Integrable systems, toric degenerations and Okounkov bodies

Abstract

Abstract. Let X be a smooth projective variety of dimension n over C equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from X to the special fiber X 0 which is a symplectomorphism on an open dense subset U . From this we are then able to construct a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions {H 1 , . . . , Hn} on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U . Moreover, our integrable system in fact generates a Hamiltonian torus action on U . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the 'moment map' µ = (H 1 , . . . , Hn) : X → R n is precisely the Newton-Okounkov body ∆ = ∆(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Convexity and Thimm’s Trick

“…Nishinou-Nohara-Ueda proved that the classical Gelfand-Zeitlin systems on U (n) coadjoint orbits can be constructed by toric degeneration [45]. This was later generalized by Harada-Kaveh who proved that, under various technical assumptions, a toric degeneration of a smooth projective variety endows an open dense subset with the structure of a proper toric manifold [25,Theorem B]. These results were applied in [34] to prove lower bounds for the Gromov width of smooth projective varieties in terms of their NewtonOkounkov bodies and in [22] and [14] to finish the proof of tight lower bounds for the Gromov width of coadjoint orbits of compact simple Lie groups (tight upper bounds were proven [11]).…”

Toric degenerations and symplectic geometry of smooth projective varieties

“…A crucial assumption in this construction is that an associated semigroup of lattice points is finitely generated (the value semigroup of ). In , given a toric degeneration for a smooth projective variety as above, the authors construct a maximal dimensional Hamiltonian torus action on by pulling back the moment map of the torus action on the toric variety (to do this one needs a compatible Kähler structure on the family ). Since in general the special fiber is non‐smooth, the pull‐back is defined only on an open subset (in the usual classical topology) of .…”

“…The fact that the toric variety has the same symplectic volume as implies that this open subset is moreover dense. A main result in is that the pull‐back moment map extends continuously to the whole . The image of this moment map is the Newton–Okounkov body associated to and the valuation (used in building the toric degeneration).…”

“…The construction in the present paper works in a much more general setting. Here we do not require the technical and restrictive assumption of ‘finite generation of the value semigroup’ needed in . We note that instead of a projective toric variety, our special fiber is just the torus , and moreover it can have a smaller symplectic volume than that of the original variety .…”

Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

“…These convex bodies generalize Newton polytopes for toric varieties to arbitrary projective varieties, and have various kinds of information about corresponding projective varieties; for instance, we can systematically construct a series of toric degenerations (see [HK,Corollary 3.14] and [A,Theorem 1]). Hence it is important to describe the explicit form of a Newton-Okounkov convex body.…”