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25
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
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Cited by 36 (2 self)
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To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
Toric vector bundles, branched covers of fans, and the resolution property
"... Abstract. We associate to each toric vector bundle on a toric variety X(∆) a “branched cover ” of the fan ∆ together with a piecewiselinear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewiselinear functions. We apply ..."
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Cited by 14 (3 self)
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Abstract. We associate to each toric vector bundle on a toric variety X(∆) a “branched cover ” of the fan ∆ together with a piecewiselinear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewiselinear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently. Contents
Algebraic groups over the field with one element
"... ABSTRACT. Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory of split reductive groups over F1, the socalled field with one element, should look like. Namely, every split reductive group over Z should descend to F1, and its group of F1rational points should be its Weyl g ..."
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Cited by 10 (3 self)
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ABSTRACT. Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory of split reductive groups over F1, the socalled field with one element, should look like. Namely, every split reductive group over Z should descend to F1, and its group of F1rational points should be its Weyl group. We connect the notion of a torified variety to the notion of F1schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive groups as F1schemes. We endow the class of F1schemes with two classes of morphisms, one leading to a satisfying notion of F1rational points, the other leading to the notion of an algebraic group over F1 such that every split reductive group is defined as an algebraic group over F1. Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.
Moduli of toric vector bundles
"... Abstract. We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partia ..."
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Cited by 9 (3 self)
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Abstract. We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s Law, in the sense of Vakil. The preliminary sections of the paper give a selfcontained
On the moduli stack of commutative, 1parameter formal Lie groups
, 2007
"... Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, ..."
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Cited by 8 (1 self)
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Abstract. We attempt to develop a general algebrogeometric study of the moduli stack of commutative, 1parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the proalgebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of nbuds, and these latter stacks are algebraic. Our main theorems pertain to the height stratification relative to fixed prime p on the stacks of formal Lie groups and of nbuds. Notably, we show that the stack of nbuds of height ≥ h is smooth and universally closed over Fp of dimension −h; we characterize the stratum of nbuds of (exact) height h and the stratum of formal Lie groups of (exact) height h as classifying stacks of certain groups, smooth algebraic in the bud case; and we obtain some structure results on these groups. We also obtain a second characterization of the stratum of formal Lie groups of height h as an inverse limit of classifying stacks of certain finite étale algebraic groups.
KERNEL ALGEBRAS AND GENERALIZED FOURIERMUKAI TRANSFORMS
, 810
"... Abstract. We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as Dmodules, equivariant sheaves a ..."
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Cited by 7 (2 self)
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Abstract. We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as Dmodules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of [39] concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon [29] concerning an analog of the Fourier transform for derived categories of quasicoherent sheaves on a dual pair of generalized 1motives.
FERMIONIC FORMS AND QUIVER VARIETIES
, 2006
"... Abstract. We prove a formula relating the fermionic forms associated to a finite quiver and the Poincaré polynomials of quiver varieties associated to the same quiver. Applied to quivers of type ADE, our result implies a version of the fermionic Lusztig conjecture. 1. ..."
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Cited by 4 (1 self)
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Abstract. We prove a formula relating the fermionic forms associated to a finite quiver and the Poincaré polynomials of quiver varieties associated to the same quiver. Applied to quivers of type ADE, our result implies a version of the fermionic Lusztig conjecture. 1.