Torus quotients of homogeneous spaces. (English) Zbl 0941.14018
From the paper: One of the classical problems in invariant theory is the study of binary quantics. The main object was to give an explicit description and study the geometric properties of \(SL_2\) quotients of the projective space for a suitable choice of linearization. The aim of this paper is to begin the study of a natural generalization of this classical question.
Let \(k\) be an algebraically closed field. Let \(G\) be a semisimple algebraic group over \(k\), \(T\) a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\), \(N\) the normalizer of \(T\) in \(G\), \(W= N/T\), the Weyl group. Consider the quotient variety \(N\setminus \setminus G/B\). In fact the aim is to study more generally the variety \(N\setminus \setminus G/Q\), where \(Q\) is any parabolic subgroup of \(G\) containing \(B\).
We study torus quotients of these homogeneous spaces. We classify the Grassmannians for which semi-stable = stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring of \(T\) invariants of the homogeneous coordinate ring of the Grassmannian \(G_{2,n}\) (\(n\) odd) over the ring generated by \(R_1\), the first graded part of the ring of \(T\) invariants.
We prove the following results:
(a) the varieties \(T\setminus \setminus G/Q\) and \(N\setminus \setminus G/Q\) are Frobenius split and as an application the vanishing theorems for higher cohomologies of these varieties;
(b) as a part of result (a), we prove the vanishing of the higher cohomology groups for the binary quantics;
(c) for the line bundle \(L\) on \(G_{r,n}\) associated to the fundamental weight \(\varpi_r\), \((G_{r,n})_T^{ss} (L)= (G_{r,n})_T^s(L)\) if and only if \(r\) and \(n\) are coprime;
(d) existence of smooth projective varieties as quotients of certain \(G/Q\) modulo a maximal torus \(T\) (in the case of \(G= SL_n)\);
(e) for \(n\) odd, a partial result about \(R_1\) generation of the graded ring \(k\widehat {[G_{2,n}]}^T= \bigoplus_{d\geq 0}R_d\).
Let \(k\) be an algebraically closed field. Let \(G\) be a semisimple algebraic group over \(k\), \(T\) a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\), \(N\) the normalizer of \(T\) in \(G\), \(W= N/T\), the Weyl group. Consider the quotient variety \(N\setminus \setminus G/B\). In fact the aim is to study more generally the variety \(N\setminus \setminus G/Q\), where \(Q\) is any parabolic subgroup of \(G\) containing \(B\).
We study torus quotients of these homogeneous spaces. We classify the Grassmannians for which semi-stable = stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring of \(T\) invariants of the homogeneous coordinate ring of the Grassmannian \(G_{2,n}\) (\(n\) odd) over the ring generated by \(R_1\), the first graded part of the ring of \(T\) invariants.
We prove the following results:
(a) the varieties \(T\setminus \setminus G/Q\) and \(N\setminus \setminus G/Q\) are Frobenius split and as an application the vanishing theorems for higher cohomologies of these varieties;
(b) as a part of result (a), we prove the vanishing of the higher cohomology groups for the binary quantics;
(c) for the line bundle \(L\) on \(G_{r,n}\) associated to the fundamental weight \(\varpi_r\), \((G_{r,n})_T^{ss} (L)= (G_{r,n})_T^s(L)\) if and only if \(r\) and \(n\) are coprime;
(d) existence of smooth projective varieties as quotients of certain \(G/Q\) modulo a maximal torus \(T\) (in the case of \(G= SL_n)\);
(e) for \(n\) odd, a partial result about \(R_1\) generation of the graded ring \(k\widehat {[G_{2,n}]}^T= \bigoplus_{d\geq 0}R_d\).
MSC:
| 14M17 | Homogeneous spaces and generalizations |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14F17 | Vanishing theorems in algebraic geometry |
References:
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| [2] | [GIT] Mumford D, Fogarty J and Kirwan F,Geometric Invariant Theory (third edition) (New York: Springer-Verlag, Berlin, Heidelberg) |
| [3] | [S] Schrijver A,Theory of Linear and Integer Programming (John Wiley and Sons Ltd.) (1986) · Zbl 0665.90063 |
| [4] | [CSS I] Seshadri C S, Quotient spaces modulo reductive algebraic groups,Ann. Math. 95 (1972) 511–556 · Zbl 0241.14024 · doi:10.2307/1970870 |
| [5] | [CSS II] Seshadri C S,Introduction to the theory of standard monomials, Brandeis Lecture Notes 4, June 1985 |
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