Wednesday, February 24, 2010

Eseen 2010 5th day, Algebraic Geometry and Arithmetic

2010-Feb-20th was the final day of the official day of Essen 2010, whose schedule was adapted to the carnival that I mentioned previously.

Thelene (CNRS & Paris) and Lazarsfield were the speakers of this day. Thelene talked about the universal invariant and algebraic cycles of higher codimension. The main result was

Z^{2i} (X) = H^{2i}_{Hodge} (X, ¥mathbb{Z}) / H^{2i}_{alg} (X, ¥mathbb{Z})

(where i = 1: Lefshectz, d-1, or 2). He considered Bloch-Ogus theorem and Betti cohomology.

Lazarsfield talked about the positivity (Eckart Viehweg and Esnault) of cycles on abelian varieties. I. Review of positivity for divisors of X = smooth projective variety / ¥mathbb{C}.

-- 1960's (Kodaira and Kleiman) for numerical theory of positivity.

II. Higher codimension:

-- product structure on the nef class (after the letter of Grothendieck to Mumford in the collection Vol. II.), especially on whether nef class is in the pseudo-effective divisors.

III. Positivity of Abelian varieties of real (k,k)-forms on V, where we set B = V / ¥Lambda for abelian variety. He stated the notion of strong / weak positivity of (k,k)- differential form.

After the conference, some of the participants are guided to the coal mine called Zeche Zollverein, which was about 20 minutes far from the Berliner Platz station by "tram".

I could have talked to famous professors as well as young fellows in this week. Therefore I appreciated this occasion of conference on the memory of Prof. Eckard Viehweg.

Eseen 2010 4th day, Algebraic Geometry and Arithmetic

The 4th day (February 19) morning was by Takuro Mochizuki, Venjakob, Fantechi. Mochizuki was on the recent development on the irregular singularity after Sabbah. The motivation came from \mathbb{Q}-holonomic D-module \mathcal{M} / X. Regular singularity case has Riemann-Hilbert correspondence for f_{\ast}, f_!, f^{\ast}, f^!, \otimes, RHom, \psi_g (nearby cycle), \phi_g (vanishing cycle). [Saito-Terasoma: \mathcal{M} regular singular meromorphic bundle, Beilinson-Bloch-Deligne-Esnault: regular, dim X = 1]. Then, rather than naive hope for "pre-Betti structure", he worked on an extra condition of "canonical Betti strucutre" for functoriality of "Stokes structure" (Stokes filration rather than Deligne filtration) after gluing of \mathbb{C}-perverse sheaves.

Venjakov was on the non-commutative Iwasawa main conjecture for elliptic curves with complex multiplication (CM). The definition of Iwasawa was on p-adic zeta function \zeta_p up to unit, and L_p for elliptic curve. The CM case was E \ \mathbb{Q} with \mathcal{O}_K = End (E).

Fantechi (SISSA) talked about "Twists or logs as you please", Degeneration form for Gromov-Witten invariants. She defined the Gromov-Witten invariants, which were numerical invariants of smooth projective variety over \mathbb{C}. Rather than computing symplectic invariants of Li-Ruan (and relative version by Ionel-Parker), here came the algebraic geometry of Jun Li (2001). After the talks, I went to the student restaurant and a cafe, and explained to participants about the recent understanding of mirror symmetry conjecture on Gromov-Witten potential and generating function of correlation function of cohomological field theory (and Frobenius structure on the Gauss-Manin integrable connection).

Afternoon talk was only by Prof. Hida on analytic number theory of 'big' Hecke algebra. As is often the case with automorphic forms, he introduced the space of cusp forms S_{k+1} with weight k+1. He explained something important for definition of analytic L-function by Greenberg's definition written in his books.

Saturday, February 20, 2010

Eseen 2010 3rd day, Algebraic Geometry and Arithmetic

3rd day morning (Feb.18) was by Kramer, Geisser, and Kim. Kramer's talk was on the Arakelov geometry (a la Gillet et Soule', Faltings, Bismut) by Quillen metric and arithmetic Riemann-Roch theorem by Burger, Kuehn, utilizing heat kernel regulator. The main goal was on the log singular-metric \mu on modular curves and he assumed the Peterson metric induced by \mu on the relative dualizing sheaf. The spectral zeta function worked for the regularization, which seems to be the anomaly formula of string theory. However, I do not know about how supersymmetry can work to regularize the trace and the determinant bundle.

Geisser was on Suslin homology (which is a Torsion of simplicial complexes with coefficient A: abelian group, assuming the resolution of singularity) and his recent preprint on Parshin's conjecture. Let k be a perfect field, and X / k separated of finite type. We divided the cohomology paring into torsion-free part and torsion part. Then the Suslin-Voevodsky theorem asserts the equivalence of Suslin and etale cohomology with coefficient \mathbb{Z}/m, where k is algebraically closed, char k doesn't divide m. Then he worked on base finite fields. Under the use of Weil group generated by Frobenius, the Conjecture P_0 (by Parshin's conjecture) is the vanishing of non-zero Suslin homology of rational coefficient for smooth proper scheme. Tate conjecture and Beilinson conjecture deduce Parshin's conjecture, and if further assume Kimura-O'Sullivan finite dimensnionality, conjecture P_0 holds. Then he finally asserted the equivalence of Conj. P_0 and generalized Kato conjecutre, in which for smooth X, we have the abelianization of the fundamental group.

Minhhyang Kim was on the Diophantine geometry and Galois theory. 1. Abelian case (elliptic curves, Abelian motives: Fontaine-Mazure & Beilinson & Bloch-Kato), 2. Non-abelian Albanase map (Pro-finite version, the section conjecture of Grothendieck, motivic version).

After the morning talk, we took a group photo which is available on the web. We went to the Mensa restaurant and ate a frankfur sausage with vegetable.

Afternoon was by Beilinson and Huybrechts. Beilinson was on the Ziv Ran space's contractiblity, which is used in the arc space (formal loop space) formalism of factorization algebra. For X: topological space, R(X) := the set of finite irreducible subsets of X. The proposition was that if X is any connected cell complex (CW complex), then R(X) is contractible. Then the story was on the category $\mathcal{S}$ of finite non-empty sets and surjectivity for the index set I and Ran space R(X). The claim was that homotopy colimit of M_{X^I} in \mathcal{S} is homotopic to Rat(X,Y) (presheaf of rational map from X to Y). Then, for X a curve, Maps (X \ {x_i}, Y) is an ind-affine ind-scheme, and conctractible if Y = Z \subset \neq \mathbb{A}^d.

Huybrechts was on the stability conditions on derived categoies of (polarized) K3 surfaces, after Douglas and Bridgeland's pi-stability condition, which is a modern reformulation of Harder-Narasimhan property and t-structure on derived category of coherent sheaves with heart. I asked him about the quintic 3-fold, but we do not know about the compact Calabi-Yau 3-folds, except for the local Calabi-Yau 3-fold by Nekrasov-Okounkov, algorithms of Landau-Ginzburg model for quiver variety with path algebra, (and recent work of Walcher for quintic 3-fold and its real Langrangian submanifold).

Thursday, February 18, 2010

Eseen 2010 2nd day, Algebraic Geometry and Arithmetic

Yesterday was the banquet day. But, before reporting the party, I will make an academic stuff.

Edinxhoven was on the polynomial time algorithm to compute 2-dimensional Galois representations. To be more precise, if we fix the weight k, we obtain a (deterministic) computation of the Hecke operator T_p within a polynomial time in log p. If we further assume the generalized Riemann hypothesis, it is also in poly. time in the weight k. Jannsen was on excellent schemes of dimension 2 and their strong resolution. This was after the recent progress of posotive characteristic case by Prof. Hironaka Jannsen mentioned the history of canonical, non-embedding, and functorial resolutions of sigularities. Laza was on the minimal model program (Hatching, de Jong-Oort), ADE hypersurface singularity, and GIT quotient.

Afternoon was by Nicaise and Abramovich. Nicaise talked about Neron model's existence, Grothendieck ring, motivic integration for monodromy of tame ramifications utilizing Chai's base change conductor. Abramovich talked about "Varieties with a twist" rather than "Geometry & moduli of stacks". He mentioned one of the books of Viehweg (Q.P. model of var. of general type) and a new ongoing book by Kollar. He talked about some family (moduli) of stable varieties. He finally mentioned the recent paper by Hacon-Mckernan on the log minimal model program.

Wednesday, February 17, 2010

Eseen 2010 1st day, Algebraic Geometry and Arithmetic

Yesterday was the 1st day of the conference on algebraic geometry and arithmetic in Essen, Germany. The internet connection (WiFi) was not established until now, so that I read the e-mail for the first time in Germany at the conference room.

Before the lecture, I had to look for how to reach the yellow building. While I was thinking about a public street (and the tram station) around the campus, it was found out that the lecture room is not at the math department, bur rather at the center of Duisburg-Essen University.

The morning talks were delivered by Vistoli and Chenever. Before that the organizer delivered a speech on the passing away of Prof. Viehweg. Vistoli was on the "essential dimension" of a curve and the genericity theorem. After the definition of Merkuriev, the theorem of Brousnan-Retchtustein-Vistov holds for any ground filed with characteristic 0 (genus 0: 2, genus 1: +infty, genus 2: 5, genus >= 3, 3g - 3). After the theorem on smooth connected Deligne-Mumford stack of finite type over a field, the genericity theorem (Beosnan-Reichstein-Vistoli) holds. After thinking about the Artin stack and stable curves with n points, one conjectured that the automorphism group G should be "extremely reductive". Then we switched the case for d & n (especially for whether 3 divides d).

After the first talk, I could see Prof. Beilinson and talked about my research projects and his recent works with Gaitsogry. I reported my current updates at the topological methods (Jacob Lurie et al.) in mathematical physics at the Komaba compus and the Kavli IPMU of the University of Tokyo.

Chenever was on the analytic number theoretical methods to the modularity problem of complex multiplication. Mazur's results for p > 13 (p not 631) for Ramanujan delta function were stated, and the theorem of Graber-Mazur was that the quasi-modular points are Zariski dense. After the theorem on the first cohomology on generic situation, its kernel is isomorphic to 2nd cohomology.

The lunch was at the shopping mall, and I took a Thai-Reise Spaghetti, which was very hot and I feel refreshed. After the lunch, we went to a cafe and we saw a carnival with bright costume.

The afternoon session was by Ngo and Brosnan. Ngo talked about the Langlands program. It started from some classical results about Godement, Jacquet, Authur on the L-funciton. The endoscopy and analytic continuation were analyzed on the pole. After the Beilinson-Drinfeld Grassmanian (n=1 case and geometric Langlands), he showed the intersection cohomology of perverse sheaf (Hecke) on the Grassmanian. For the adelization of the trace formula, he utilized the Lie group and its Haar measure, Tamagawa measure on the coset. Then the Poisson resummation formula enables us to make the summation absolutely convergent.