Algebraic Geometry Seminar-Fall 2021

Title: On semi-orthogonal decompositions of flag varieties over the integers

• Timing(Via Zoom):December 17, 2PM


• Speaker: A. Samokhin, IITP-RAS


• Abstract:We will discuss semi-orthogonal decompositions of flag varieties over the integers and some applications to modular representation theory.

Title:

• Timing(Via Zoom):December 10


• Speaker: Available


• Abstract:

Title:Lagrangian fibrations on hyper-Kähler fourfolds

• Timing(Via Zoom):December 3, 3:30PM


• Speaker: Emmanuele Macri, Université Paris-Saclay


• Abstract:We will present joint work in progress with Olivier Debarre, Daniel Huybrechts and Claire Voisin on the SYZ hyper-Kähler conjecture for fourfolds under certain topological assumptions.

Title:Stability of zeta statistics

• Timing(Via Zoom):Friday November 26, 2PM


• Speaker:Ronno Das, University of Copenhagen


• Abstract:Many natural families of varieties, for example spaces of zero cycles or spaces arising from Bertini problems, demonstrate stability of both point counts over finite fields and motivic classes. However, neither of these two forms of stability implies the other. We provide a common framework for the two forms of stability and prove that the stronger property of ``Hadamard convergence'' is satisfied by many examples arising from zero cycles. Time permitting, I will also talk about connections with cohomological stability and a unified approach to proofs of these different forms of stability. This talk is based on joint work with Margaret Bilu and Sean Howe.

Title: Symplectic orbits of unimodular row

• Timing(Via Zoom): Friday October 29, 2PM


• Speaker: Tariq Syed, PDMI, St Petersburg


• Abstract:The generalized Serre question asks whether vector bundles over topologically contractible smooth affine complex varieties are always trivial. The question has been answered affirmatively in dimensions 1 and 2 and classification results for vector bundles over threefolds give a necessary and sufficient criterion in dimension 3. By studying symplectic orbits of unimodular rows, it is possible to give a sufficient criterion for the triviality of all vector bundles over smooth affine complex fourfolds. Assuming a conjecture by Asok-Østvær, this criterion would yield a positive answer to the generalized Serre question in dimension 4. In my talk, I will survey the results on the generalized Serre question in lower dimensions and explain how to prove the result in dimension 4.

Title:Borel-Laplace multi-transform, and integral representations of solutions of qDEs

• Timing(Via Zoom):Friday October 22, 2pm


• Speaker: Giordano Cotti, University of Lisbon


• Abstract:The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of its solutions conjecturally rules also the topology and complex geometry of X. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type, and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs. Based on arXiv:2005.08262.

Title: Rational points on symmetric powers of projective spaces

• Timing(Via Zoom):Friday October 8, 2:30 PM


• Speaker: Kevin Destagnol, Université Paris-Saclay, Orsay, France


• Abstract:Recently, Ellenberg, Satriano and Zureick-Brown proposed a unified stacky version of Manin and Malle's conjectures where Manin's conjecture predicts the quantitative behavior of rational points of bounded height on Fano varieties in terms of geometric invariants of the underlying variety and Malle's conjecture gives predictions about the number of number fields with prescribed Galois group of bounded discriminant. In this talk, we provide elements in favour of this new conjecture by studying rational points on some symmetric powers of projective spaces.

Title: Ordinary Enriques surfaces

• Timing(Via Zoom):Friday October 1, 2:30 PM


• Speaker: Sofia Tirabassi, University of Stockholm University


• Abstract: We introduce the notion of ordinary Enriques surfaces and we we use it to prove Torelli type theorems which are, roughly speaking, an equivariant version of Nygaard and Taelman's Torelli theorems for ordinary K3 surfaces over finite fields. This is a joint work with R. Laface.

Title: Subvarieties of geometric genus zero of a very general hypersurface

• Timing(Via Zoom):Friday September 24, 2PM


• Speaker: Takeshi Abe, Kumamoto University


• Abstract:Let X be a very general hypersurface of degree d in a n-dimensional complex projective space P^n and Y be a k-dimensional subvariety of X with geometric genus zero. Clemens-Ran showed that if d>= (3n-k+2)/2, then Y is contained in the union of lines in X. In the talk, we will consider the case d>(7n-2k+18)/5.

Title:Nonexistence of semi-orthogonal decompositions and the base locus of the paracanonical systems

• Timing(Via Zoom):Friday September 17, 2PM


• Speaker: Xun Lin, Tsinghua University


• Abstract:In the seminar, I will introduce a base locus of the paracanonical systems, and shows that the nonexistence of semi-orthogonal decompositions is governed by this new base locus. As an application, I will complete the picture of the indecomposability of derived categories of the symmetric product of curves.

Title:Some results on Seshadri constants.

• Timing(Via Zoom):Thursday September 9, 4 PM


• Speaker: Krishna Hanumanthu, Chennai Mathematical Institute


• Abstract:Let X be a projective variety and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is the infimum, over all curves C passing through x, of the ratios (L.C)/m, where (L.C) denotes the intersection product of L and C and m is the multiplicity of C at x. These constants were defined by J.-P. Demailly in 1990 and they shed light on the local behaviour of L at x and even say something about the nature of L and X. Their study is now an active area of research. We will give an overview of the current work in this area and discuss some recent results on Grassmann bundles over curves and Bott towers.

Spring 2021

Title:Calabi problem for smooth Fano threefolds

• Timing(Via Zoom):Wednesday May 12, 3PM


• Speaker: Ivan Cheltsov, University of Edinburgh


• Abstract:In this talk I will explain which three-dimensional complex Fano manifolds admit Kahler-Einstein metrics.

Title: On very stable bundles over curves

• Timing(Via Zoom):Friday April 23, 3:00PM


• Speaker: Christian Pauly, University de Nice Sophia-Antipolis


• Abstract:A very stable vector bundle over a curve is a vector bundle having no non-zero nilpotent Higgs fields. They were introduced by Drinfeld and studied by Laumon in connection with the nilpotent cone of the Hitchin system. According to Drinfeld non-very stable bundles, also called wobbly bundles, form a divisor in the moduli space of vector bundles. In this talk I will try to explain the motivations for studying the properties of wobbly divisors, with a special focus on the rank-2 (joint work with S. Pal) and rank-3 case (joint work with A. Peon-Nieto).

Title:A Thurston compactification for stability conditions

• Timing(Via Zoom):Friday April 16, 2:00 PM


• Speaker: Anand Deopurkar, Australian National University


• Abstract:There is a fascinating parallel between curves on a surface and objects in a triangulated category. This parallel allows us to transfer intuition, techniques, and theorems between the two seemingly distant worlds. I will explain a construction (joint with Asilata Bapat and Tony Licata) of the space of Bridgeland stability conditions on a suitable triangulated category, motivated by the Thurston compactification of the Teichmuller space.

Title: GAGA type conjecture for the Brauer group via derived geometry

• Timing(Via Zoom):Friday April 9, 2PM


• Speaker: Federico Binda, University of Milan


• Abstract:In Brauer III, Grothendieck considered the problem of comparing the cohomological Brauer group Br(X) = H^2_{et}(X,G_m) of a scheme X, proper and flat over a henselian DVR R, and the inverse limit of the Brauer groups \varprojlim_nBr(X_n), where X_n = X\otimes_R R/m^n. He proved that the canonical map Br(X) \to \varprojlim_n Br(X_n) is injective under a number of restrictions, notably when X is regular and if the inverse system Pic(X_n) satisfies a Mittag-Leffler condition (for example if the relative Picard scheme PicX/S of X/S is smooth). Nevertheless, he left as an open problem the question on whether the formal injectivity holds under less restrictive assumptions. Thanks to the machinery of derived algebraic geometry and the results of Toen on derived Azumaya algebras and derived Morita theory, we are able to rephrase Grothendieck's question in terms of a formal GAGA-type problem for smooth and proper categories, enriched over the \infty-category QCoh(X) of quasi-coherent O_X-modules. In this framework we can show that Grothendieck's injectivity conjecture always holds (even without flatness assumption on the structure morphism) for a proper derived scheme X \to S where S is the spectrum of any complete Noetherian local ring, if we are willing to replace the inverse limit \varprojlim_n Br(X_n) with the Brauer group Br(X) of the formal scheme \mathfrak{X} given by the colimit of the thickenings X_n. The obstruction involving the inverse system Pic(X_n) considered by Grothendieck appears naturally in the Milnor sequence for a certain tower of spaces. This is a joint work in progress with Mauro Porta (IRMA, Strasbourg).

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• Timing(Via Zoom):Friday March 26


• Speaker: Available


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Title:Homological projective duality and categorical joins

• Timing(Via Zoom):March 19, 2:00PM


• Speaker:Alexander Kuznetsov, HSE-Moscow


• Abstract: I will overview the somewhat old theory of homological projective duality and the relatively new construction of categorical joins. Both allow to construct and describe interesting semiorthogonal decompositions for algebraic varieties and I will try to illustrate the theory by examples.

Title:A non-archimedean definable Chow theorem

• Timing(Via Zoom):March 12, 10:00AM


• Speaker: Abhishek Oswal, IAS, Princeton


• Abstract:In recent years, o-minimality has found some striking applications to diophantine geometry. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame' property is the following surprising generalization of Chow's theorem proved by Peterzil and Starchenko - A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this talk, we shall explore a non-archimedean analogue of an o-minimal structure and a version of the definable Chow theorem in this context.

Title:Title: Counting (twisted) Fourier Mukai partners of an Ordinary K3 surface.

• Timing(Via Zoom):March 5, 3:00PM


• Speaker:Tanya Kaushal Srivastava, IST Austria


• Abstract:In this talk, I will give counting formulas for (twisted) Fourier-Mukai partners of an ordinary K3 surface and relate some of these counts with interesting open questions in Number theory. If time permits, I will also discuss some results about the derived autoequivalence group of K3 surfaces. This talk is based on the work done in my PhD thesis and joint work(in progress) with Sofia Tirabassi and Piotr Achinger.

Title:Arakelov Geometry of Modular Curves X_0(p^2)

• Timing(Via Zoom):Friday February 26


• Speaker:Chitrabhanu Chaudhury, NISER, Bhubaneshwar


• Abstract::I shall outline the construction of a semisimple and minimal regular model for X_0(p^2) over an appropriate number field. This will be a regular scheme over the spectrum of the ring of integers of that number field, such that the fibres are complete curves with at worst nodal singularities and satisfying certain stability conditions. The generic fibre of the model is isomorphic to X_0(p^2). The purpose of this construction is to use the theory developed by Shou-Wu Zhang, using Arakelov theory, for proving an effective version of a conjecture by Bogomolov in this special case of modular curves X_0(p^2).

Title: Motivic-to-Chow isomorphisms for quotient stacks

• Timing(Via Zoom):Friday February 19, 3:00PM


• Speaker: Neeraj Deshmukh, IISER Pune


• Abstract:For smooth schemes over a field, Voevodsky proved comparison isomorphisms between motivic cohomology and Bloch's higher Chow groups. Until recently, such a theorem for quotient stacks was known only with rational coefficients by the work of R. Joshua. In this talk, we will discuss how to extend these motivic-to-chow isomorphisms to quotient stacks with integer coefficients. To prove such a theorem, we will define the motive of an algebraic stack in the Nisnevich topology, and also construct a presentation for this motive by simplicial schemes in the (unstable) A^1-homotopy category (and, therefore, the triangulated category of mixed motives). This is joint work with Utsav Choudhury and Amit Hogadi.

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• Timing(Via Zoom):Friday February 12


• Speaker: Available


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Title:Equivariant Virtual Classes

• Timing(Via Zoom):February 5, 3:00PM


• Speaker:Bhamidi Sreedhar, KIAS


• Abstract:For a scheme with a group scheme action and an equivariant perfect obstruction theory we shall outline a proof of a virtual equivariant Riemann-Roch theorem. This is an extension of a result of Fantechi-Göttsche to the equivariant context. We shall further discuss a non-abelian virtual localization theorem, relating the virtual classes of the stack and virtual classes of the associated inertia stack for a naturally induced perfect obstruction theory. This talk is based on joint work with Charanya Ravi.

Title:Projective manifolds whose tangent bundle contains a strictly nef subsheaf

• Timing(Via Zoom):Friday January 29, 3:00PM


• Speaker: Wenhao Ou, Academy of Mathematics and System Sciences, Beijing


• Abstract:After a theorem of Andreatta and Wisniewski, if the tangent bundle of a projective manifold X contains an ample subsheaf, then X is isomorphic to the projective space. We show that, if the tangent bundle contains a strictly nef subsheaf, then X is a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group of X is virtually abelian, then X is isomorphic to a projective space. This is joint with Jie Liu and Xiaokui Yang.

Title:Compactifications of moduli of points and lines in the projective plane

• Timing(Via Zoom):Friday January 22, 3:00PM


• Speaker:Luca Schaffler, KTH, Stockhom


• Abstract:Projective duality identifies the moduli space B_n parametrizing configurations of $n$ general points in the projective plane with X(3,n), parametrizing configurations of n general lines in the dual projective plane. When considering degenerations of such objects, it is interesting to compare different compactifications of the above moduli spaces. In this work, we consider Gerritzen-Piwek's compactification of B_n and Kapranov's Chow quotient compactification of X(3,n), and we show that they have isomorphic normalizations. We also construct an alternative compactification parametrizing all possible n-pointed central fibers of Mustafin joins associated to one-parameter degenerations of n points in the projective plane, which was proposed by Gerritzen and Piwek. We fully describe this alternative compactification for n=5,6 . This is joint work with Jenia Tevelev.

Title:Irregular fibrations and derived categories

• Timing(Via Zoom):Friday January 15, 3:00PM


• Speaker:Luigi Lombardi, University of Milan


• Abstract:In this seminar I will show that an equivalence of derived categories of sheaves of smooth projective varieties preserves some specific classes of fibrations over varieties of maximal Albanese dimension. These types of fibrations, called chi-positive higher irrational pencils, can be thought as an extension to higher-dimension of the notion of a irrational pencil over a smooth curve of genus greater or equal to two. This is a joint work with F. Caucci and G. Pareschi.

Fall 2020

Title: Blown-up toric surfaces with non-polyhedral effective cone

• Timing(Via Zoom):Friday December 18, 3:00 PM.


• Speaker: Ana-Maria Castravet, Universite de Versailles


• Abstract:I will report on recent joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia. We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudoeffective cone, both in characteristic 0 and in prime characteristic. As a consequence, we prove that the pseudo-effective cone of the Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, is not polyhedral if n>=10 in characteristic 0 and in positive characteristic for an infinite set of primes of positive density. In particular, these moduli spaces are not Mori dream spaces even in positive characteristic.

Title:Delta Geometry and Group Schemes

• Timing(Via Zoom):Friday December 11, 3:00PM.


• Speaker: Arnab Saha, IIT-Gandhinagar


• Abstract:Delta geometry was initially developed by A. Buium as an analogue of differential algebra. In this theory, a non-linear operator \delta with respect to a non-zero prime \mathfrak{p} on a fixed number ring plays the analogous role of a derivation in the case of function rings. Such a \delta comes from the ring of Witt vectors and its associated lift of Frobenius. As an example, \delta on the ring of integers \mathbb{Z} is the Fermat quotient operator given by \delta x = \frac{x-x^p}{p}. Now given a group scheme G defined over a ring with a \delta on it, for every n, one can canonically define the arithmetic jet space J^nG which is an extension of G by another group scheme N^n. In this talk, we will discuss the structure of N^n in detail. We will then look into the application of the above in p-adic Hodge theory in the case when G is an abelian scheme.

Title: On existence of weak Zariski decomposition on projective bundles

• Timing(Via Zoom):Friday December 4, 3:00 PM.


• Speaker: Snehajit Misra, TIFR


• Abstract: Zariski introduced a decomposition on surfaces to dismantle orthogonally (pseudo-)effective divisors into positive and negative parts, both of which enjoy the useful geometric properties. Many attempts has been made to generalise this decomposition on higher dimensional varieties, for instance the Fujita-Zariski decomposition and the CKM-Zariski decomposition. Recently, a weeker notion of this decomposition has been suggested by Birkar. In this talk, we will discuss about existence of weak Zariski decomposition of pseudo-effective divisors on projective bundles over higher dimensional varieties in order to answer a question posed by Birkar.

Title: Algebraic cycles and refined unramified cohomology

• Timing(Via Zoom):Friday November 27, 2:30 PM.


• Speaker: Stefan Schreieder, University of Hannover.


• Abstract:We introduce refined unramified cohomology groups. This notion allows us to give in arbitrary degree a cohomological interpretation of the failure of integral Hodge- or Tate-type conjectures, of l-adic Griffiths groups, and of the subgroup of the Griffiths group that consists of torsion classes with trivial transcendental Abel--Jacobi invariant. Our approach simplifies and generalizes to cycles of arbitrary codimension previous results of Bloch-Ogus, Colliot-Thelene-Voisin, Voisin, and Ma that concerned cycles of codimension two or three. As an application, we give for any i>2 the first example of a uniruled smooth complex projective variety for which the integral Hodge conjecture fails for codimension i-cycles in a way that cannot be explained by the failure on any lower-dimensional variety.

Title: The cobordism ring of algebraic involutions

• Timing(Via Zoom):Friday November 20, 3PM


• Speaker: Olivier Haution, LMU, Munich


• Abstract:I will provide an elementary definition of the cobordism ring of involutions of smooth projective varieties over a field (of characteristic not 2). I will describe its structure, and give explicit "stable" polynomial generators. I will draw some concrete consequences concerning the geometry of fixed loci of involutions, in terms of Chern numbers. I will in particular mention an algebraic version of Boardman's five halves theorem.

Title:Holomorphic 1-forms and geometry

• Timing(Via Zoom):Friday November 13. 4:30PM.


• Speaker: Yajnaseni Dutta, University of Bonn


• Abstract: In this talk I will discuss various consequences of zeros global holomorphic 1-forms on smooth projective varieties. It has been indicated by plethora of results that a lot of the geometry is dictated by 1-forms that admit zeros. I will present some old and new results in this direction. Later I will focus on a special set of such 1-forms which arise out of both the generic vanishing theory and the decomposition theorem. This is a joint work with Feng Hao and Yongqiang Liu.

Title: Geometric invariants and geometric consistency of Manin's conjecture.

• Timing(Via Zoom):Friday November 6, 5:30PM.


• Speaker: Akash Kumar Sengupta, Columbia University


• Abstract:Let X be a Fano variety with an associated height function defined over a number field. Manin's conjecture predicts that, after removing a thin set, the asymptotic growth of the number of rational points of bounded height on X is controlled by certain geometric invariants (e.g. the Fujita invariant of X). I will talk about how to use birational geometric methods to study the behaviour of these invariants and propose a geometric description of the thin set in Manin's conjecture. Part of this is joint work with Brian Lehmann and Sho Tanimoto.

Title: Residual categories of Grassmannians

• Timing(Via Zoom):Friday Oct 30, 3:00PM.


• Speaker: Maxim Smirnov, University of Augsburg


• Abstract:Exceptional collections in derived categories of coherent sheaves have a long history going back to the pioneering work of A. Beilinson. After recalling the general setup, I will concentrate on some recent developments inspired by the homological mirror symmetry. Namely, I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.

Title: Semiorthogonal decompositions for singular varieties

• Timing(Via Zoom):Friday Oct 23, 3:30PM.


• Speaker: Evgeny Shinder, University of Sheffield


• Abstract:I will define a semiorthogonal decomposition for derived categories of singular projective varieties due to Kawamata, into finite-dimensional algebras, generalizing the concept of an exceptional collection in the smooth case. I will present known constructions of these for nodal curves (Burban), torsion-free toric surfaces (Karmazyn-Kuznetsov-Shinder) and two nodal threefolds (Kawamata). Finally, I will explain obstructions coming from the K_{-1} group, and how it translates to maximal nonfactoriality in the nodal threefold case. This is joint work with M. Kalck and N. Pavic.