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Regular version of the site
Of all publications in the section: 22
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Article
Kuznetsov A. G. Compositio Mathematica. 2011. Vol. 147. P. 852-876.

Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T \to S$. Given an admissible subcategory $\CA$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X\times_S T$, called the base change of $\CA$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived category of the fiber product. As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on $X$ and of the category of perfect complexes on $X$. As an application we prove that the projection functors of a semiorthogonal decomposition are kernel functors.

Added: Jun 6, 2011
Article
Feigin B. L., Loktev S., Miwa T. et al. Compositio Mathematica. 2002. Vol. 134. No. 2. P. 193-241.
Added: May 31, 2010
Article
Amerik E., Verbitsky M. Compositio Mathematica. 2017. Vol. 153. No. 8. P. 1610-1621.

Let M be an irreducible holomorphic symplectic (hyperkähler) manifold. If b 2 (M ) > 5, we construct a deformation M 0 of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b 2 (M ) > 14, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

Added: Nov 6, 2017
Article
Cheltsov I., Park J., Won J. Compositio Mathematica. 2016. Vol. 152. P. 1198-1224.

For each del Pezzo surface S with du Val singularities, we determine whether it admits a (−K S )-polar cylinder or not. If it allows one, then we present an effective Q-divisor D that is Q-linearly equivalent to −K S and such that the open set S\Supp(D) is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial G a -actions on their affine cones defined by their anticanonical divisors.

Added: Aug 31, 2016
Article
Kuznetsov A. G., Perry A. Compositio Mathematica. 2018. Vol. 154. No. 7. P. 1362-1406.

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

Added: Sep 13, 2018
Article
Finkelberg M. V., Bezrukavnikov R., Mirkovic I. Compositio Mathematica. 2005. Vol. 141. No. 3. P. 746-768.
Added: Oct 12, 2012
Article
Esterov A. I. Compositio Mathematica. 2019. Vol. 155. No. 2. P. 229-245.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.  In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus.  The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

Added: Feb 5, 2019
Article
Katzarkov L., Blanc A., Pandit P. Compositio Mathematica. 2018. Vol. 154. P. 2055-2089.

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a k-linear 1-category for a eld k. Our main result states that if C is a k-linear 1-category which has a compact generator whose groups of self-extensions vanish for suciently high positive degrees, then every formal deformation of C has zero curvature and moreover admits a compact generator.

Added: Nov 29, 2018
Article
Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Added: Mar 17, 2013
Article
Yuri Prokhorov, Constantin Shramov. Compositio Mathematica. 2014. Vol. 150. No. 12. P. 2054-2072.
Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over Q have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field k of characteristic zero.
Added: Jan 29, 2015
Article
Gritsenko V., Hulek K., Sankaran G. Compositio Mathematica. 2010. Vol. 146. No. 2. P. 404-434.

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12.

To resolve some geometric problems  we give a new, clear, formulation of Siegel's formula for the  number of representationс natural numbers by positive definite quadratic forms of  odd rank. It may be expressed either in terms of Zagier L-functions or in terms of the H.~Cohen numbers.  

 
Added: Mar 3, 2015
Article
Polishchuk A., Dyer B. Compositio Mathematica. 2019. Vol. 155. No. 4. P. 681-710.

In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.

Added: Jun 4, 2019
Article
Losev Ivan. Compositio Mathematica. 2017. Vol. 153. No. 12. P. 2445-2481.

In this paper we study categories O over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden, Licata, Proudfoot and Webster. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories O. We use these structures to study shuffling functors of Braden, Licata, Proudfoot and Webster (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

Added: Oct 15, 2017
Article
Positselski L., Bezrukavnikov R. Compositio Mathematica. 2010. Vol. 146. No. 2. P. 480-496.
Added: Oct 10, 2011
Article
Vologodsky V., Stewart A. Compositio Mathematica. 2013. Vol. 149. P. 63-80.
We compute the center of the ring of PD differential operators on a smooth variety over $\bZ/p^n\bZ$ confirming a conjecture of Kaledin. More generally, given an associative algebra A0 over $\bF_p$ and its flat deformation An over $\bZ/p^{n+1}\bZ$ we prove that under a certain non-degeneracy condition the center of An is isomorphic to the ring of length n+1 Witt vectors over the center of A0.
Added: Sep 28, 2015
Article
Verbitsky M. Compositio Mathematica. 2007. Vol. 143. No. 6. P. 1576-1592.
Added: Oct 4, 2010
Article
Bogomolov F. A., Amerik E., Rovinsky M. Compositio Mathematica. 2011. Vol. 148. No. 6. P. 1819-1842.
Added: Oct 13, 2011
Article
Rubtsov V., Enriquez B., Feigin B. L. Compositio Mathematica. 1998. Vol. 110. No. 1. P. 1-16.
Added: Jun 1, 2010
Article
Hacking P., Prokhorov Y. Compositio Mathematica. 2010. Vol. 146. No. 1. P. 169-192.
Added: Feb 18, 2011
Article
Cheltsov I., Dubouloz A., Park J. Compositio Mathematica. 2018. Vol. 154. No. 11. P. 2462-2484.

We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.

Added: Oct 17, 2018
Article
Brav C. I., Thomas H. Compositio Mathematica. 2014. Vol. 150. No. 3. P. 343-333.

We show that some hypergeometric monodromy groups in Sp(4,Z) split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank 2. In particular, we show that the monodromy of the natural quotient of the Dwork family of quintic threefolds in P^{4} splits as Z*Z/5. As a consequence, for a smooth quintic threefold X we show that a certain group of autoequivalences of the bounded derived category of coherent sheaves is an Artin group of dihedral type.

Added: Sep 29, 2014
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