Algebraic Geometry conference
Dates: 10th December to 16th December 2015
Schedule of talks (tentative)
Abstracts
Name: Donu Arapura
Title: Numerically perfect cycles.
Abstract: Given a singular complex projective variety $X$, we define a kind of algebraic cycle on it which is "cohomological" in nature.
In codmension one, this is essentially the same thing as a numerically Cartier divisor in the sense of Boucksom, de Fernex and Favre. The space of
these cycles, which lie in $H^{2i}(X)/W_{2i}$, enjoy many good properties. In particular, it would coincide with the space of all the Hodge cycles there assuming
the (usual) Hodge conjecture.
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Name: Lior BarySoroker
Title: Geometric vs Arithmetic Ramification.
Abstract: In this talk we will consider a branched covering $f:C \to \mathbb{P}^1$ defined over $\mathbb{Q}$.
For a rational number a, the fiber $f^{1}(a)$ gives rise to a number field (in fact, \'{e}tale algebra) which, loosely speaking, is generated by the coordinates of the points in the fiber.
The main focus of this talk is the study of the number of ramified prime numbers in these number fields. I will present two results:
\noi 1. a central limit theorem, obtained with Franois Legrand, which answer the question what is the typical number of ramification.
\noi 2. sharp upper bounds (joint with Tomer Schlank).
The underlining idea behind these results is that the geometric branch locus "controls" the arithmetic one.
If time permits, some applications, e.g. to the minimal ramification problem will be discussed.
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Name: Indranil Biswas
Title: BohrSommerfeld Lagrangians of moduli spaces of Higgs bundles.
Abstract: Let $X$ be a compact connected Riemann surface of genus at least two. Let $M_H(r,d)$ denote the moduli space of semistable Higgs
bundles on $X$ of rank $r$ and degree $d$. We prove that the compact
complex BohrSommerfeld Lagrangians of $M_H(r,d)$ are precisely the
irreducible components of the nilpotent cone in $M_H(r,d)$. This
generalizes to Higgs $G$bundles and also to the parabolic Higgs
bundles. (Joint work with Niels Leth Gammelgaard and
Marina Logares.)
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Name: Andre Chatzistamatiou
Title: Decomposition of the diagonal for hypersurfaces
Abstract:This is joint work in progress with Marc Levine. Smooth hypersurfaces of sufficiently small degree in projective space are Fano varieties. One consequence is that they admit a decomposition of the diagonal, that is, a multiple of the diagonal $\Delta$, considered as a cycle in the Chow group, is essentially supported on a divisor. Let $N$ be the minimal positive integer such that $N\cdot \Delta$ is essentially supported on a divisor. We will present some results on the divisibility of $N$ by integers less than the degree of the hypersurface.
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Name: Ajneet Dhillon
Title: Essential dimension of coherent sheaves.
Abstract: The dimension of an algebraic variety can defined to be the transcendence degree of its function field over a base field. When
this definition is lifted to algebraic stacks one arrives at the notion of the essential dimension of the algebraic stack. I will discuss
some recent joint work with I. Biswas and N. Hoffmann on the essential dimension of the moduli stack of vector bundles.
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Name: Chandan Singh Dalawat
Title: The compositum of all wildly ramified extensions of prime degree
Abstract: The Galois group of a finite extension of local fields comes with a lot of extra structure, for example the filtration by higher ramification subgroups. We will motivate and describe this structure, and compute it for a naturally occurring Galois extension, namely the compositum of all wildly ramified extensions of prime degree.
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Name: Kenichiro Kimura
Title: Semialgebraic chains and the Hodge realization of mixed Tate motives.
Abstract: The motivation of this work is to understand clearly the Hodge realization
functor of the category of mixed Tate motives defined by Bloch and Kriz. Using semialgabraic
triangulations of products of $\mathbb{P}^1_\mathbb{C}$, we construct a certain complex $TC$ which
satisfies several properties, including a generalized Cauchy formula about the integrals
of certain differential forms with logarithmic poles. As an application, we construct
a Hodge realization functor on BlochKriz mixed Tate motives via period integrals.
This is a joint work with Masaki Hanamura and Tomohide Terasoma.
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Name: Satoshi Kondo
Title: On modular symbols for $PGL_d$ in positive characteristic.
Abstract:(Joint work with Seidai Yasuda) We study the homology and
the BorelMoore homology with rational coefficients of a quotient of
the BruhatTits building of $PGL_d$ of a nonArchimedean local field of
positive characteristic by an arithmetic subgroup. We define an analogue of modular symbols in this context.
Our theorem states that the image of the canonical map
from homology to BorelMoore homology is contained
in the subspace generated by the modular symbols.
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Name: Amalendu Krishna
Title: Zero cycles on affine varieties
Abstract: The aim of this talk is to study the Affine Roitman
torsion problem for 0cycles on singular varieties. We show
that the affine Roitman torsion problem has positive solution
for all affine varieties over an algebraically closed field.
We shall describe some applications of this solution. In
particular, we shall show that an old open question of Murthy has
a positive solution.
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Name: James Lewis
Title: The regulator map from Bloch's simplicial higher Chow groups to Deligne cohomology.
Abstract: An explicit formula for the Bloch cycle class map from his higher Chow groups to Deligne cohomology, was provided
by Kerr/Lewis/M{u}llerStach [KLM] (Compositio Math 142) for projective algebraic manifolds, and in the general case by KerrLewis (invent. math. (2007)),
based on a cubical description of these groups. We provide an explicit formula for the simplicial version of these groups. This is based on joint work with Matt Kerr
and Patrick Lopatto.
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Name: Joseph Lipman
Title: About the fundamental class of a flat schememap.
Abstract: For a flat schememap $f \colon X \to Y$, the fundamental class is a canonical derivedcategory map $c_f$ from the Hochschild complex $H_f$ to the relative dualizing complex $f^!O_Y$. For example, when f is smooth of relative dimension d, $c_f$ composed with the natural map $\Omega_f^d[d] \to H_f$ is Verdier's isomorphism from $\Omega_f^d[d]$ to $f^!O_Y$; and this should provide a natural bridge between abstract and concrete Grothendieck duality. If $f$ is also proper, $c_f$ corresponds to a map $Rf_*H_f \to O_Y$ which is, conjecturally, related to a known trace map for Hochschild complexes which generalizes the wellknow trace map for differential forms. These
and perhaps other related open questions will be discussed.
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Name: Sukhendu Mehrotra
Title: Hidden symmetries of moduli spaces of sheaves on K3 surfaces
Abstract:Let $X$ be a $K3$ surface, and $Y$ the Hilbert scheme of $g$ points on it. It follows from results of Addington and MarkmanMehrotra that the derived category $D(Y)$ carries an exotic autoequivalence constructed from the universal ideal sheaf. Addington has conjectured that any moduli space of sheaves on $X$ should carry such a derived symmetry; in fact, it should arise from the same construction using the universal (twisted) sheaf. This was confirmed by him for a class of moduli spaces in recent work with Donovan and Meachan. Here, we discuss another class of moduli spaces which was worked out jointly with Eyal Markman.
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Name: Arvind Nair
Title: Mixed motives in $A_g$
Abstract:We will discuss the computation of some simple mixed motives appearing in the cohomology of the moduli space of principally polarized Abelian varieties and of
its Satake compactification.
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Name: Amnon Neeman
Title: Separable monoids in $D_{qc}(X)$.
Abstract: It is possible to define separable monoids in any monoidal category, and in the special case where the monoidal category is modules over a commutative ring the study of the subject goes back to the 1960s. In the last decade several people, mostly around Balmer, have studied another interesting class of examples, the ones that come from tensor triangulated categories.
It is easy to show that smashing Bousfield localizations always give rise to examples. In the special case where X is a noetherian scheme and the monoidal category is $D_{qc}(X)$, another class of examples comes from etale maps $U\to X$. The main theorem is that these are the only examples.
In the talk we'll review the history, state the results, and discuss the (many) open problems that remain.
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Name: Nitin Nitsure
Title: Schematic HarderNarasimhan stratification for families of principal bundles in higher dimensions
Abstract: The parameter scheme of a family of principal bundles on a projectivevariety has a settheoretic stratification by the HarderNarasimhan type. We show that these strata can be endowed with natural scheme structures, with an appropriate universal property. Consequently, principal bundles of a fixed HarderNarasimhan type form an algebraic stack.
Joint work with Sudarshan Gurjar.
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Name: Andrew Obus
Title: Toward a classification of local Oort groups.
Abstract: The local lifting problem asks, for $k$ an algebraically closed field of characteristic $p$, which finite group actions on $k[[t]]$ by $k$automorphisms ("local actions") lift to characteristic zero? It turns out that answering this question is enough to decide which smooth projective curves in characteristic $p$ (with a group action) lift to characteristic zero. A group such that every such action lifts is called a "local Oort group" (for $p$). It is easy to show that cyclic groups of primeto$p$ order are local Oort, but the situation when there is wild ramification is much more complicated. We will summarize previous work on classifying these groups, and give a sense of the questions still unanswered.
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Name: Noriyuki Otsubo
Title: CM periods, CM regulators and hypergeometric functions.
Abstract: Due to the GrossDeligne period conjecture, CM periods for abelian fields are written as a product of values of the gamma function at rational numbers which reflect the Hodge type. We give examples of fibrations with a group action for which the conjecture holds. For those fibrations, we express the Beilinson regulators in terms of values of generalised hypergeometric functions at rational numbers, which are natural generalisation of gamma values and of polylogarithm values. This is a joint work with Masanori Asakura.
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Name: Deepam Patel
Title: Motivic structures on higher homotopy of nonnilpotent spaces
Abstract: In his fundamental paper on the projective line minus three points, Deligne constructed certain extensions of mixed Tate motives arising from the fundamental group of the projective line minus three points. Since then motivic structures on homotopy groups have been studied by many authors. In this talk, we will show how to construct a motivic structure on the (nilpotent completion of) $n^{th}$ homotopy group of $\mathbb{P}^n$ minus $n+2$ hyperplanes in general position.
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Name: B.P. Purnaprajna
Title: On Higher Dimensional Extremal Varieties of General Type.
Abstract:Relations among fundamental invariants plays an important role in algebraic geometry. In this talk, we consider the relations between canonical volume and genus for varieties of general type. We prove an inequality for a $n$dimensional minimal Gorenstein variety of general type and investigate the compelling extremal case, when the inequality is an equality. These extremal varieties are natural higher dimensional analogue of Horikawa's surfaces whose invariants satisfy the equality in Noether's inequality. We prove that for extremal varieties of general type of arbitrary dimension, their canonical linear systems are base point free. We give a characterization of these varieties. Moreover, we show that the deformation of these varieties remain in the same type. It is also proved that these extremal varieties of general type are simply connected, and are pluriregular (in the smooth case). Optimal results on projective normality of pluricanonical linear systems will also be dealt in this talk. These results give a complete generalization of Horikawa's results in the Annals for all dimensions!
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Name: Kay Rülling
Title: Higher Chow groups with modulus and relative Milnor Ktheory
Abstract: For a pair (X,D) consisting of a smooth variety X over a field and aneffective Catier divisor D, S. Saito and K. Kato defined in the 80's a relative version of Milnor Ktheory in order to study higher dimensional geometric class field theory. Recently M. Kerz and S. Saito gave a different approach using Chow groups of zero cycles of (X,D). In this talk I will explain a link between these two approaches in the case where the support of D is a simple normal crossings divisor. Namely, in this case we construct a cycle map from the motivic complex of (X,D) in weight r, introduced by S. Saito and F. Binda, to the relative Milnor Ksheaf of (X,D) in degree r and shifted by [r]. This map induces an isomorphism between the Nisnevich cohomology groups in degrees greater equal r+ dim X. This is joint work with S. Saito.
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Name: Pramathanath Sastry
Title: Residues and Traces via Verdier's isomorphism.
Abstract: In 1968, Verdier outlined an approach to Grothendieck duality based on earlier work of Deligne, and showed that flat base change for Òupper shriekÓ yields the fact that for a smooth map, the (top) relative differentials control duality. However, other explicit aspects of duality (residues and traces) seemed intractable via this approach (in spite of VerdierÕs claims to the contrary). The talk will be on joint work with Suresh Nayak showing how one can obtain residues and traces via this approach. Critical to our approach is the work by Alonso, Jeremas and Lipman on duality on formal schemes, especially their flat base change for duality on formal schemes.
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Name: Ronnie Sebastian
Title: Deligne's conjectures, on the automorphic side
Abstract: The modest aim of my talk will be to explain the conjectures of Deligne on special values of Lfunctions associated to motives defined over a number field, and to explain how this motivates some similar questions on the automorphic side. Consequently I shall explain how to define periods for some automorphic representations, and prove results about special values of Lfunctions attached to these. This is work in progress with Harald Grobner.
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Name: Liran Shaul
Title: The twisted inverse image pseudofunctor over commutative DGalgebras via rigid dualizing DGmodules.
Abstract: Let $K$ be a base Gorenstein noetherian ring of finite Krull dimension and consider the category of commutative nonpositive differential gradedalgebras $A$ over $K$,
such that $H^0(A)$ is essentially of finite type over $K$ and such that $A$ has finite flat dimension over $K$.
In this talk, we will explain how to extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup.
We will define rigid dualizing DGmodules, discuss their functorial properties with respect to cohomologically finite and cohomologically essentially smooth maps, show existence and uniqueness, and use them to construct $f^!$.
As an application, we will deduce a derived base change result for the twisted inverse image of an essentially finite type map between ordinary commutative noetherian rings in this category.
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Name: Vasudevan Srinivas
Title: Frobenius action on local cohomology and the Hodge filtration
Abstract: In this talk, based on joint work with S. Takagi, I will discuss a variation of the ``weak ordinarity conjecture'' (which is in fact a
consequence of it). For a $d$ dimensional isolated CM singularity in char. 0, this ``new'' conjecture relates the nilpotency of the Frobenius action on local cohomology in degrees $< d$ of the mod $p$ reductions, and the vanishing of the $0^{th}$ graded piece of the Hodge filtration on the vanishing cycle cohomology, in char. $0$. I will also discuss some known cases.
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Name: Angelo Vistoli
Title: Fundamental Gerbes
Abstract:Let $X$ be a connected and geometrically reduced variety over a field k, with a fixed rational point $x_0$ in $X(k)$. Nori defined a profinite group scheme $N(X,x_0)$, usually called Nori's fundamental group scheme, with the property that homomorphisms $N(X,x_0)$ to a fixed finite group scheme $G$ correspond to $G$torsors $P \to X$, with a fixed rational point in the inverse image of $x_0 \in P$. If $k$ is algebraically closed of characteristic $0$ this coincides with Grothendieck's fundamental group, but is in general very different.
Nori's main theorem is that if $X$ is complete, the category of finitedimensional representations of $N(X,x_0)$ is equivalent to an abelian subcategory of the category of vector bundles on $X$, the category of essentially finite bundles.
Furthermore, Nori defined a similar fundamental groups with unipotent group schemes, whose category of finitedimensional representations is equivalent to the category of vector bundles admitting a filtration with trivial quotients.
In my talk I will discuss my joint work with Niels Borne, in which we remove the dependence on the base point, substituting Nori's fundamental group with a gerbe (in characteristic $0$ this had already been done by Deligne). Furthermore I will explain our work in progress in which we find some new classes of groups scheme, such as abelian affine group scheme, for which fundamental gerbes exist.
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