On the Geometry of the Anti-canonical Bundle of the Bott-Samelson-Demazure-Hansen Varieties

Let G be a semi-simple simply connected algebraic group over the field ℂ of complex numbers. Let T be a maximal torus of G, and let W be the Weyl group of G with respect to T. Let Z(w, i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple i associated to a reduced expression of an element w ∈ W. We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w, the anti-canonical line bundle on Z(w, i) is globally generated. As consequence, we prove that Z(w, i) is weak Fano.

Assume that G is a simple algebraic group whose type is different from A₂. Let S = {α₁, …, α_n} be the set of simple roots. Let w be such that support of w is equal to S. We prove that Z(w, i) is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and \({w^{ - 1}}(\sum\nolimits_{t = 1}^n {{\alpha _t}) \in - S} \) $$w^{-1}({\displaystyle {\sum }_{t=1}^n{\alpha }_t)\in -S}$$ .