Fano Varieties and Extremal Laurent Polynomials

I will shortly post my notes from the workshop on Extremal Laurent Polynomials at Imperial College London on 17-19 September 2010.  I do not have notes from the talk by Katzarkov, and my notes from Batyrev’s talk are incomplete.  If you have good notes from these talks then please post them here.

Here are my notes from Manivel’s talk in Bonn:

Manivel_Bonn_Sep_2010

Summary:

• geometric Satake after Drinfeld and Ginzburg
• example:
• the quantum Chevalley formula (Fulton–Woodward)
• cohomology of G/P in general
• cohomology of G/P: improvements in the miniscule case
• semisimplicity of (Chaput–Manivel–Perrin) and connections to work of Kostant (Gorbunov–Petrov)
• quantum Satake for miniscule type-D Grassmannians

Here are my notes from Golyshev’s second talk in Bonn:

Golyshev_2_Bonn_Sep_2010

Summary:

• the “spin representation of K3” and its quantum analog
• some evidence: the Bertram–Ciocan-Fontanine–Kim result for
• the geometric Satake correspondence (Drinfeld–Ginzburg)
• miniscule Grassmannians as smooth -orbits in the affine Grassmannian
• a quantum analog of geometric Satake in this case

Here are my notes from Kresch’s talk in Bonn:

Kresch_Bonn_Sep_2010

Summary:

• the orthogonal Grassmannian
• Schubert varieties are indexed by strict partitions
• special Schubert classes
• ring presentations for and
• the connection to algebraic combinatorics via Praguez-Rataiski polynomials
• quantum Giambelli and quantum Pieri (Kresch-Tamvarkis)

Here are my notes from Katzarkov’s talk in Bonn:

Katzarkov_Bonn_Sep_2010

Summary:

• Clemens-Griffiths showed that the 3-dimensional cubic is not rational by showing that its intermediate Jacobian is not the Jacobian of a curve; we suggest analogs of this.
• Homological Mirror Symmetry and the perverse sheaf of vanishing cycles
• detecting rationality via monodromy properties on the LG mirror (Gross-Katzarkov, Pryzalkowski, Golyshev)
• spectra of triangulated categories; examples
• Theorem: is rational of dimension implies that the spectrum of has no gaps of size greaeter than
• the outlook for 4-dimensional cubics

Here are my notes from Golyshev’s talk in Bonn.

Golyshev_1_Bonn_Sep_2010

Summary:

• a historical analogy: the study of Fano varieties now versus algebraic number theory prior to the proof of the Weil Conjectures
• linearization and motives
• Tannakian categories
• the “quantum Tannakian category”
• realizations in algebraic number theory: one can detect algebraic varieties by detecting their motives (in practice by detecting their L-functions via Selberg-Stark)
• the “quantum Tate realization” and quantum detection
• the Satake correspondence and its quantum parallel

I will shortly post my notes from the workshop at the Max-Planck-Institut für Mathematik in Bonn (Quantum motives: realizations, detection, applications; 13-14 September 2010).  I do not have notes from the talks by van Straten, Altmann, or Coates: if you have good notes from these then please post them here.

Here are all the Minkowski period sequences, cross-linked with useful data.

Here are all the 3D reflexive polytopes, with face decompositions and other useful data.

Let X be a smooth Fano threefold with Picard number P = .

Then subring of algebraic (even) cycles in X is (2+2P)-dimensional, and its Lefschetz decomposition has P blocks: 1 block of length 4 and (P-1) blocks of length 2. So its image in cohomologies of anticanonical section (K3 surface) is (2+2P – P) = (2+P)-dimensional.

For “general” Fano threefold with Picard number P we expect
regularized quantum differential equation (RQDE) to be of degree (2+P) in
and to have (2+2P) singular points. Nevertheless degree in t may be more than number of singular points
due to apparent singularities.

It turns out that condition for general is not very general in practice.

Assume Fano threefold X has action of finite group G in one of the 4 ways:
a. G acts on X by regular (algebraic) transformations,
b. G acts on X by symplectic transformations,
c. X is defined over non-algebraically closed field k and G is Galois group Gal(k),
d. X is a fiber of a smooth family over some base B and fundamental group acts on via monodromy.

For cases a,b,c consider the induced action of G on cohomology of X.

Let p = be $G$-invariant Picard number.
G-invariant part of cohomologies is (2+2p)-dimensional.

Define of as subring generated by and C[t].
It is easy to see is contained in .

This implies that regularized I-series is annihilated by
differential operator of degree (2+p).

So it is natural to ask about possible G-actions on Fano threefolds.
First (numerical) step is to see the possible automorphisms of Mori cone or Kaehler cone.
We have some structures on :
a. lattice ) and element inside the lattice,
b. rational polyhedral cone of numerically effective divisors,
c. nondegenerate integral quadratic form (Lefschetz pairing) : .
We call this information .

Group of automorphisms of Mori structure is finite, and for any action
G-invariant Picard number is not less than dimension of invariants of
with respect to whole group of automorphisms of Mori structure.

As far as I remember (but cannot find a reference) for all Fano threefolds one may find some moduli
and some kind of G-action such that G-invariant Picard group coincides with invariant part of $H^2$ with respect to automorphisms of Mori structure.

The standard reference for automorphisms of Mori structure is probably:
Kenji Matsuki, “Weyl groups and birational transformations among minimal models”, AMS 1995

He studies slightly different problem, but has a similar answer. Unfortunately I haven’t a copy of this book, but copied one page from google books.

He says automorphisms of Mori structures turn out to be Weyl groups.

He claims the following Fano threefolds have nontrivial automorphisms:

P – Picard number, then list of Mori-Mukai numbers with the given Picard number

P=2:
: 2, 6, 12, 21, 32 (these should be G-Fano, but number 2 is suspicious)
other have p=2

P=3:
: 1, 27 (G-Fano, suspicious that 13 is in the next line)
: 3, 7, 9, 10, 13, 17, 19, 20, 25, 31 (should correspond to p=2)
other have p=3

P=4:
: 1 (G-Fano)
: 6 (p=2)
: 2 (p=2)
: 3, 4, 7, 8, 10, 12 (p=3)
trivial – 5,9,11 (should have p=4)
missing number 13 from Erratum

P=5:
: 3 (p=2)
: 1 (p=3)
: 2 (p=4)

For cases our threefolds are products of a line and del Pezzo surface . They all have Weyl group of type and p=2.

So the distribution in p is the following (case 4.13 is missing):
p is always less than 5;
p=4 – 4 varieties: 5.2; 4.5, 4.9, 4.11
p=3 – 26 varieties: 5.1; 4.3, 4.4, 4.7, 4.8, 4.10, 4.12; and 19 with P=3
p=2 – 50 varieties
p=1 – 25 varieties (or 26 if 2.2 is there)

This means just 4 varieties should have N=6, and other have even less.

The obvious thing to do is to recompute ourselves the Mori structure and its automorphisms
(in particular discriminant of Lefschetz quadratic form is an important invariant that we need anyway).

FOCs are frequently occuring congruences (Ramanujan’s 691s). These should be made usual suspects when searching for mirror duals of polarization structures (“degrees”). For instance, the fact that there is a del Pezzo of deg 5 signals that 5 must be a FOC for elliptic curves,  i.e. a_p=p+1 mod 5 for all p’s for many curves. In the same manner, the existence of a V_22 means that 11 is a FOC in a suitable problem on 3-dim’l Galois reps.

I attach a PARI script that produces FOCs for elliptic curves. When run from within the dir with elliptic curves data in the standard distribution, it inputs the vec of ell curves with small conductors and outputs the congruences. Adjust SLOWDOWN according to the speed of your computer.

Do report on your findings.

default(parisize,120000000);

SLOWDOWN=10^4;
e=read(“ell0″);

congruences(E)=
{
res=0;
forprime(p=10^5,10^5+SLOWDOWN,res=gcd(res,1+p-ellap(E,p)));
if(7==7,print(res,”\t”,”\t”));

}

cong(ii)=
{
num_curves=matsize(e[ii])[2];
for(jj=2,num_curves,EE=ellinit(e[ii][jj][2]);congruences(EE));
}

for(ii=1,matsize(e)[2],cong(ii));