Smooth Projective Scheme Research Articles

Zero-cycles in families of rationally connected varieties

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár (Publ. Res. Inst. Math. Sci. 40(3):689–708, 2004).

Deformation theory of orthogonal and symplectic sheaves

We show that the deformation functor of an orthogonal (resp. symplectic) sheaf over a smooth projective scheme admits a miniversal pro-family, identifying its space of first-order deformations with the first hypercohomology space of a complex which is naturally constructed out of the orthogonal (resp. symplectic) sheaf. We also provide an obstruction theory of these objects whose target is the second hypercohomology space of this complex.

A functorial approach to regular homomorphisms

Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, e.g., families of schemes parameterized by a smooth variety over a given field. In that general setting, we establish the existence of an initial regular homomorphism, going by the name of algebraic representative, for codimension-2 cycles on a smooth proper scheme over the base. This extends a result of Murre for codimension-2 cycles on a smooth projective scheme over an algebraically closed field. In addition, we prove base change results for algebraic representatives as well as descent properties for algebraic representatives along separable field extensions. In the case where the base is a smooth variety over a subfield of the complex numbers we identify the algebraic representative for relative codimension-2 cycles with a subtorus of the intermediate Jacobian fibration which was constructed in previous work. At the heart of our descent arguments is a base change result along separable field extensions for Albanese torsors of separated, geometrically integral schemes of finite type over a field.

On the p-curvatures of t-connections over a relative smooth projective scheme

Let be a global function of an integral -scheme and be a smooth projective scheme over . We show that the characteristic polynomial of the curvature of an integrable connection over is horizontal.

Proper connective differential graded algebras and their geometric realizations

We show that every proper connective dg algebra A admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. If A is moreover smooth, we compute the noncommutative Chow motive of A. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective dg algebras.

New examples of non-Fourier–Mukai functors

A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension$\ge 3$equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes.

ZERO-CYCLES ON NORMAL PROJECTIVE VARIETIES

AbstractWe prove an extension of the Kato–Saito unramified class field theory for smooth projective schemes over a finite field to a class of normal projective schemes. As an application, we obtain Bloch’s formula for the Chow groups of $0$ -cycles on such schemes. We identify the Chow group of $0$ -cycles on a normal projective scheme over an algebraically closed field to the Suslin homology of its regular locus. Our final result is a Roitman torsion theorem for smooth quasiprojective schemes over algebraically closed fields. This completes the missing p-part in the torsion theorem of Spieß and Szamuely.

An asymptotic vanishing theorem for the cohomology of thickenings

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let \(X_t\) denote the t-th thickening of X in Y. Fix an ample line bundle \(\mathcal {O}_Y(1)\) on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers \(t \geqslant 1\), the cohomology group \(H^k(X_t,\mathcal {O}_{X_t}(j))\) vanishes for \(k < \dim X\) and \(j < -ct\). Note that there are no restrictions on the characteristic of the field, or on the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.

Equivariant Hodge theory and noncommutative geometry

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

Characteristic cycles and the conductor of direct image

We prove the functoriality for a proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms to curves. The conductor formula in the constant coefficient case gives the geometric case of a formula conjectured by Bloch.

On the Voevodsky Motive of the Moduli Stack of Vector Bundles on a Curve

AbstractWe define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

Liftable derived equivalences and objective categories

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain projective scheme is triangle-objective, that is, any triangle autoequivalence on it, which preserves the the isomorphism classes of complexes, is necessarily isomorphic to the identity functor.

Finite-dimensional differential graded algebras and their geometric realizations

We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable semi-exceptional collection. Moreover, we also show that it gives a characterization of such categories assuming that a subcategory is idempotent complete and has a classical generator.

A note on Grothendieck’s standard conjectures of type 𝐶⁺ and 𝐷 in positive characteristic

Making use of topological periodic cyclic homology, we extend Grothendieck’s standard conjectures of type C + \mathrm {C}^+ and D \mathrm {D} (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of Kontsevich. As a first application, we prove Grothendieck’s original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck’s (generalized) conjectures in the new cases of “low-dimensional” orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot’s cohomological interpretation of the classical zeta function and of Grothendieck’s conditional approach to “half” of the Riemann hypothesis. Along the way, following Scholze, we prove that the topological periodic cyclic homology of a smooth proper scheme X X agrees with the crystalline cohomology theory of X X (after inverting the characteristic of the base field).

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

Let k be the algebraic closure of a finite field of odd characteristic p and X a smooth projective scheme over the Witt ring W(k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs–de Rham flow and prove that the category of periodic Higgs–de Rham flows over X/W(k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group \pi_1(X_K) of the generic fiber of X , after Fontaine–Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber X_k of X of rank \leq p initiates a semistable Higgs–de Rham flow and thus those of rank \leq p-1 with trivial Chern classes induce k -representations of \pi_1(X_K) . A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p , it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

Algebraization for zero-cycles and the $p$-adic cycle class map

Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is surjective. We relate this restriction map to the $p$-adic cycle class map.

Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of special type. Different properties of both noncommutative schemes and morphisms between them are reviewed and discussed. In addition, the concept of a geometric realization for a derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. Also studied are the construction of gluings of noncommutative schemes via morphisms, along with certain new phenomena such as phantoms, quasi-phantoms, and Krull–Schmidt partners which arise in the world of noncommutative schemes and which enable us to find new noncommutative schemes. The last sections consider noncommutative schemes connected with basic finite- dimensional algebras. It is proved that such noncommutative schemes have special geometric realizations under which the algebra goes to a vector bundle on a smooth projective scheme. Such realizations are constructed in two steps, the first of which is the well-known construction of Auslander, while the second step is connected with the new concept of a well-formed quasi-hereditary algebra, for which there are very special geometric realizations sending standard modules to line bundles. Bibliography: 50 titles.

Chow motives of abelian type over a base

Let S be a Noetherian scheme, let X be a smooth projective scheme over S, whose fibres are connected curves of genus g, and let J be the Jacobian scheme of the relative curve X over S. We generalise the theorem due to Rolph Schwarzenberger and prove that if S is integral and normal, and the structural morphism admits a section, then there exists a locally free sheaf on J, such that the relative symmetric power is isomorphic to the projective bundle over J, provided , and the ample divisor is mathrm{{Sym}}^{d-1}(X/S), embedded into by the section of the structural morphism from X to S. Then we use this result to generalise the theorem due to Shun-Ichi Kimura: if S is an integral regular scheme, separated and of finite type over a Dedekind domain, then all relative Chow motives of abelian type over S are finite-dimensional.

Rost nilpotence and étale motivic cohomology

A smooth projective scheme X over a field k is said to satisfy the Rost nilpotence principle if any endomorphism of X in the category of Chow motives that vanishes on an extension of the base field k is nilpotent. We show that an étale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to obtain an elegant proof of Rost nilpotence for surfaces, as well as for birationally ruled threefolds over a field of characteristic 0.

A note on Grothendieck’s standard conjectures of type $C^+$ and $D$

Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, linear sections of determinantal varieties, and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.