mathbb{P}^{n}\times\mathbb{P}^{n}\times\mathbb{P}^{1}$ consists of 2 components

• $\iota)$

a flag manifold $\mathbb{F}(1,n)$ cut out by the equation $\sum x_{i}y_{i}=0$ in $\mathbb{P}^{n}\times\mathbb{P}^{n}$ for any value of $[t,s]\in\mathbb{P}^{1}$, thus forming a component that is a $\mathbb{P}^{1}$ bundle over $\mathbb{F}(1,n)$, and

• $\iota\iota)$

when $t=0$, we have a second component $\mathbb{P}^{n}\times\mathbb{P}^{n}\times{[0,1]}$.

These 2 components intersect in $\mathbb{F}(1,n)\times{[0,1]}$.

The fibre at infinity $Z_{\infty}$ of the map $w\colon Z\rightarrow\mathbb{P}^{1}$ is formed by the solutions of (7) inside $Z$, that is, $Z_{\infty}=Z\cap V$. Since the point at infinity is $[t,s]=[1,0]$, we can not at the same time have that $t=0$, so that the entire second component, described in item $\iota\iota)$ can not belong to $Z_{\infty}$. Taking $s=0$ we obtain precisely the proper transform of the blow-up of $\mathcal{I}$ inside $Z$, and we conclude that $Z_{\infty}=w^{-1}([1,0])\simeq\mathbb{F}(1,n)$.

Lemma 5.2**.**

$(Z,w)$* is a weakly tame compatification of $(\mathcal{O}_{n},f_{H})$.*

**Proof: **By construction $Z$ compactifies $\mathcal{O}_{n}$ and by Lemma 5.1 $w$ is an extension of $f_{H}$. Writing the boundary divisor $D_{Z}$ as a sum of 2 two irreducible components $D^{v}$ and $D^{h}$, we have that the vertical component $D^{v}\simeq\mathbb{F}(1,n)$ is a smooth irreducible divisor which is the fibre of the map $w$ over infinity as described in Remark 5.1. The horizontal component $D^{h}$ of $D^{Z}$ is the exceptional divisor of the blow up, therefore a $\mathbb{P}^{1}$- bundle over $\mathcal{I}$, where $\mathcal{I}\subset\mathbb{P}^{n}\times\mathbb{P}^{n}$ is the indeterminacy locus of the rational map $R_{H}$ as in Definition 4.2.

Therefore, $D_{Z}=D^{h}+D^{v}$ is a strict normal crossing divisor, such that $D^{h}$ is smooth and surjective over $\mathbb{P}^{1}$ and $D^{v}$ is the scheme theoretic fibre over infinity. Since $\mathcal{O}_{n}$ has trivial canonical class, one can find a unique anticanonical divisor $-K_{Z}$ with support $D_{Z}$. Such divisor fails the condition in Definition 2.2 that requires that this anticanonical divisor be equal to $D_{Z}$; instead, it is a combination of $D^{h}$ and $D^{v}$ with positive coefficients:

[TABLE]

and because of these nontrivial multiplicities we refer to our compactification as weakly tame.

Remark 5.2**.**

Given our choice of smooth compactification and the fact that the fibre at infinity is smooth, as explained in remark 5.1, we obtain that the monodromy around infinity is trivial, that is, the map (3) is the identity. Consequently, the grading of the cohomologies to be considered in definition 2.5 is also trivial. However, notice that it does not imply that all the invariants $h^{p,q}(Y,w)$ are zero. Indeed,

[TABLE]

by lemma  6.3.

6 Hodge structures on minimal adjoint orbits

The following results will be used to show that the adjoint orbits have pure Hodge structures. Consider $X$ smooth projective, and $Y$ smooth projective of codimension 1 in $X$. To describe the Hodge structure on $U=X\setminus Y$, we consider the Gysin sequence

[TABLE]

It turns out that to make the homomorphism $\operatorname{\mathrm{H}}^{k-2}(Y)\rightarrow\operatorname{\mathrm{H}}^{k}(X)$ a map of Hodge structures, it is sufficient to shift the weights up by (1,1) (Ho, , Sec. 2.1). So we get the Gysin map

[TABLE]

which is a map of Hodge structures of weight $k$, and thus induces Hodge structures on $\ker\delta_{k}$ and $\operatorname{coker}\delta_{k}$, see (Ho, , Thm. 10). Furthermore, the Hodge filtrations on $\ker\delta_{k+1}$ and $\operatorname{coker}\delta_{k}$ are the same as those induced by the Hodge filtration on $\operatorname{\mathrm{H}}^{k}(U)$ via the short exact sequence (arising from the Gysin sequence)

[TABLE]

Thus, $\operatorname{\mathrm{H}}^{n}(U)$ admits a natural mixed Hodge structure with weight filtration

[TABLE]

and Hodge filtration $F^{p}\operatorname{\mathrm{H}}^{n}(U)$ given by classes represented by $\geq$ $p$-holomorphic logarithmic differential forms such that

[TABLE]

where the kernel and cokernel of $\delta_{k}$ are given their natural Hodge structures (Ho, , Cor. 11).

6.1 The Hodge structure on $\mathcal{O}_{n}$

We consider the case when $U=\mathcal{O}_{n}$ is the minimal adjoint orbit as in definition 1.1 compactified to $\mathbb{P}^{n}\times{\mathbb{P}^{n}}$ so that $\mathbb{P}^{n}\times{\mathbb{P}^{n}}\setminus U=\mathbb{F}(1,n)$ is a flag variety.

Example 6.1**.**

Taking $U=\mathcal{O}_{n}$ and $X=\mathbb{P}^{n}\times\mathbb{P}^{n}$ in (8) we get

[TABLE]

and obtain a trivial weight filtration

[TABLE]

Thus, the Hodge structures on the minimal adjoint orbits $\mathcal{O}_{n}$ are pure.

We show that our $LG$ models are Hodge–Tate. Let $Z=\mathrm{Bl}_{\mathcal{I}}{\mathbb{P}^{n}\times\mathbb{P}^{n}}$ as in section 5 with $E$ the exceptional divisor. So, we have that $Z\backslash E=\mathbb{P}^{n}\times\mathbb{P}^{n}\backslash\mathcal{I}$. We compute the Hodge numbers of $E$.

Lemma 6.1**.**

The Hodge numbers of the exceptional divisor $E$ are $h^{p,q}(E)=0$ unless $p=q$.

**Proof: **Firstly, observe that $\mathbb{F}(1,n)$ is a $\mathbb{P}^{1}$-bundle over $\mathcal{I}$. Secondly, observe that $E$ is a $\mathbb{P}^{1}$-bundle over $\mathcal{I}$. So we have the Hodge polynomials

[TABLE]

On the other hand $\mathbb{F}(1,n)$ fibres over $\mathbb{P}^{n}$ with fibre $\mathbb{P}^{n-1}$. Thus, $\alpha(E)=(1+uv+\cdots+u^{n-1}v^{n-1})(1+uv+\cdots+u^{n}v^{n}).$

Thus, we conclude that $E$ has only $(p,p)$ cohomology.

Lemma 6.2**.**

If $p\neq q$ then the KKP numbers of $LG_{n}$ vanish, that is, $f^{p,q}(Y,w)=h^{p,q}(Y,w)=i^{p,q}(Y,w)=0$.

**Proof: **Example 6.1 describes the trivial weight filtration on the orbit $Y=\mathcal{O}_{n}$ showing that the Hodge structure of $\mathcal{O}_{n}$ is pure. By lemma 6.1 we have that the Hodge numbers of $E$ vanish for $p\neq q$, and similarly for the compactification $Z$ of $\mathcal{O}_{n}$ obtained by blowing up $\mathbb{P}^{n}\times\mathbb{P}^{n}$ at $\mathcal{I}$. Therefore, the same is also true for orbit $\mathcal{O}_{n}$. Lemma 5.1 shows that the compactified Landau–Ginzburg model $w\colon Z\rightarrow\mathbb{P}^{1}$ has no critical points over $E$, implying in particular that is does not acquire extra critical points at infinity, see remark 5.1. It then follows that the logarithm of the monodromy operator at infinity $N$ is trivial for all cohomology groups, forcing the first Hodge theoretical invariants, $h^{p,q}(Y,w)$, to vanish when $p\neq q$.

The numbers $f^{p,q}(Y,w)$ can be computed from the dimensions of the Hodge graded pieces of the canonical mixed Hodge structure on the relative cohomology $\operatorname{\mathrm{H}}^{m}(Y,Y_{b})$ using results from Ha or Sh . For the cases we consider here, the equality $f^{p,q}(Y,w)=h^{p,q}(Y,w)$ follows from (Ha, , Thm. 3.1), which shows that the $f^{p,q}$ Hodge numbers are computed from the dimensions of the Hodge graded pieces of the pair $(Y,w)$, implying that $f^{p,q}(Y,w)=h^{p,q}(Y,w)$.

Since our compactified Lefschetz fibration $w\colon Z\rightarrow\mathbb{P}^{1}$ has no critical points at infinity, it has the same critical points as $f$, see lemma 5.1. Consequently, by standard results for vanishing cycles in Lefschetz theory, we know that there exist $n+1$ vanishing cycles in middle cohomology for $w$, one for each of its critical points. Furthermore, we also know that the relative cohomologies $\operatorname{\mathrm{H}}^{m}(Y,Y_{b}\mathchar 24635\relax\;\mathbb{C})$ vanish for $m\neq\frac{1}{2}\dim Y$, and $\operatorname{\mathrm{H}}^{2n}(Y,Y_{b}\mathchar 24635\relax\;\mathbb{C})=n+1$ where $2n$ equals half the real dimension of $Z$ (see lemma 6.3 for an alternative proof). To complete the proof showing that the third invariant, $i^{p,q}(Y,w)=0$ unless $p=q=n$, we work in the opposite direction. Katzarkov–Kontsevich–Pantev KKP proved the equality

[TABLE]

Therefore, it remains to verify that $i^{n,n}(Y,w)=n+1$. Let $\lambda_{i}$ be a critical value of $w$. Since $w$ is a Lefschetz fibration and $\dim Z=2n$, the potential $w$ can be written in the form $z_{1}^{2}+\dots+z_{2n}^{2}$ around the critical point. Hence, the monodromy around the singular fibre $Y_{\lambda_{i}}$ is a Dehn twist with local monodromy given by a matrix $T=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ applied to local coordinates of $\mathbb{C}^{n}\oplus\mathbb{C}^{n}$. Consequently, $N=\log T=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ and $N^{2}=0$. It then follows that the weight filtration corresponding to this monodromy around $Y_{\lambda_{i}}$ has only one step, and the critical point with critical value $\lambda_{i}$ contributes with a $+1$ towards the value of $i^{n,n}(Y,w)$. Summing over all critical points we then get $i^{n,n}(Y,w)=n+1$.

Lemma 6.3**.**

\displaystyle h^{2k}(Y,Y_{b}\mathchar 24635\relax\;\mathbb{C})=\left\{\begin{array}[]{ll}k+1&\textup{if }k=n,\\ 0&\textup{otherwise}.\end{array}\right.**

**Proof: **Using (GGSM2, , Thm. 2.1), we have that $Y=\mathcal{O}_{n}\sim\mathbb{P}^{n}$ ($\sim$ denotes homotopy equivalence) and using (GGSM1, , Cor. 3.4) and the fact that $f_{H}$ has $n+1$ critical points, we get that $Y_{b}\sim\mathbb{P}^{n}\setminus\{P_{1},\ldots,P_{n+1}\}$ where $P_{1},\ldots,P_{n+1}$ are points in $\mathbb{P}^{n}$. Using the Mayer–Vietoris sequence we obtain

[TABLE]

Then, the long exact sequence of the pair

[TABLE]

gives

[TABLE]

Theorem 6.1**.**

The LG model $(\mathcal{O}_{n},f_{H})$ admits a weakly tame compactification and satisfies the KKP conjecture.

**Proof: **Lemmas 6.2 and 6.3 together with the equality (1) imply that the three invariants coincide on $\mathcal{O}_{n}$. We depict their values in the KKP diamond:

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Acknowledgements We thank the referee for many useful suggestions. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy). Gasparim was partially supported by the Vicerrectoría de Investigación y Desarrollo Tecnológico de la Universidad Católica del Norte (Chile). Rubilar acknowledges support from Beca Doctorado Nacional – Folio 21170589. Gasparim and Rubilar were partially supported by Network NT8 of the Office of External Activities of ICTP (Italy).

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