International Congress of Mathematicians

Spectral Problems 603line X a : x a = 0. Since T-orbit of po is dense in P 2 , we can vary t £ T so that tpotends to p a . Then for any vector e £ E = £(p 0 ), we have te £ £(tpo) and can trythe limitlim (te)which either exists or not. Let us denote by E a (0) the set of vectors e £ E forwhich the limit exists:L; a (0) := {e G E\ lim (te) exists}.Evidently E a (0) is a vector subspace of E, independent of po and p Q .An easy modification of the previous construction allows to define for integerm £ Z, the subspaceE a (m):=\e£E\ lim ( —) -(te) exists > .(^ tpo^p a \tß) JRoughly speaking E a (m) consists of vectors e £ E for which te vanishes up to orderm as tpo tends to coordinate line X a . The subspaces E a (m) form a non-increasingexhaustive Z-filtration:E a :••• D E a (m - 1) D B a (m) D E a (m + 1) D • • • ,E a (m) = 0, for ro>0, (3.2)E a (m) = E, for m«0.Applying this construction to other coordinate lines, we get a triple of filtrationsE a , E ß , E 1 in generic fiber E = £(po), associated with toric bundle £.Theorem 3.1. The correspondence£^(E a ,E ß ,E r ) (3.3)establishes an equivalence between category of toric vector bundles on P 2 and categoryof triply filtered vector spaces.We'll use notation £(E a ,E ßfiltrations E a ,E ß ,E~>.3.2. Stability,E~>) for toric bundle corresponding to triplet ofThe previous theorem tells that every property or invariant of a vector bundlehas its counterpart on the level of filtrations. For application to spectral problemsthe notion of stability of a vector bundle £ is crucial. Recall that £ —¥ P 2 is said tobe Alumford-Takemoto stable iffÇi(T)_ ci(£)_ ( ,(àA)rk^ < rk£for every proper subsheaf T C £, and semistable if weak inequalities hold. Hereci(£) = deg det £ is the first Chern class. Donaldson theorem [7] brings a deepgeometrical meaning to this seemingly artificial definition: Every stable bundlecarries unique Hermit-Einstein metric (with Ricci curvature proportional to metric).