Morphisms from projective spaces to $G/P$

Let $G=SL(n,\mathbb{C})$. Let $B$ denote the Borel subgroup of upper triangular matrices and $T$ be the maximal torus of diagonal matrices in $G$. The projective homogeneous variety $G/B$ is called the generalised flag variety (or simply flag variety). Let $W$ denote the Weyl group for $(G,T)$. In our case, it is $S_{n}$, the symmetric group in $n$ letter. For $w\in W$, let $X(w)$ be the corresponding Schubert variety inside $G/B$ which is defined to be the closure of $B$ orbit passing through the $T$-fixed point $wB$. The $B$-orbit is called a Schubert cell and is denoted by $C(w)$. The study of cohomology ring of the generalised flag variety $H^{\bullet}(X,\mathbb{Z})$ goes back to Ehresmann [6]. He also showed that the flag variety has a partition in terms of Schubert cells. In [3], Borel gave a presentation of the ring. Let $x_{1},x_{2},\ldots,x_{n}$ denote $n$ indeterminates. Let $\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$ denote the graded polynomial ring with degree of $x_{i}$ being $2$ for each $i$. $S_{n}$ acts on indeterminates by permuting the indices. Let ${\mathcal{}{I}}=\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]_{\plus}^{S_{n}}$ denote the ideal of $\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$ generated by $S_{n}$ invariants in positive degrees. In [3], Borel showed that there is an graded isomorphism

In [4], Chow showed that there is a basis of $H^{\bullet}(G/B,\mathbb{Z})$ given by the classes of the Schubert varieties inside the Grassmannian. From [8], we know that the cohomology ring $H^{\bullet}(G/B,\mathbb{Z})$ is also isomorphic to the Chow ring $A^{\bullet}(G/B)$.

Let $P$ be a parabolic subgroup containing $B$, let $W_{P}$ denote subgroup of $W$ generated by simple reflections that appears in the Bruhat decomposition of $P$. Let $W^{P}=W/W_{P}$ denote the set of minimal length coset representives. In [13], Reiner–Woo–Yong gave a presentation of the cohomology ring $H^{\bullet}(G/P,\mathbb{Z})$. They show that

where $H^{\bullet}(G/B,\mathbb{Z})^{W_{P}}$ denotes the ring of invariants of $H^{\bullet}(G/B,\mathbb{Z})$ under the $W_{P}$ action. In the same paper, they also study a presentation of the cohomology ring $H^{\bullet}(X(w),\mathbb{Z})$ and get a relatively shorter presentation of the cohomology ring of the Schubert varieties appearing in the flag varieties.

Let $k\leq n$ be two positive integers. Let $Gr(k,n)$ denote the Grassmannian of $k$-dimensional subspaces of an $n$ dimensional complex vector space. Note that the Grassmannian variety is also a partial flag variety $G/P$ for a maximal parabolic subgroup $P$. Maps from projective spaces to Grassmannian have been studied extensively by Tango in a series of papers [16, 18, 19]. In [16], he showed that there is no map from $\mathbb{P}^{m}$ to $Gr(k,n)$ for $m\geq n$. In [17], he also showed that there is an indecomposable vector bundle of rank $n-1$ on $\mathbb{P}^{n}$. While answering Lazarsfeld’s problems [9], under the assumption $1<k<n-k$ Paranjape–Srinivas in [12] show that there is a finite surjective morphism $Gr(k,n)$ to $Gr(l,m)$ if and only if $k=l$ and $n=m$, in which case they are isomorphic. This naturally led to the study of maps between various projective homogeneous spaces.

The study of cohomology ring plays a crucial role in attempting many of these questions. Tango’s original proof compared the Chow ring of the two varieties. In [10], Muñoz–Occhetta–Conde showed that a weaker property of the cohomology ring is required to obtain Tango type result which they called effective good divisibility which was an improvement of the notion of effective good divisibility introduced by Pan in [11]. Using effective good divisibility, Naldi–Occhetta in [1] were able to extend Tango’s result and they show that every morphism between $Gr(k,n)$ to $Gr(l,m)$ for $n>m$ is constant. In the paper they show that the effective good divisibility of the Grassmannian is $n$. In [14], Muñoz–Occhetta–Conde studied the maps between rational homogenous variety using effective good divisibility. They show that there is no map from a projective variety to a rational homogeneous variety where the effective good divisibility of the projective variety is higher than that of a rational homogeneous variety.

In our paper, we study maps from projective spaces to certain partial flag varieties. We first observe the following:

Let $P$ be a parabolic subgroup of $G$ containing $B$. Let $m$ denote the rank of the Picard group $Pic(G/P)$. We call it the rank of $G/P$. For instance, rank of the Grassmannian $Gr(k,n)$ is $1$ and the rank of the full flag variety $G/B$ is $n-1$. Similarly, we have rank of $G/P$ is $n-2$, for a minimal parabolic subgroup $P$. From 1.1 and Tango’s result it is natural to ask the following question:

We study the question when $P$ is a minimal parabolic subgroup. Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{n-1}$ denote the set of simple roots. Let $s_{\alpha_{j}}$, $1\leq j\leq n-1$ denote the corresponding simple roots. To each such $\alpha_{j}$ we can associate a minimal parabolic subgroup $P_{\alpha_{j}}:=B\cup Bs_{\alpha_{j}}B$. We obtain the following two theorems.

After we had put up the first version in the arXiv, it was communicated to us via an email from Yanjie Li that the proof of Theorem 5.1 has an error. Formula (8) in the proof of the theorem is wrong. To overcome that, we are giving a completely elementary proof of theorem 1.4. Due to the same error, we have now removed Theorem 6.1 from the arxiv version where we state that there is no morphism from $\mathbb{P}^{4}$ to $G/P_{\alpha_{j}}$ for $1\leq j\leq n-1$. We have some evidence for the statement and we are working on it currently. After putting up our first version we have been communicated of a generalisation of some of our results by Fang and Ren (see, [7]).

Acknowledgement We are very thankful to Shrawan Kumar for informing us about his conjecture, and various helpful discussions. We are thankful to Rohith Varma for stimulating discussions. We are also thankful to Senthamarai Kannan for suggesting the article [14].

In this section we discuss Kumar’s conjecture. This conjecture is due to Shrawan Kumar (see, [15]). Let $X$ be a complex connected homogeneous projective variety. Then $X$ can be written as $H/P$ for a complex semisimple connected algebraic group $H$ and $P\subset H$ a parabolic subgroup. Observe that the $H$ and $P$ are not necessarily unique. For instance, the projective space $\mathbb{P}^{2n-1}$ is a homogeneous space for $SL(2n,\mathbb{C})$ as well as $Sp(2n,\mathbb{C})$. Assume also that $X$ is indecomposable, ie $X$ cannot be written as $X_{1}\times X_{2}$ where none of $X_{1}$ or $X_{2}$ is singleton.

For instance, $\mathbb{P}^{2n-1}$ has minss rank $n-1$ because it is realisable as a homogeneous space of $Sp(2n,\mathbb{C})$ whereas the maxss rank is $2n-2$ when it is realised as a quotient of $SL(2n,\mathbb{C})$. Note that we have minss rank $X=0$ if and only if $X$ has a realisation of the form $H/B$ for a Borel subgroup $B$ inside $H$. In this case, the maxss rank is also $0$. We also observe that the minss rank $X$ is same as maxss rank $X$ in most cases, however the cases where it doesn’t hold can be found in [5, §2].

Kumar in [15] has proved the conjecture when $X^{\prime}$ can be written as $H^{\prime}/B^{\prime}$ for a simple algebraic group $H^{\prime}$ with $B^{\prime}$ a Borel subgroup. Note that this is equivalent to the case when semisimple rank of $X^{\prime}$ is $0$. However, to what generality the conjecture is solved remains unclear, and our paper is a step towards understanding this conjecture. We look at the conjecture when $X^{\prime}$ has rank $0$ or $X^{\prime}$ is a rank $1$ homogeneous space of $SL(n,\mathbb{C})$. In the former case, we verify conjecture (a) when $X$ is homogeneous spaces of $SL(n,\mathbb{C})$. In the later, we verify the conjecture for any projective space. Some of the major computations in this paper analyses the map from $\mathbb{P}^{3}$ which has minss rank $1$ to $SL(n,\mathbb{C})/P$ for a minimal parabolic subgroup $P$ (which has maxss rank $1$ as well).

Let $G=SL(n,\mathbb{C})$ denote the set of all $n\times n$ matrices with determinant $1$. Let $B$ denote the Borel subgroup of upper triangular matrices, and $T$ denote the maximal torus consisting of diagonal matrices inside $G$. Denote $R$ the root system of $(G,T)$. Let $R^{+}$ denote the subset of $R$ consisting of positive roots. Let $\epsilon_{i}$ denote the character of $T$ which sends $diag(t_{1},t_{2},\ldots,t_{n})$ to $t_{i}$. Let $\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}$. Then a subset $S=\{\alpha_{1},\ldots,\alpha_{n-1}\}$ of $R^{+}$ gives a set of simple roots. The Weyl group $W$ is the group generated by the simple reflections $s_{\alpha}$, $\alpha\in S$. In our case, $W$ is the symmetric group in $n$ letters $S_{n}$. The simple reflections $s_{\alpha_{i}}$ can be thought of as the transposition of $i$-th and $i+1$-th letter. We would use the one-line notation $(w(1),w(2),\ldots,w(n))$ to denote the permutation $w$ in $S_{n}$.

Let $J$ be a subset of $S$. Let $W_{J}$ denote the subgroup of $W$ generated by $s_{\alpha}$, $\alpha\in J$. For every $J$ we associate a parabolic subgroup $P_{J}$ as follows

The set $W^{J}=W/W_{J}$ is called the set of minimal length coset representatives. Alternatively, we have (see, [2, Section 2.5])

The full flag variety is by definition the variety $G/B$. The projective homogeneous space $G/P_{J}$ is called a partial flag variety and its Bruhat decomposition is given by

Whenever $W_{J}$ is generated by one element $s_{\alpha}$ for $\alpha\in S$, we call the associated parabolic subgroup a minimal parabolic subgroup and we denote it as $P_{\alpha}$. Note that, $P_{\alpha}/B$ is isomorphic to $\mathbb{P}^{1}$. Whenever $J$ is obtained from $S$ by removing one simple root $\alpha_{k}$, we call the associated parabolic subgroup a maximal parabolic subgroup and we denote it by $P_{\hat{\alpha_{k}}}$. We recall that the Grassmannian variety $Gr(k,n)$ of $k$ dimensional subspaces of a $n$-dimensional complex vector space is isomorphic to $G/{P_{\hat{\alpha_{k}}}}$. Let

Let $w=(i_{1},i_{2},\ldots,i_{k})\in I(k,n)$. Let $e_{1},e_{2},\ldots,e_{n}$ be the standard basis of $\mathbb{C}^{n}$. Let $M_{i}$ denote the vector space spanned by $e_{1},e_{2},\ldots,e_{i}$. The Schubert cell $C(w)$ in the Grassmannian is defined as

The dimension of such a Schubert cell $C(w)$ is given by $\sum_{j}(i_{j}-j)$. The Schubert variety $X(w)$ which is the closure of $C(w)$ in Grassmannian can be seen to be

Let $R$ denote the polynomial ring $\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$ in $n$ variables with degree of $x_{i}$ being $2$. We recall that $S_{n}$ acts on the variables as

The action extends to an action of $S_{n}$ on $R$. A polynomial $f(x_{1},x_{2},\ldots,x_{n})$ in $R$ is symmetric if and only if

for all $\sigma\in S_{n}$. The power sum symmetric polynomial $p_{k}(x_{1},x_{2}\ldots,x_{k})$ is defined as

We recall that the subring of invariants $R^{S_{n}}$ of $R$ is a graded subring and is generated by symmetric polynomials. Let ${\mathcal{}{I}}$ denote the ideal generated by symmetric polynomials in positive degree. The power sum symmetric polynomials $p_{k}(x_{1},x_{2},\ldots,x_{n})$ for $1\leq k\leq n$ form a set of generators for ${\mathcal{}{I}}$.

Let $X$ be a projective variety. Let $H^{\bullet}(X)=\bigoplus\limits_{d=1}^{n}H^{d}(X)$ denote the cohomology ring of the variety with integer coefficients. Let $A^{\bullet}(X)=\bigoplus\limits_{d=1}^{n}A^{d}(X)$ denote its Chow ring. We recall from [8, Chapter 19] that there exists a cycle map

Whenever $X$ is a partial flag variety the map $cy$ is an isomorphism (see, [8, Example 19.1.11]) and the cohomologies in odd degrees vanish. When $X$ is the full flag variety $G/B$ we recall

In [3], Borel, gave a presentation of the cohomology ring using the polynomial ring $R$ and the ideal $I$

The results were extended for $G/P$, where $P$ is a parabolic subgroup of $G$ containing $B$ in [13]. Let $J\subset S$ such that $P=P_{J}$. We have $W_{J}$ the subgroup of Weyl group generated by $J$ as above. Since $W_{J}$ is subgroup of $W$ it also acts on $H^{\bullet}(G/B)$. Reiner–Woo–Yong show that,

We observe that

As in the previous section, we have $G=SL(n,\mathbb{C})$, $B$ denotes the Borel subgroup consisting of the diagonal matrices in $G$. We will begin this section by proving the following:

Let $V$ be a vector space of dimension $n$ and

be a sequence of integers. We define $G(i_{1},i_{2},\ldots,i_{k})$ the partial flag variety $G/P$ consisting of linear subspaces $L_{i_{1}},L_{i_{2}},\ldots,L_{i_{k}}$ of $V$ such that $L_{i_{j}}\subset L_{i_{j+1}}$ and $\mathrm{dim}(L_{i_{j}})=i_{j}$.

We assume the notations from the previous sections. We thus have $P_{\alpha}$ the minimal parabolic subgroup $B\cup Bs_{\alpha}B$. When $\alpha=\alpha_{1}$ we will show that there is no non constant map from $\mathbb{P}^{3}$ to $G/{P_{\alpha}}$. Since $G/P_{\alpha_{1}}\cong G/P_{\alpha_{n-1}}$ we conclude that there is no non constant map from $\mathbb{P}^{3}$ to $G/{P_{\alpha_{n-1}}}$ as well. However, when we have any other minimal parabolic subgroup $P_{\alpha_{j}}$, $j\neq 1,n-1$ we will show that there are non constant maps from $\mathbb{P}^{3}$ to $G/{P_{\alpha_{j}}}$.

Fix a basis $e_{1},e_{2},\ldots,e_{n}$ of $V$. Define the subspaces $M_{i}$ to the span of $e_{1},e_{2}\ldots,e_{i}$. Let $D_{k}$ denote the Schubert divisor in the $Gr(k,n)$ which is defined as

We define the following two codimension 2 Schubert subvarieties of the Grassmannian $Gr(k,n)$:

We note that $D_{n,n+1}$ and $D_{1,0}$ are empty sets. We prove the following lemmas.

Let $E_{1}$ denote the codimension $3$ Schubert cycle defined by the Schubert variety $\{F\in Gr(2,n)|~{}F\cap M_{n-3}\neq 0~{}\rm{and}~{}F\subset M_{n-1}\}$. Let $E_{2}$ be the codimension $3$ Schubert cycle defined by the Schubert variety $\{F\in Gr(2,n)|~{}F\cap M_{n-4}\neq 0\}$.

Since the map $H^{*}(Gr(k,n))$ to $H^{*}(G/B)$ is injective the above relations holds in $H^{*}(G/B)$ as well. We use the same notations $D_{i}$ and $D_{i,j}$ to define the Schubert classes in $H^{*}(G/B)$. Note that the above relations can also be deduced from Monk’s formula.