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POSITIVITY AND KLEIMAN TRANSVERSALITY IN EQUIVARIANT KTHEORY OF HOMOGENEOUS SPACES
"... Abstract. We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth– Ram [GrRa04] concerning the alternation of signs in the structure constants for torusequivariant Ktheory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman trans ..."
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Abstract. We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth– Ram [GrRa04] concerning the alternation of signs in the structure constants for torusequivariant Ktheory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant Kclass of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for nontransitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term—the top one—with a welldefined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary Ktheory that brings Kawamata–Viehweg vanishing to bear. 1.
A Matroid Invariant via the KTheory of the Grassmannian
, 2006
"... Let G(d,n) denote the Grassmannian of dplanes in Cn and let T be the torus (C∗) n /diag(C ∗ ) which acts on G(d,n). Let x be a point of G(d,n) and let Tx be the closure of the Torbit through x. Then the class of the structure sheaf of Tx in the Ktheory of G(d,n) depends only on which Plücker coor ..."
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Cited by 17 (2 self)
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Let G(d,n) denote the Grassmannian of dplanes in Cn and let T be the torus (C∗) n /diag(C ∗ ) which acts on G(d,n). Let x be a point of G(d,n) and let Tx be the closure of the Torbit through x. Then the class of the structure sheaf of Tx in the Ktheory of G(d,n) depends only on which Plücker coordinates of x are nonzero – combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K ◦ (G(d,n)) to Z[t]. Letting gx(t) denote the image of (−1) n−dim Tx [OTx], gx behaves nicely under the standard constructions of matroid theory. Specifically, gx1⊕x2 (t) = gx1 (t)gx2(t), gx1+2 x2 (t) = gx1 (t)gx2(t)/t, gx(t) = gx⊥(t) and gx is unaltered by series and parallel extensions. Furthermore, the coefficients of gx are nonnegative. The existence of this map implies bounds on (essentially equivalently) the complexity of Kapranov’s Lie complexes [13], Hacking, Keel and Tevelev’s very stable pairs [11] and the author’s tropical linear spaces when they are realizable in characteristic zero [25]. Namely, in characteristic zero, a Lie complex or the underlying d − 1 dimensional scheme of a very stable pair can have at (n−i−1)! (d−i)!(n−d−i)!(i−1)! most strata of dimensions n − i and d − i respectively and a tropical linear space realizable in characteristic zero can have at most this many idimensional bounded faces.
Lifting Tropical Intersections
 DOCUMENTA MATH.
, 2013
"... We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient su ..."
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We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over nonnoetherian valuation rings of rank 1. In particular, we prove subadditivity of codimension and a principle of continuity for intersections in smooth schemes over such rings, generalizing wellknown theorems over regular local rings. An appendix on the topology
QUANTUM KTHEORY OF GRASSMANNIANS
, 2008
"... We show that (equivariant) Ktheoretic 3point GromovWitten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) Ktheory of a twostep flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we sho ..."
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Cited by 12 (5 self)
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We show that (equivariant) Ktheoretic 3point GromovWitten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) Ktheory of a twostep flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the GromovWitten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum Ktheory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for GromovWitten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.
A general homological KleimanBertini Theorem, 0705.0055v1
 ANDERS S. BUCH AND LEONARDO C. MIHALCEA
"... Abstract. Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against Gorbits vanish, then for generic g ∈ G, the sheaf Tor X j (gF, E) vanishes for all j ≥ 1. This generalizes a result of Miller and Speyer f ..."
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Abstract. Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against Gorbits vanish, then for generic g ∈ G, the sheaf Tor X j (gF, E) vanishes for all j ≥ 1. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical KleimanBertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0. 1.
Cohomology of Coherent Sheaves and Series of Supernatural Bundles
, 2009
"... We show that the cohomology table of any coherent sheaf on projective space is a convergent—but possibly infinite—sum of positive real multiples of the cohomology tables of what we call supernatural sheaves. ..."
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We show that the cohomology table of any coherent sheaf on projective space is a convergent—but possibly infinite—sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.
Geometric idealizers
"... Abstract. Let X be a projective variety, σ an automorphism of X, L a σample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring B = B(X, L, σ), let I be the right ideal of sections vanishing at Z. We study the subring R = k + I of B. Under mild condi ..."
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Abstract. Let X be a projective variety, σ an automorphism of X, L a σample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring B = B(X, L, σ), let I be the right ideal of sections vanishing at Z. We study the subring R = k + I of B. Under mild conditions on Z and σ, R is the idealizer of I in B: the maximal subring of B in which I is a twosided ideal. We give geometric conditions on Z and σ that determine the algebraic properties of R, and show that if Z and σ are sufficiently general, in a sense we make precise, then R is left and right noetherian, has finite left and right cohomological dimension, is strongly right noetherian but not strongly left noetherian, and satisfies right χd (where d = codim Z) but fails left χ1. We also give an example of a right noetherian ring with infinite right cohomological dimension, partially answering a question of Stafford and Van den Bergh. This