Peterson Isomorphism in K-theory and Relativistic Toda Lattice

The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

Hitchin pairs on reducible curves

We define semistable generalized parabolic Hitchin pairs (GPH) on a disjoint union [Formula: see text] of integral smooth curves and construct their moduli spaces. We define a Hitchin map on the moduli space of GPH and show that it is a proper map. We construct moduli spaces of semistable Hitchin pairs on a reducible projective curve [Formula: see text]. When [Formula: see text] is the normalization of [Formula: see text], we give a birational morphism [Formula: see text] from the moduli space [Formula: see text] of good GPH on [Formula: see text] to the moduli space [Formula: see text] of Hitchin pairs on [Formula: see text] and show that the Hitchin map on [Formula: see text] induces a proper Hitchin map on [Formula: see text]. We determine the fibers of the Hitchin maps. We study the relationship between representations of the (topological) fundamental group of [Formula: see text] and Higgs bundles on [Formula: see text]. We show that if all the irreducible components of [Formula: see text] are smooth, then the Hitchin map is defined on the entire moduli space [Formula: see text].

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

Minimal rational threefolds II

The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by ℓ(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in ℓ(G, X): for (G, Z), (G, W) ∊ ℓ(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.