Title: Non-existence of phantoms on some rational surfaces

• Timing: Monday, 02 September, 2024 at 04:00 pm


• Speaker: Swarnava Mukhopadhyay


• Abstract:A phantom on a smooth projective variety is a non-trivial admissible subcategory of the derived category of coherent sheaves that is invisible to invariants like the Grothendieck group or Hochschild homology. There had been a general expectation that varieties admitting a full exceptional collection should not have phantoms however this was upended by the work of Efimov. More recently, Krah’s elegant example exhibits phantoms on the blowup of the projective plane at ten points in a general position. It is natural to wonder whether phantoms still exist if one takes the blowup at points that are not in general position. In this talk, following the recent work of Borisov-Kimboi (arXiv.2405.01683), we consider the above question for surfaces that are blowups of the projective plane at a finite set of points on a smooth cubic curve, in a very general position on the curve.

Title: Categorical absorption of singularities

• Timing: Monday, 09 September, 2024 at 04:00 pm


• Speaker: Umesh Dubey


• Abstract: The idea of categorically absorbing singularities was introduced by Kuznetsov and Shinder for singular varieties as admissible subcategories of derived categories whose complement is smooth and proper. The authors have described the absorption of singularities in some special cases in terms of $\mathbb{P}^{\infty}$-objects. In this special case, it was referred to as a categorical ordinary double point. In this seminar, we will briefly overview certain sections of the pre-print (arXiv:2207.06477v5 [math.AG]) by A. Kuznetsov and E. Shinder.

Title: An equivariant Gabber presentation lemma

• Timing: Monday, 30 September, 2024 at 04:00 pm


• Speaker:Anand Sawant


• Abstract:Gabber's presentation lemma is an important tool in motivic algebraic topology over a field. For instance, it is used in a crucial way to explicitly write down resolutions of certain Nisnevich sheaves with nice $\mathbb A^1$-invariance properties. In this talk, I will present an equivariant version of Gabber's presentation lemma in the setting of schemes over a field of characteristic different from 2 endowed with an action of the finite group of order 2. The talk will be based on the preprint https://arxiv.org/abs/2310.08125 by Tom Bachmann.

Title: Modified Derksen invariant

• Timing: Monday, 7 October, 2024 at 04:00 pm


• Speaker: Parnashree Ghosh


• Abstract: Abstract: Let k be a field of characteristic zero. In this talk we will introduce an invariant related to G a -actions on affine varieties over k, namely the modified Derksen invariant. We will prove that when X is a cylinder over an affine variety Y , then there are only three possible structures of this invariant. We will see examples of varieties of each kind. This is based on the following work of Boldyrev, Gaifullin and Shafarevich: Modified Derksen invariant, https://arxiv.org/abs/2312.08421v1.

Title: Naive A^1-connectedness of retract rational varieties

• Timing: Monday, 14 October, 2024 at 04:00 pm


• Speaker: Biman Roy


• Abstract: In A^1-homotopy theory, introduced by Morel-Voevodsky, A^1-connected component π0^{A1}(X) of a smooth variety X plays the role of topological connected component of a topological space. A crude approximation of π0^{A1}(X) is the A1-chain connected component sheaf S(X) which involves the naive A1-homotopies. Asok-Morel, Balwe-Hogadi-Sawant showed that a smooth proper variety over a field k is A1-connected if and only if it is A1-chain connected i.e. S(X)(Spec F) is trivial, for every finitely generated separable field extension F/k. Asok-Morel, Kahn-Sujatha proved that a smooth retract rational proper variety over a field k is A1-chain connected. In a recent work Balwe-Rani improved this result by showing that a smooth retract rational proper variety over an infinite field k is naively A1-connected i.e. S(X) is the trivial sheaf. This talk will be based on the article https://doi.org/10.48550/arXiv.2307.04371 by Chetan Balwe and Bandana Rani.

Title: Period-index problem for hyperelliptic curves

• Timing: Monday, 21 October, 2024 at 04:00 pm


• Speaker: Saurabh Gosavi


• Abstract: Let $F$ be a field and $\alpha$ be in the Brauer group $Br(F)$. The period of $\alpha$ is its order in $Br(F)$, and the index of $\alpha$ is the square root of the dimension of the (unique) underlying division algebra of $\alpha$. It is well-known that the period divides the index and that they share the same prime factors, and so there exists $N(\alpha) \geq 1$ such that $ind(\alpha)$ divides $per(\alpha)^{N(\alpha)}$. When $F$ is the function field of a curve over a number-field, it is expected that there should exist a uniformly bound for the exponent $N(\alpha)$, i.e., there should exist an integer $N \geq 1$ such that for every $\alpha$ in $Br(F)$, one has $ind(\alpha)$ divides $per(\alpha)^{N}$. This is still an open problem. In this preprint, R. Parimala and J. Iyer show that when $C$ is a hyperelliptic curve over a totally imaginary number field and $\alpha$ is a period two Brauer class that is locally trivial, i.e., $\alpha$ becomes trivial when restricted to every completion of $F$, index of $\alpha$ also equals two. Following M. Lieblich, they prove this by showing that there is a rational point in the moduli space of stable $\alpha$-twisted locally free sheaves of rank two on $C$. Based on https://arxiv.org/pdf/2201.12780

Title: On decompositions for Fano schemes of intersections of two quadrics

• Timing: Monday, 28 October, 2024 at 04:00 pm


• Speaker: Gopinath Sahoo


• Abstract: In the preprint - https://arxiv.org/abs/2403.12517, the authors propose conjectural semi-orthogonal decompositions for Fano schemes of linear subspaces on intersections of two quadrics, in terms of symmetric powers of the associated curves. They provide evidence for the conjectures by obtaining formulae for the Hodge numbers of these Fano schemes. In this talk, I will focus on the odd-dimensional case and give an overview of these conjectures and briefly explain the evidence provided by the authors.

Title: Free summands of stably free modules.

• Timing: Monday, 11 November, 2024 at 04:00 pm


• Speaker: Sandeep S


• Abstract: In this preprint by Ben Williams and W S Gant, (https://arxiv.org/abs/2409.15445), the authors show that a stably free module $P$ over a ring $R$ satisfying "P \oplus R \cong R^{24m}$ has a free direct summand of rank 2 when $\mathbb{Q}\subset R$. This gives a partial converse to a theorem of Raynaud. The proof uses the methods of $\mathbb{A}^1$- homotopy theory. We will state the main results and outline the proof.

Title: Existence of global Neron models beyond semi-abelian varieties

• Timing: Monday, 18 November, 2024 at 04:00 pm


• Speaker: Subhadip Majumder


• Abstract: Let $S$ be an excellent Dedekind scheme with function field $K$. For a smooth connected commutative group scheme $G_K$ over $K$, it was conjectured in the book ``N\'eron Models" by Bosch, L\"utkebohmert and Raynaud that a N\'eron-lift model for $G_K$ over $S$ exists if it doesn't contain $\mathbb{G}_a$. This conjecture is known to be true if $S$ is local or $char \ S=0$. However, if $char \ S=p>0$, then this is an open question. In the preprint arxiv.org/abs/2310.14567v2, the authors showed that the conjecture holds when the residue fields of $S$ at closed points are perfect, but is false in general. In this talk, I will discuss the proof of this result.

Title:

• Timing: Monday, 09 December, 2024 at 04:00 pm


• Speaker: Samiron


• Abstract:

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Title: Geometrically pure G_a-actions

• Timing:Tuesday, 23 April, 2024 at 04:00 pm


• Speaker: Buddhadev Hajra


• Abstract:In the recent article doi.org/10.1016/j.jalgebra.2024.02.041, Masayoshi Miyanishi defined when an action of the additive algebraic group scheme G_a on an affine variety Y is said to be geometrically pure. Such a G_a-action guarantees the existence of a geometric quotient of Y by that action whenever Y is assumed to be normal. Namely, there exists the quotient morphism q: Y \to X to a normal affine variety X such that the graph morphism \Phi: G_a \times Y \to Y \times_{X} Y is an isomorphism. The geometric pure-ness of the given G_a-action is the first criterion ever to ensure the existence of a geometric quotient Y/G_a. As a consequence, an algebraic characterization of the affine 3-space is obtained in the above article. In my lecture, I will present this content through certain interesting properties of pure subrings of commutative rings.

Title:The Singer-Hopf conjecture for Algebraic Varieties.

• Timing:Tuesday, 16 April, 2024 at 04:00 pm


• Speaker: Ashutosh Roychoudhury


• Abstract:In the 1930s, Hopf conjectured that a closed 2d-dimensional compact Riemannian manifold X with non-positive sectional curvature satisfies $(-1)^d \xi(X) \ge 0$, later strengthened by Singer for aspherical manifolds (i.e., those with contractible universal cover). In the context of algebraic varieties, one can ask much more general questions about the euler characteristic of perverse sheaves even over positive characteristic. In this talk I would like to start with a survey of the known results and methods for Abelian Varieties and Tori(here an interesting perspective is offered by the Fourier-Mellin transform on Constructible Sheaves) following which, I'd like to present some results in the paper: Perverse Sheaves on Varieties with large fundamental group (https://arxiv.org/pdf/2109.07887.pdf) by D.Arapura and B.Wang.

Title:Cohomological flatness in dimension 0

• Timing:Tuesday, 2 April, 2024 at 04:00 pm


• Speaker: Yogish Holla


• Abstract: In this lecture we will study the notion of Cohomological flatness in dimension 0 and give sufficient conditions numerically for it over discrete valuation rings following the results of Offer Gabber and Rémi Lodh (arXiv: 2403.01831v1 [math.AG] 4 March 2024), generalising a classical result of Raynaud.

Title: On log motivic cohomology for semistable varieties

• Timing:Tuesday, 12 March, 2024 at 04:00 pm


• Speaker: Subhodip Majumder


• Abstract:Motivic cohomology is a cohomology theory for schemes whose existence was conjectured by Beilinson and Lichtenbaum in mid 80's and was formaily defined by Voevodsky in mid 90's for smooth schemes. These cohomology groups are closely related to algebraic K-theory via some Atiyah-Hirzebruch type spectral sequence. Motivic cohomology theory for singular schemes has also been developed by various authors throughout the years. In the preprint "https://arxiv.org/abs/2108.02845", the authors have defined a logarithamic version of motivic cohomology theory for a semistable variety considering the log structure associated with it. In this talk we are going to discuss the importance of the theory, its constructions and main results of the preprint.

Title: Fully faithful functors, skyscraper sheaves and birational equivalence

• Timing:Tuesday, 5 March, 2024 at 04:00 pm


• Speaker: Sandeep S


• Abstract:For a smooth projective variety $X$, the bounded derived category $D^b(X)$ encodes a great deal of information about $X$ by the results of Bondal, Orlov, Kawamata, etc. The preprint https://arxiv.org/abs/2402.1257 by Li, Lin and Zhao shows that if there is a fully faithful exact functor between $D^b(X)$ and $D^b(Y)$ sending some skyscraper sheaf to a skyscraper sheaf, then $X$ and $Y$ are birational. In the talk, we will state the main results and outline the proofs.

Title: The relative Du Bois complex

• Timing:Tuesday, 20 February, 2024 at 04:00 pm


• Speaker: Arnab Roy


• Abstract:For a smooth projective variety X over complex numbers, one gets a Hodge structure(pure) on the singular cohomology of X in complex coefficients. Which can be obtained by the spectral sequence associated to the algebraic De Rham complex of X. In case of nonsingular varieties the singular cohomology admits Deligne's Mixed Hodge structure and the Du Bois complex (introduced by P. Du Bois in 1981) is the complex whose associated spectral sequence recovers the mixed Hodge structure. In this sense the Du Bois complex is a generalization of the algebraic De Rham complex. The Du Bois complex is a filtered complex, whose graded parts are in D^b_coh(X), these graded pieces are interesting on their own, as noted in Steenbrink(1985) these pieces provide generalization of Kodaira type vanishing theorems for singular spaces. The preprint named 'The relative Du Bois complex -- on a question of S. Zucker' { https://arxiv.org/abs/2307.07192}by Sándor J Kovács, Behrouz Taji takes the first step to generalize the Du Bois complex for a family (More precisely they construct the relative Du Bois complex when the base is a smooth curve). Similar generalisation of the graded pieces was already done by the first named author in the paper "smooth families over rational and elliptic curves". I will discuss the construction of the relative complex following the preprint and try to motivate why one should care for such a construction, If time permits I would like to discuss some applications of the relative graded pieces.

Title:Combing a hedgehog over a field

• Timing:Tuesday, 13 February, 2024 at 04:00 pm


• Speaker: Anand Sawant


• Abstract:The title refers to the eponymous preprint {\tt https://arxiv.org/abs/2311.07486} by A. Anayevskiy and M. Levine, which investigates the question of the existence of a non-vanishing section of the tangent bundle on a smooth affine quadric hypersurface over a perfect field of characteristic different from 2. I will outline the proofs of the main results in the above preprint. The main idea is to employ appropriate quadratic enrichments of the main ingredients in the classical proof of the hedgehog (or the hairy ball) theorem for real spheres.

Title:On the Grothendieck Serre Conjecture for smooth schemes over Pr\"ufer bases

• Timing: February 6, 4PM


• Speaker: Ritankar Nath


• Abstract:The Grothendieck-Serre Conjecture tells us that given a regular local ring $R$ and a reductive $R$-group scheme $G$, every non-trivial $G$-torsor on $R$ remains non-trivial over its fraction field. In the talk, several known cases of the Grothendieck-Serre conjecture will be discussed, particularly its analogue when $R$ is a semilocal Pr\"ufer domain. Then, we will explicitly prove the conjecture for some cases when $R$ is the localisation at finitely many points of a smooth scheme over a semilocal Pr\"ufer domain. We will mainly follow the preprint https://arxiv.org/abs/2301.12460 titled "Grothendieck-Serre for constant reductive group schemes" by Ning Guo and Fei Liu.

Title:Cancelled

• Timing:December 12, 4PM


• Speaker: Ritankar Nath


• Abstract:

Title: Cancelled

• Timing: December 5, 4PM


• Speaker: Sourav Sen


• Abstract:

Title:

• Timing:November 28, 4PM


• Speaker: Debojyoti Bhattacharya


• Abstract:

Title: Bounding Geometrically Integral del Pezzo Surfaces

• Timing:November 21, 4PM


• Speaker: Omprokash Das


• Abstract:This is a 2023 article by Fabio Bernasconi and Gebhard Martin (https://arxiv.org/pdf/2306.07000.pdf). They prove various boundedness statements of singular del Pezzo surfaces defined over a non-algebraically closed field of positive characteristics. These kinds of results are very useful for studying Mori fiber spaces of dimension 3 in positive characteristic (defined over an algebraically closed field) and in general the BAB Conjecture in positive characteristic.

Title: Saturation bounds for smooth varieties

• Timing:November 7, 4PM


• Speaker: Vijaylaxmi Trivedi


• Abstract: In the above titled paper (L. Ein, H.T. Ha and R. Lazarsfeld, Algebra and Number Theory, Volume 16, 2022, No. 6) the authors give bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example they show that a clasical statement due to Macaulay for zero dimensional complete intersection ideals holds for smooth variety. For curves, they give bound on the saturation degree of powers in terms of the regularity. Here we give some background and sketch proof of some of the results of the paper.

Title: HFD objects and their application to the indecomposability of derived categories

• Timing: October 17, 4PM


• Speaker: Gopinath Sahu


• Abstract:Orlov introduced a useful finiteness property called homologically finite for objects of a triangulated category. The subcategory of homologically finite objects behaves well with respect to admissible semi-orthogonal decompositions. He proved that for Noetherian separated schemes of finite Krull dimension with enough locally free sheaves, the homologically finite objects of the bounded derived category of coherent sheaves are precisely the perfect complexes. In the preprint: https://arxiv.org/abs/2211.09418v2, Kuznetsov, and Shinder suggest a modification of this notion. Given a small dg-enhanced triangulated category T, they introduce homologically finite-dimensional objects of the derived category of dg-modules over T. Unlike the notion of Orlov the new construction can be iterated, using which they have introduced reflexivity, and some other properties for triangulated categories. They have shown that the triangulated categories arising in geometric and algebraic contexts are reflexive. They have proved that the indecomposability of a reflexive triangulated category is equivalent to the indecomposability of its HFD partner, using which they have deduced the non-existence of semi-orthogonal decomposition for derived categories of various singular varieties.

Title: Full exceptional collections of vector bundles on rank 2 linear GIT quotients.

• Timing:October 10, 4PM


• Speaker: Jagadish Pine


• Abstract:Beilinson proved that the derived category of coherent sheaves $D^b_{coh}(\mathbb{P}^N)$ admits a full exceptional collection of line bundles. Viewing $\mathbb{P}^N$ as a GIT quotient $\mathbb{C}^{N+1}//G_m$, one can pose the general question: Given a split reductive group $G$, and a linear representation $X$, when does a GIT quotient $X^{ss}//G$ admit a full exceptional collection of vector bundles? Since the variety $X^{ss}//G$ might have singularities, one must work with quotient stacks [X^{ss}/G]. When the rank of $G$ is $2$, under certain assumptions, D^b_{coh}[X^{ss}/G] will provide a positive answer to this question. Using this result as building blocks, one can construct many examples that admit a full exceptional collection of vector bundles using quiver varieties. The talk is based on the arxiv preprint by Halpern-Leistner and Kimoi Kemboi titled "Full exceptional collections of vector bundles on rank $2$ linear GIT quotients". https://arxiv.org/abs/2202.12876

Title: Morphisms between homogeneous varieties

• Timing:October 3, 4PM


• Speaker: A. J. Parameswaran


• Abstract:There are many classical and obvious results on morphisms: like P^n cannot map (nonconstant) to P^m if m < n, every nonconstant map of P^n is finite and so on. We will racall some results of Tango on morphisms of Grassmannians. We will sketch a proof of the fact that every map of P^2 to SL_n/B is constant and conclude that if SL_n/P to SL_m/B is a nonconstant map then the parabolic P is a Borel subgroup of SL_n. We will extend this to showing some examples of maps of P^3 to homogeneous spaces by minimal parabolic subgroups and draw the attention to the recent conjecture of Shrawan Kumar.

Title: Phantoms on surfaces and mutations of exceptional collection

• Timing:September 12, 4PM


• Speaker: Swarnava Mukhopadhyay


• Abstract:An admissible subcategory in a triangulated category is called a phantom if it has trivial Hochschild homology and Grothedieck group. In this talk, we will discuss the recent work of Johannes Krah (arXiv:2304.01269) on the existence of phantoms on a rational surface and discuss its implications on conjectures about the fullness of exceptional collections of maximal length and on transitivity of the action of the mutations and shifts on the set of maximal exceptional collections. If time permits we will discuss the origin of the example in Krah's preprint. This builds upon earlier works of Krah (arXiv:2211.07724), Elagin-Lunts (MR3598503), Kuznetsov (MR3691718), Orlov-Kuleshov (MR1286839), and Vial (MR3570150). 

Title: Geometry of canonical extensions

• Timing:September 5, 4PM


• Speaker: Priyankur Chaudhury


• Abstract:This talk is based on the recent works Fano threefolds with affine canonical extensions (https://arxiv.org/abs/2211.11261), Stein complements in compact Kähler manifolds (https://arxiv.org/abs/2111.03303) by Horing and Peternell as well as earlier work Canonical complex extensions of Kähler manifolds (https://arxiv.org/abs/1807.01223)by Greb and Wong. It will be centered around the study of complements of smooth codimension 1 subvarieties in smooth projective varieties.
  It is well known that the complement of an ample subvariety of a projective variety is affine. Conversely, specifying that a smooth divisor has affine (or more generally, Stein) complement places strong restrictions on its normal bundle. A particularly interesting example of such complements is the so-called canonical extensions, first introduced by Greb and Wong. They are defined as follows: any hyperplane class on a smooth projective variety corresponds to a nontrivial extension E of its tangent bundle TX by the structure sheaf. Letting PE and P(TX) denote the respective projectivizations, the complement of P(TX) in PE is known as the canonical extension corresponding to the given class. Varieties with affine canonical extensions (such as flag varieties) have big tangent bundles. Conversely, Horing and Peternell conjecture that any projective manifold having an affine canonical extension is rational homogeneous. We will discuss this along with a string of related conjectures and present evidence in lower dimensions.

Title:Newton-Okounkov bodies and Picard numbers on surfaces

• Timing: 11 January 4PM, AG-77


• Speaker: Aditya Subramaniam


• Abstract:In this talk, we will discuss a preprint of Fernandez-Nickel-Roe 'Newton-Okounkov bodies and Picard numbers on surfaces' https://arxiv.org/pdf/2101.05338v1.pdf
  Newton-Okounkov bodies were introduced by A. Okounkov as a tool in representation theory; later Kaveh-Khovanskii and Lazarsfeld-Mustata developed a general theory with applications to both convex and algebraic geometry. In this preprint, the authors study the shapes of all Newton-Okounkov bodies of a given big divisor on a surface S with respect to all rank 2 valuations of $K(S)$. They obtain upper bounds for, and in many cases determine exactly, the possible numbers of vertices of these bodies. The upper bounds are expressed in terms of Picard numbers. They also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor determines the Picard number of S, and proves that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.

Title:Strictly nef divisors on K-trivial fourfolds

• Timing:18 January 4PM, AG-77


• Speaker: Priyankur Chaudhury


• Abstract:This talk will be based on the paper "Strictly nef divisors on K-trivial fourfolds" by Haidong Liu and Shinichi Matsumura : [2105.07259] Strictly nef divisors on K-trivial fourfolds (arxiv.org) A Cartier divisor L on a projective variety X is called strictly nef if its intersection number with every curve in X is strictly positive. Such divisors need not be ample; there are examples due to Mumford, Ramanujam and Subramanian. A conjecture of Serrano from the 90's predicts however that if X is smooth and projective, then the canonical divisor K of X added to any large enough multiple of L will be ample. The hardest case of this conjecture is the K-trivial case, where it is not even completely known in dimension 3. In the above paper, the authors confirm this conjecture for K-trivial fourfolds with vanishing irregularity, i.e any strictly nef divisor on such a fourfold must be ample. The proof consists of two main parts: 1. showing that any strictly nef divisor on a K-trivial fourfold has non-negative Kodaira dimension. This part uses analytic methods: the calculus of currents and multiplier ideal sheaves. 2. showing that any effective strictly nef divisor on a K-trivial fourfold is ample. This uses some standard techniques of the minimal model program.

Title:Geometric local systems on generic curves

• Timing:25 January 4PM, AG-77


• Speaker: Najmuddin Fakhruddin


• Abstract:The talk will be based on parts of the preprint entitled "Geometric local systems on very general curves and isomonodromy" by Aaron Landesman and Daniel Litt (arXiv:2022.00039). A complex local system $L$ on a smooth projective curve $X$ over $\mathbb{C}}$ is said to be "geometric" if there exists a smooth projective morphism $f:X \to U$, with $U$ a nonempty Zariski open subset of $X$, and an integer $i$ such that $L|_U$ is a direct summand of $R^i f_* \mathbb{C}_X$. The main result that we will discuss says that if the rank of such an $L$ is small compared to the genus of $X$, then the monodromy representation associated to $L$ must have finite image; this leads to counterexamples to conjectures of Budur--Wang and Esnault--Kerz.The proof uses methods from the theory of variations of Hodge structure.

Title:Vanishing cohomology and Betti bounds for complex projective hypersurfaces

• Timing:1 February 4PM, AG-77


• Speaker: Mohit Upamanyu


• Abstract:Let V be a (possibly singular) hypersurface in $P^n(C)$. In this talk we define the Deligne vanishing cycle complex. We prove a vanishing result for the hypercohomlogy of the Deligne vanishing cycle complex, and use it to derive various results about singular cohomology of V. This is work of LAURENTIU MAXIM, LAURENTIU PAUNESCU, AND MIHAI TIBAR. The preprint can be found at https://arxiv.org/pdf/2004.07686.pdf

Title:Reconstructing a variety from the Zariski topological space-Part I

• Timing:8 February 4PM, AG-77


• Speaker: V. Srinivas


• Abstract:The talk will give an introduction to results of Kollar, based on earlier work of Lieblich and Olsson, showing that the underlying Zariski topological space of a variety determines the variety, under suitable hypotheses. I will try to state results in some generality, and sketch proofs under some extra simplifying hypotheses, which will however show the main new features. References include arxiv preprints of Kollar and others, and also a Seminaire Bourbaki presentation of Cesnavicius from April, 2021.

Title:Reconstructing a variety from the Zariski topological space-Part II

• Timing:15 February 4PM, AG-77


• Speaker:V. Srinivas


• Abstract: The talk will give an introduction to results of Kollar, based on earlier work of Lieblich and Olsson, showing that the underlying Zariski topological space of a variety determines the variety, under suitable hypotheses. I will try to state results in some generality, and sketch proofs under some extra simplifying hypotheses, which will however show the main new features. References include arxiv preprints of Kollar and others, and also a Seminaire Bourbaki presentation of Cesnavicius from April, 2021.

Title:Finite generation of certain valuation semigroups on toric surfaces

• Timing:22 February 4PM, AG-77


• Speaker: Jyoti Dasgupta


• Abstract: In this talk, we will discuss a preprint of Klaus Altmann, Christian Haase, Alex Kuronya, Karin Schaller, and Lena Walter, available at https://arxiv.org/abs/2209.06044. Finite generation of semigroups or rings arising from geometric situations has been a question of interest for a long time. In this preprint, the authors consider valuation semigroups arising from Newton-Okounkov theory in the special case of toric surfaces. They provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank.

Title:Volumes of line bundles as limits on certain generically nonreduced schemes

• Timing:1 March 4PM, AG-77


• Speaker: Suprajo Das


• Abstract:In this talk we shall discuss a recent article (arXiv$2007.12925$v$2$) by Dale Cutkosky and Roberto Nunez. Let $X$ be a proper scheme of dimension $d$ over a field $k$ and let $\mathcal{L}$ be a line bundle on $X$. The \emph{volume} of $\mathcal{L}$ is defined as $$\mathrm{vol}(\mathcal{L}):= \limsup\limits_{n\to\infty}\dfrac{\dim_k H^0(X,\mathcal{L}^{\otimes n})}{n^d/d!}.$$ A fundamental problem is to study whether the $\limsup$ in the definition of volume can be replaced by a limit. It is known to exist as a limit on generically reduced schemes. In this preprint, the authors show that the volume exists as a limit in certain classes of schemes which are not necessarily generically reduced.

Title:The Kawaguchi-Silverman conjecture for endomorphisms on affine surfaces.

• Timing:8 March 4PM, AG-77


• Speaker: Buddhadev Hajra


• Abstract:This talk is based on the preprint https://arxiv.org/pdf/2104.05339.pdf by J. Jia, T. Shibata, J. Xie, and D.-Q. Zhang. For a quasi-projective variety $X$ and a finite surjective endomorphism $f:X \longrightarrow X$ defined over $\overline{\mathbb{Q}}$, the Kawaguchi-Silverman conjecture (KSC) is a conjecture predicting the coincidence of the first dynamical degree $d_1(f)$ of $f$ and arithmetic degree $\alpha_f(P)$ at a point $P \in X$ having Zariski dense $f$-orbit. This conjecture is verified for certain algebraic varieties, but the case of an open algebraic variety is hardly verified. Assuming $X$ is a smooth affine surface such that the logarithmic Kodaira dimension of $X$ is non-negative, the authors confirm KSC (when $\deg(f) \geq 2$) in this preprint, which I will present in this talk.

Title:Bernstein-Sato theory for singular rings in positive characteristic.

• Timing:5 April 4PM, AG-77


• Speaker: Sudeshna Roy


• Abstract:The Bernstein-Sato polynomial is a classical invariant which measures the singularities of the zero-locus of a holomorphic function in a very subtle way. The theory surrounding the Bernstein-Sato polynomial is vast. However, in this talk, we will sketch some features of the theory of Bernstein-Sato polynomials relating to commutative algebra. Our main aim is to discuss a version of the theory for singular ambient rings in positive characteristic, developed in the article https://arxiv.org/abs/2110.00129v2 by Jack Jeffries, Luisz Nunez-Betancourt and Eamon Quinlan-Gallego. We will outline some finiteness and rationality results for Bernstein-Sato roots and give a brief overview of how they encode some information about other classes of numerical invariants.

Title: Higher Tate Traces of Chow motives

• Timing:12 April 4PM, AG-77


• Speaker: Srimathy Srinivasan


• Abstract:Motivic decompositions of projective varieties in the category of Chow motives are very useful to study the geometry of varieties. I will talk about the paper https://arxiv.org/pdf/2302.12311.pdf where the authors define a new motivic invariant called the Tate trace of a motive and use it to classify motives of projective homogenous varieites with finite coefficients. It is also shown that the techniques can be used to characterise motivic equivalence of semisimple groups through Tits indices.

Title:Existence of moduli spaces for algebraic stacks

• Timing:19 and 26 April 4PM, AG-77


• Speaker: Yogish Holla


• Abstract: In this lecture we will describe the proof the existence of good/adequate moduli spaces for algebraic stacks which are locally reductive. The paper under consideration is by Jarod Alper, Daniel Halpern-Leister, and Jochen Heinloth (arXiv:1812.01128v4 [math.AG] 27 Jun 2022). This theorem is a generalisation of a well known theorem of Keel and Mori. This can be applied for the construction of coarse moduli spaces for many interesting moduli spaces like principal bundles and Bridgeland semistable objects in the derived category of coherent sheaves.

Title:A stacky approach to identify the semi-stable locus of vector bundles

• Timing:10 May 4PM, AG-77


• Speaker: Arkamouli Debnath


• Abstract:This talk is based on the preprint ( https://arxiv.org/abs/2302.09245 ) by Dario Weissmann and Xucheng Zhang. It also fits in the framework of the series of papers being written at recent times by mathematicians like Jarod Alper, Jochen Heinloth etc with the idea to rewrite the available literature about moduli spaces in the language of stacks and avoiding GIT. The existence of the course moduli space of semistable vector bundles on a smooth projective curve is a well known result and the classical proof relies on the notion of semi-stable points in GIT. This paper will try to give a stacky criteria to identify the semi-stable sublocus of the stack of vector bundles. One of the main contributions of this paper is to identify when the good/adequate moduli space (as introduced by Alper, which is apriori an algebraic space) becomes a scheme.

Title:Infinite torsion in Griffiths groups

• Timing:September 13, 4PM, AG-77


• Speaker: A. Sawant


• Abstract:The Griffiths group of degree $i$ of a smooth projective complex variety is the group of homologically trivial codimension $i$ algebraic cycles on it modulo algebraic equivalence. We will outline a new method due to S. Schreieder to detect nontriviality of classes in Griffiths groups, which was used by him to show that there exist smooth projective complex varieties with infinite 2-torsion in their degree 3 Griffiths groups, addressing a question due to C. Schoen. The talk is based on the preprint https://arxiv.org/abs/2011.15047 by S. Schreieder.

Title:Abundance Conjecture on Uniruled Varieties

• Timing:September 20, 4PM, AG-77


• Speaker: Omprokash Das


• Abstract: The content of this talk is a paper by Vladimir Lazic. Abundance conjecture says that if X is a smooth projective variety such that its canonical divisor K_X is nef, i.e. K_X intersects every curve non-negatively, then there is a positive integer m such that the m-th tensor power of the canonical line bundle \omega_X^{\otimes m}\cong \mathcal{O}_X(mK_X) has non-zero global sections, and moreover, these global sections generate the line bundle \omega_X^{\otimes m}. In particular, there is a projective morphism f: X\to \mathbb{P}^N to a projective space determined by global sections of \omega_X^{\otimes m}. This morphism allows X to be seen as a fibration of Calabi-Yau varieties (i.e. varieties whose canonical classes are trivial). The Abundance conjecture is one of the most important outstanding conjectures in the minimal model program. In the paper titled ``Abundance for Uniruled Varieties which are not Rationally Connected'', Lazic shows that if (X, B) is a klt pair of dimension n such that X is uniruled but not rationally connected, and if we assume that the minimal model program holds in dimension n-1, then the Abundance conjecture holds for (X, B) is dimension n. In my talk, I will explain the main ideas and techniques of Lazic's proof .

Title:Seshadri stratification and standard monomial theory

• Timing:September 27, 4PM, AG-77


• Speaker: Sarjick Bakshi


• Abstract:In this talk we will discuss a preprint of Chirvi--Fang--Littelmann `Seshadri stratification and standard monomial theory '. https://arxiv.org/pdf/2112.03776.pdf The theory of Seshadri stratifications has been developed by Littelmann--Chirvi--Fang with the intention to build up a new geometric approach towards a standard monomial theory for embedded projective varieties with certain nice properties. The authors show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In their framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.

Title:Perfectoid covers of abelian varieties

• Timing:October 11, 4PM, AG-77


• Speaker: Ashutosh Roy Choudhury


• Abstract:Perfectoid spaces are adic spaces of special kind over certain non-archimedean fields made to compare mixed characteristic situations with purely finite characteristic ones. A basic result in the theory is that the inverse limit (more precisely, the ``tilde limit'') over multiplication by $p$ maps on an abelian variety over an algebraically closed non-archimedean field of residue characteristic $p$, turns out to be perfectoid.

  The talk will be based on https://arxiv.org/pdf/1804.04455.pdf by C.Blakestad, D. Gvirtz, B. Heuer, D. Shchedrina, K. Shimizu, P. Wear, Z. Yao where they prove the above result using the theory of Raynaud Extensions. Time permitting, I'll discuss a few applications of this result.

Title: Singular Principal Bundles

• Timing:October 18, 4PM, AG-77


• Speaker: Arusha C


• Abstract:The talk will be based on the article "Singular Principal Bundles on Reducible Curves" by Angel Luis Munoz Castaneda and Alexander Schmitt (https://mathscinet.ams.org/mathscinet-getitem?mr=4337924). Singular principal bundles on smooth projective manifolds were introduced by A. Schmitt in 2002 and were extended to a wide class of singular varieties by U.N. Bhosle. The results of the paper together with the previous works of the authors prove the existence of a universal moduli space (of semistable singular principal bundles) over the moduli space of stable curves thus providing an analogous moduli space to the one obtained by R. Pandharipande for vector bundles. In an earlier work, A. Castaneda had reduced the construction of this universal moduli space to the construction of the moduli space of swamps. I will briefly outline their methods.

Title:Real forms of some Gizatullin surfaces and Koras-Russell threefolds

• Timing:November 1, 4PM, AG-77


• Speaker: Sourav Sen


• Abstract:This talk will be based on a preprint of Jérémy Blanc, Ana Bot and Pierre-Marie Poloni. https://doi.org/10.48550/arXiv.2108.12389 Given a complex algebraic variety $X$, a real form of $X$ is a real algebraic variety whose complexification is isomorphic to $X$. In this talk, we shall discuss the isomorphism class of real forms of Gizatullin surfaces of a specific kind and of Koras-Russell threefolds of the first kind.

Title:On Inversion of Adjunction.

• Timing:November 15, 4PM, AG-77


• Speaker: Raktim Mascharak


• Abstract:Adjunction is well known for irreducible smooth divisor of smooth variety. In MMP, inversion of adjunction is a very useful technique to use inductive arguments and extract information about singularities of actual variety from singularity of smaller dimensional sub-varieties. In the paper going by similar name, by Fujino and Hashizume, they have proved inversion of adjunction for arbitrary dimensional log canonical center of any pair $(X,\Delta)$. I will present the outline of the proof.

Title: Pro-étale uniformisation of abelian varieties.

• Timing: November 22, 4PM, AG-77


• Speaker: Arnab Roy


• Abstract:Any complex g-dimensional abelian variety admits a uniformisation in terms of its topological universal cover (which is C^{g}) and a lattice. From the work of Raynaud, Bosch, Lütkebohmert we know that for non-archimedean case there exist a 'uniformization' of abelian varieties, in rigid analytic category in terms of a semi-abelian rigid space and and a discrete lattice. While in the complex uniformization, the universal cover was isomorphic for all g-dimensional abelian varieties, the rigid analytic uniformization is not even locally constant (i.e. in the moduli space of abelian varieties). In the category of diamonds introduced by P.Scholze, there exists a new kind of "pro-étale uniformization" in terms of the perfectoid tilde limit and the p-adic Tate module of the abelian variety, which remains locally constant. The talk is based on https://arxiv.org/pdf/2105.12604.pdf by Ben Heuer, where the above result is proved. I shall try to explain the main idea and techniques of the proof.

Title:Suslin's Cancellation Conjecture in the Smooth Case

• Timing:November 29, 4PM, AG-77


• Speaker: S. Sandeep


• Abstract:For a smooth affine algebra of dimension $d$ over an algebraically closed field $k$ with $d!\in k^{\times}$, it is known that stably isomorphic projective modules of rank at least $d$ are isomorphic. Also, this is known not to be true in general when the modules have rank less than $d-1$. In this paper (https://arxiv.org/abs/2111.13088) by Fasel, the above is extended to modules of rank $d-1$ using the \mathbb{A}^1-$homotopy theory.

Title:Constructing a Derived Lift of The Gillet Soule Motivic Measure

• Timing:December 6, 4PM, AG-77


• Speaker:Ritankar Nath


• Abstract:Classically, the Gillet Soule Motivic measure was defined into the Grothendieck Group of Chow Motives. In this talk, using the techniques of six functor formalism, we will construct an analogous measure into the K theory spectra of Beilinson motives which will be shown to descend down to the Gillet Soule Motivic measure at the K_0 level. The talk will be based on the preprint https://arxiv.org/abs/2109.07065v3 by Joshua Lieber.

Title: Counting maximal degree subbundles of vector bundles (usual and symplectic cases)

• Timing:December 13, 4PM, AG-77


• Speaker: Arkamouli Debnath


• Abstract:The problem of counting maximal degree subbundles of a usual vector bundle on a curve has been addressed by several Mathematicians since roughly the 1980s. The most general result was proved by Y.Holla in 2004. In their 2019 preprint "Counting Maximal Lagrangian Subbundles Over An Algebraic Curve" the authors D.Cheong, I.Choe and G.Hitching address the same question for symplectic vector bundles. The proof for the symplectic case, just like the usual case, relies on intersection theory on a certain Quot scheme and the Vafa-Intriligator formula. In this talk, I will first go through the ideas behind the proof of the usual case and then the Symplectic case. The arxiv link is : https://doi.org/10.48550/arXiv.1903.04238