Fano Varieties and Extremal Laurent Polynomials

Here are my (incomplete) notes from Batyrev’s talk in the Extremal Laurent Polynomials workshop at Imperial:

Batyrev_London_Sep_2010

If you have complete notes, please post them here.

Partial summary:

• spherical varieties
• a model family of examples: -varieties
• review: compactification and toric varieties
• spherical compactification in these examples
• valuations and coloured cones
• finding toric degenerations

Here are my notes from Altmann’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Altmann_London_Sep_2010

Summary:

• an example due to Pinkham
• deformations and Minkoski sums; a lattice condition
• constructing the deformation corresponding to a Minkwoski decomposition
• the versal deformation space and the moduli space of generalized Minkowski summands of Q
• Example: the cone over a hexagon
• the equations defining the versal deformation space in the isolated Gorenstein case
• a new point of view: double divisors

Here are my notes from Siebert’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Siebert_London_Sep_2010

Summary:

• An overview of toric degenerations and the Gross–Siebert picture
• Examples: a pencil of quartics in and a pencil of elliptic curves in
• the Gross–Siebert Reconstruction Theorem
• Mirror Symmetry and the discrete Legendre Transform
• Landau–Ginzburg models; the Hori–Vafa mirror
• how to extend the superpotential from the central fiber to the whole of the mirror family in such a way that the resulting superpotential is proper
• Example: , flattening the boundary of the polyhedral complex
• Broken lines and scattering

Here are my notes from Golshev’s talk at the Extremal Laurent Polynomials workshop at Imperial:

Golyshev_London_Sep_2010

Summary:

• a historical analogy: the study of Fano varieties now versus the study of algebraic varieties in the early 1970s
• the key idea: classifying Fanos by detecting and classifying Fano quantum motives and their realizations
• possible approaches
• the Tannakian picture
• the quantum Satake correspondence (Golyshev–Manivel)
• two steps in this direction:  Ueda’s proof of the Dubrovin Conjecture for ; Galkin–Golyshev–Iritani’s proof that Apery=Gamma
• irregular monodromy data for Dubrovin’s quantum connection
• the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis
• the Extended Dubrovin Conjecture and the Apery=Gamma hypothesis hold for

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