Conference on Geometry - Local and Global 2018 - Schedule and Abstracts of Talks - School of Mathematics Tata Institute of Fundamental Research ...

Conference on
Geometry - Local and Global - 2018
            1 – 5 October, 2018

 Schedule and Abstracts of Talks

         School of Mathematics
 Tata Institute of Fundamental Research

Title of Talks

Donu Arapura           Vanishing theorems for Higgs bundles
Joseph Ayoub           New realisations for motives over positive characteristic
                       fields
Luca Barbieri Viale    A note on periods
Prakash Belkale        Triviality properties of principal G-bundles on singular
                       curves and conformal blocks
Spencer Bloch          Motivic Gamma functions
Steven Cutkosky        Local Uniformization, defect and associated graded rings
Wilberd van der Kallen Cohomological Finite Generation and the Identity
                       Correspondence
N Mohan Kumar          Some remarks on Berger’s conjecture
Mircea Mustaţă       An overview of Hodge ideals
Amnon Neeman           Approximable triangulated categories
Kapil Paranjape        CM K3 surfaces and the Intermediate Jacobian of some
                       Threefolds
Deepam Patel           An abelian category of Hypergeometric Motives
Piotr Pragacz          Flag bundles, Segre polynomials, and push-forwards
Andreas Rosenschon     Etale motivic cohomology
Kay Rülling           Reciprocity sheaves and conductors
Shuji Saito            Rigid analytic K-theory and p-adic Chern character
Anand Sawant             Strict vs. genuine A1 -homology
Vijaylaxmi Trivedi       Hilbert-Kunz invariants and their applications
Sinan Ünver             Infinitesimal Chow Dilogarithm of higher modulus
Olivier Wittenberg       Massey products in the Galois cohomology of number fields

                                 1

Abstracts

               Monday, 01 October 2018 (11:00-12:00)
  Speaker : Andreas Rosenschon
  Title   : Etale motivic cohomology
This is a survey talk, covering general properties of etale motivic cohomology, as
well as applications and results obtained in joint work with V. Srinivas and Anand
Sawant.
                     Monday, 01 October 2018 (12:15-13:15)
  Speaker : Vijaylaxmi Trivedi
  Title       : Hilbert-Kunz invariants and their applications
We give a brief survey of Hilbert-Kunz multiplicities and density functions. These
are characteristic p-singularity invariants for commutative Noetherian rings. The
HK density function is a compactly supported continuous function and was in-
troduced to study the numerical invariant eHK (HK multiplicity). This idea of
replacing a number (eHK ) by a function (HK density) seems to be an effective
technique to handle the notoriously difficult invariant eHK .
    The HK multiplicity characterizes seemingly unrelated invariants like Frobe-
nius semistability of the vector bundles over curves, and the tiling property of
rational convex polytopes. On the other hand (1) the integral of the HKd function
is the HK multiplicity and (2) the maximum support of the HKd function for an
ideal I happens to be the F-threshold (an invariant introduced by Mustaţă-Takagi-
Watanabe) of the maximal ideal at that ideal I (in many interesting cases). This
talk is partially based on joint works with K.I. Watanabe and M. Mondal.
               Monday, 01 October 2018 (15:00-16:00)
  Speaker : Donu Arapura
  Title   : Vanishing theorems for Higgs bundles
I want to describe a vanishing theorem for semi stable (parabolic) Higgs bundles
inspired by Saito???s vanishing theorem for Hodge modules. I will outline the
proof, where the main reduction step is in positive characteristic. If there is time, I
will say something about an analogue of Hodge modules in positive characteristic.
Part of this work was done with F. Hao and H. Li.
                    Monday, 01 October 2018 (16:30-17:30)
   Speaker : Anand Sawant
   Title      : Strict vs. genuine A1 -homology
We will introduce a new version of A1 -homology, which is often entirely com-
putable. We will describe some explicit computations of these homology groups
for classifying spaces, reductive groups and generalized flag varieties and also
discuss some applications. The talk is based on joint work with Fabien Morel.

                                          2

Tuesday, 02 October 2018 (09:30-10:30)
  Speaker : Shuji Saito
  Title   : Rigid analytic K-theory and p-adic Chern character

I will explain a joint work with Moritz Kerz and Georg Tamme on a newly de-
veloped theory of analytic K-theory K an (X ) for rigid spaces X over a complete
discrete valuation field κ. The plan of the talk is as follows.
   Part I: (i) the construction of a pro-spectrum K an (X ), first for affinoids via
“pro-homotopization” and “analytic Bass delooping” of the connective algebraic
K-theory, and then its globalization using descent for admissible coverings. (ii) If
X is a scheme separated of finite type over the integer ring O ⊂ κ and X := Xbrig is
the rigid space associated to the formal completion Xb of X along the special fiber,
K an (X ) is compared with the continuous K-theory pro-spectrum
                       K cont (X) := “ lim ” K(X ⊗O O /(πm )),
                                         m

where π ∈ O is a prime element. Thus the algebrization problem for K cont (X) is
rephrased by the same sort of a problem for K an (X ).
   Part II: Assume ch(κ) = 0 and the residue field of O is perfect of characteristic
p > 0. Let X be a smooth scheme of relative dimension d over O and X := Xbrig .
For integers 0 ≤ i < p − 2 − d and n > 0, we construct the p-adic Chern character
isomorphism:
                                                  2r−r
                   Kian (X , Z/pn Z) '
                                         M
                                                 HNis  (Xn , Sn (r)Nis ),
                                         r≤d+i

where Sn (r)Nis = τ≤r Rε∗ Sn (r) with Sn (r) the log syntomic complex on the syn-
tomic site (Xn )syn with Xn = X ⊗O O /(pn ), introduced by Fontaine-Messing, Kato
and Tsuji, and ε : (Xn )syn → (Xn )Nis is the natural map of sites. As a consequence,
the algebrization problem is related to a Bloch-Kato conjecture on the image of
p-adic regulator maps for motivic cohomology of the generic fiber of X .
   Let K be a field with a complete non-archimedean absolute value | · | and O =
{x ∈ K | |x| ≤ 1} and fix π ∈ O with |π| < 1. Let X be a proper smooth scheme
over O and write Xn = X ⊗O O /(πn+1 ) for n ≥ 0. The continuous K-groups of X
are defined as
                             K̂icont (X) := (i ∈ Z),
where Ki (Xn ) are the algebraic K-groups of Xn . The algebrization problem for
K0cont (X) concerns                                

                             Coker Ki (X) → K̂i (X) .

                                             3

It is motivated by works of Bloch-Esnault-Kerz and Morrow, which reduces Grothendieck’s
variational Hodge conjecture to the algebrization problem for K0cont (X) (in case
O = C[[t]]).

              Tuesday, 02 October 2018 (11:00-12:00)
  Speaker : Amnon Neeman
  Title   : Approximable triangulated categories

We will begin the talk with five new theorems in algebraic geometry, statements
about the derived categories Dbcoh (X) and D per f (X) for schemes X. Each of these
theorems represents a major improvement over what was known. For each of
the five theorems special cases go back to results of Bondal and Van den Bergh,
Rickard and Rouquier. The short summary is that the old results all assumed
equal characteristic, the new results are the first to work in mixed characteris-
tic. The more detailed story is that, even in equal characteristic, our results are
a sharp improvement over what was known. It turns out that all these results are
relatively straightforward corollaries of the theorem that the category Dqc (X) is
approximable. This is a new notion we will explain, and then illustrate its power.

               Tuesday, 02 October 2018 (12:15-13:15)
  Speaker : Mircea Mustaţă
  Title   : An overview of Hodge ideals

I will discuss some invariants of singularities of Q-divisors on complex smooth
varieties, that generalize the multiplier ideals from birational geometry, and which
come naturally out of Saito’s theory of mixed Hodge modules. This is based on
joint work with Mihnea Popa.

               Tuesday, 02 October 2018 (15:00-16:00)
  Speaker : Kay Rülling
  Title   : Reciprocity sheaves and conductors

Reciprocity sheaves were recently introduced by Kahn-Saito-Yamazaki. Exam-
ples are homotopy invariant Nisnevich sheaves with transfers, smooth commuta-
tive group schemes, K??hler differentials, the group of lisse rank one Q̄` sheaves
with finite monodromy, and the group of irreducible connections of rank one in
characteristic zero. In this talk I will define the motivic conductor for an abstract
reciprocity sheaf and show that in many examples it coincides with classical (non-
log) conductors, such as the pole order, the irregularity, or the Artin conductor.
As a consequence we obtain new and unified proofs for certain properties of these
conductors from general results about reciprocity sheaves. This is joint work with
Shuji Saito.

                                         4

Wednesday, 03 October 2018 (09:30-10:30)
  Speaker : Olivier Wittenberg
  Title   : Massey products in the Galois cohomology of number fields

Let k be a field and p be a prime. Massey products of n > 2 classes in H 1 (k, Z/pZ)
are expected to vanish whenever they are defined. We establish this conjecture
when k is a number field, for any n. This constraint on the absolute Galois group
of k was previously known to hold when n = 3 and whenn = 4, p = 2. (Joint work
with Yonatan Harpaz.)

              Wednesday, 03 October 2018 (11:00-12:00)
  Speaker : Wilberd van der Kallen
  Title   : Cohomological Finite Generation and the Identity Corre-
            spondence

We discuss the finite generation of rational cohomology of reductive groups when
the the algebra of coefficients is finitely generated as an algebra. We recall inter-
esting precursors of this result. We also explain how resolution of the diagonal is
used in one stage of the proof.

              Wednesday, 03 October 2018 (12:15-13:15)
  Speaker : Luca Barbieri Viale
  Title   : A note on periods

After showing the existence of a period regulator for motivic cohomology of an
algebraic scheme, a period conjecture over the field of algebraic numbers can be
formulated by saying that this period regulator is surjective. Showing that a suit-
able Betti???de Rham realization of 1-motives is fully faithful we can verify this
period conjecture in several cases. The divisibility properties of motivic coho-
mology imply that this period conjecture is a neat generalization of the classical
Grothendieck period conjecture for smooth and proper schemes. This talk is the
report of a joint paper with F. Andreatta and A. Bertapelle (along with an appendix
by B. Kahn on the named divisibility properties).

                                         5

Thursday, 04 October 2018 (09:30-10:30)
  Speaker : Joseph Ayoub
  Title   : New realisations for motives over positive characteristic
            fields

Let K be an algebraically closed field of characteristic zero endowed with a rank
one valuation having residue field k of characteristic p > 0. Fixing some extra
data, we associate a realisation functor

                                R : DM(k) → D(A)
from the category of Voevodsky motives to the derived category of A-modules
having the following properties. (1) The ring A is a K-algebra which has an explicit
description reminiscent to the description of algebras of abstract periods. (2) The
functor R is monoidal and takes a geometric motive to a perfect complex of A-
modules. (3) There are comparison maps relating R with the classical realisations.
If time permits, we also state some conjectures about the ring A and the realisation
functor R.

               Thursday, 04 October 2018 (11:00-12:00)
  Speaker : Piotr Pragacz
  Title   : Flag bundles, Segre polynomials, and push-forwards

We give Gysin formulas for all flag bundles of types A, B, C, D. The formulas
(and also the proofs) involve only the Segre classes of the original vector bun-
dles and characteristic classes of universal bundles. As an application we provide
new determinantal formulas. We also establish Gysin formulas for Kempf-Laksov
bundles and their isotropic analogs. This is a joint work with Lionel Darondeau.

               Thursday, 04 October 2018 (12:15-13:15)
  Speaker : Prakash Belkale
  Title   : Triviality properties of principal G-bundles on singular
            curves and conformal blocks

Principal bundles for semisimple groups over smooth affine curves over alge-
braically closed fields are trivial by a result of Harder. This result (and an ex-
tension to families due to Drinfeld and Simpson) allows for a comparison between
representation theory (conformal blocks) and geometry (moduli of bundles) for
smooth curves. I will describe joint work with N. Fakhruddin in which many of
these results are extended to the case of singular curves.

                                         6

Thursday, 04 October 2018 (15:00-16:00)
  Speaker : Deepam Patel
  Title   : An abelian category of Hypergeometric Motives

Given an semi-abelian scheme A we construct a category of hypergeometric mo-
tives over A. In the case of tori, and at the level of realizations, this category
contains various types of hypergeometric local systems studied by many authors
in the literature. If time remains, we will also discuss the period isomorphism in
this context. This is based on joint work with Madhav Nori.

               Thursday, 04 October 2018 (16:30-17:30)
  Speaker : Sinan Ünver
  Title   : Infinitesimal Chow Dilogarithm of higher modulus

Let Cm be a smooth and projective curve over the truncated polynomial ring k[t]/(t m )
over k, a field of characteristic 0. Given non-zero rational functions f , g, and h on
Cm 4, and m < r < 2m, I will define an invariant r ( f gh)k. This is an analog of the
real analytic Chow dilogarithm and the extension to non-linear cycles of the ad-
ditive dilogarithm. Using this construction I will state and prove an infinitesimal
version of the strong reciprocity conjecture of Goncharov and give an extension
of Park’s regulator for cycles with modulus.

                                         7

Friday, 05 October 2018 (09:30-10:30)
  Speaker : N Mohan Kumar
  Title   : Some remarks on Berger’s conjecture

A conjecture made by R. Berger several decades ago states that a curve with tor-
sion free Kahler differentials is in fact smooth. We relate this to some interesting
questions about Hilbert schemes of points.

                Friday, 05 October 2018 (11:00-12:00)
  Speaker : Steven Dale Cutkosky
  Title   : Local Uniformization, defect and associated graded rings

We begin by giving an overview of Abhyankar’s approach to local uniformization
(resolution of singularities along a valuation) and the role of defect in an extension
of valued fields as an obstruction to local uniformization. We discuss how defect
can be detected through lack of finite generation of extensions of associated graded
rings along the valuation. We then raise a question on eventual finite generation of
extensions of associated graded rings in defectless extensions, which we answer
positively in dimension two and equicharacteristic zero.

               Friday, 05 October 2018 (12:15-13:15)
  Speaker : Kapil Paranjape
  Title   : CM K3 surfaces and the Intermediate Jacobian of some
            Threefolds

To be announced

                Friday, 05 October 2018 (15:00-16:00)
  Speaker : Spencer Bloch
  Title   : Motivic Gamma functions

Motivic gamma functions are Mellin transforms of period functions. They interpo-
late the coefficients of Picard-Fuchs differential equations. I will focus on Taylor
series coeficients for motivic gamma functions and relations to variation in mon-
odromy. A natural tool to study these periods is the theory of relative completions
as developed by Brown and Hain. (Joint with M. Vlasenko and F. Brown)

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