Research diary

I am generally interested in higher dimensional geometry, minimal model program, theory of singularities (characteristic zero or positive characteristics), and Gromov-Witten theory.

Birational Geometry

-September 16th, 2012: The divison theorem seems to work better for varieties of general type. We shall consider numerical dimensions instead.
-September 10-15th, 2012: Study jet schemes and arc spaces. Jets/Arcs along a divisor (or a closed subscheme)? Study Prof. Ishii's paper.
-September 6th, 2012: Recall N. Nakayama's theorem: If h^0(mK_X+A) is a bounded function of m, i.e., \kappa_\sigma(X)=0, then \kappa(X)=0. The proof uses addition of numerical dimension \kappa_\sigma(-) for Albanese morphism with an argument analogue to Kawamata's theorem on Characterization of Abelian varieties.

Prof. Abhyankar: Resolution of singularities in dim > 3 of char(k)=p>6=(\dim X)! (Albanese's trick) and resolution of singularities in dim > 2 for arithmetic cases are open! (Resoultion of singularities in any dimension of char(k)=0 is done by Hironaka.)

-September 5th, 2012: Can we adapt the ``Theorem of division''?

-August 30th, 2012: Can we adapt Siu's proof on extension theorem?

-August 29th, 2012: Study a vanishing theorem of [dF-E] that generalizes a theorem of [BEL] and has recently been generalized by [L-S]. This vanishing theorem relates to regularity of subschemes in P^n, or in general multi-regularities. The proof uses Inversion of adjunction (for l.c.i.), multiplier ideal sheaves, and Nadel vanishing theorem.

-August 27th, 2012: How do we study the distribution of degrees of the line bundle summand of the direct images of m-th pluricanonical sheaves?

-August 24th, 2012: One step closer to the nonvanishing conjecture in dimension two! We show that the degree must be at least -1 for some m.

-August 23th, 2012: We want to show that not all direct summand of f_*\omega_X^\otimes m have degree -2 for m>1. The first step is to show that H^0(\omega_X^\otimes m\0times f^*O(1))=\=0. Assume not, then this lead to the following question:
What kind of properties do the nontrivial extensions 0->\omega_X^\otimes m->E_m->O_X-> have for X a smooth projective surface (non-minimal) of k(X)=0 with K_X^2<0?

-August 22th, 2012: Brian's paper on cone of nef curves on JAG is very interesting. It says that Borisov-Alexeev-Borisov Conjecture can help to prove that coextremal rays are generated by rational curves. I shall gather those problems related to BAB Conjecture.

-August 21th, 2012: The almost positivity result of the direct image of m-th pluricanonical sheaf (for m>1) in my thesis is correct. Instead of using Kawamata-Viehweg vanishing, we should quote the vanishing of Bogomolov-Sommese type for the higher direct images from 6.11(a) Esnault-Viehweg.

The idea of using Lefschetz pencils and the associated algebraic fiber spaces (X\rightarrow P^1) to prove the non-vanishing conjecture probably would not work. Even though we get stonger positivity of direct image of m-th pluricanonical sheaf (for m>1), it seems to be a disguise of the fact that we are using sufficiently ample linear system, for which the fibers of algebraic fiber spaces belong to. Resoving a pencil complicates the geometry of ambient space. On the contrary, minimal model program aims to simplify the geomtry. Note that an AFS X\rightarrow P^1 with K_X PSEF cannot be minimal.

A general idea is to create numerical trivial fibration of a given variety X with PSEF K_X. This shall lead to a tower of varieties with fibers of numerically trivial canonical class which ends with a variety of general type.

Gromov-Witten Theory

-September 4th, 2012: Read Katz's book on a first course to enumerative geometry. Action and a glance of Lagrange in mathematical classic mechanics.
-August 29th, 2012: Read Katz's book on a first course to enumerative geometry. First time see excess intersection.
-August 27th, 2012: We study Givental's oscillating integral on equivariant CP^n.
-August 23th, 2012: Landau-Ginzberg A-model(?) <=> B-model of smooth projective manifolds = Hodge theory. Landau-Ginzberg B-model = Frobenius algebra of isolated hypersurface singularities <=> A-model of smooth projective manifolds = Gromov-Witten theory.

-August 22th, 2012: Read part of the Teleman's note on TFT. I get the definition of TFT(=TQFT=QFT) and know that CohFT is a generalization of TFT in the sense that it is a family of TFT's. E.g., 2D TFT is equivalent to the Frobenius algebra associated to the complex vector space A correspoind to S^1, and quantum cohomology is a family of Frobenius algebras. Is TCohFT the same as CohFT? The field theory is originated from physics, but how originally do we mathematically formulate them in such a way? What is the Conformal field theory?

This is taken from Teleman's note: A Cohomological Field theory generalises the notion of a Frobenius algebra to the situation where (real) surfaces vary in families, and the values of the theory are (matrices in) the cohomology classes in the base space of the family, instead of numbers. The functor has to be functorial on the base for the considering family, in particular it sufficies to study the universal curve C_{g,n} over M_{g,n}. In [K_M], the CohFT on a finite dimensional k-vector space H equiped with a non-degenerate bilinear symmetry pairing is defined to be linear functions I_n:H^\otimes n\rightarrow H^*(M_{0,n},k) satisfying axioms the same as those of GWI's (S_n-invariant, multilinear, WDVV, etc). The goal of [K-M] is to show that genus zero 2D CohFT's are equivalent to germs of Frobenius manifolds.

Gromov-Witten Theory is a CohFT (a (1+1)-dim'l TQFT over M_{g,n}?) and is governed by all genus CohFT. Ancestors in GWT, which lacks one or two pointed curve counting, requires the correspoding Frobenius algebra to carry a calibration in order to answer most enumerative problem of rational curves. The calibration is provided by the 1-pointed J-function. There several interesting and important conjectures concerning semisimple GWT's:
Givental's conjecture: Reconstruction for semisimple GWT's.
Dubrovin's conjecture: A smooth projective manifold X has semisimple QH^*(X) if and only if D^b(X) contains a complete exceptional collection. (Optimistically this follows from MS: LG B-model of the mirror partner of X has semisimple Frobenius algebra and whose Fukaya-Seidal category(?) contains a complete exceptional collection. The opposite MS provides the required complete exceptional collection in D^b(X).) This is confirmed by Kawamata's theorem ([K])for existence of complete exceptional collection for projective toric varieties and knowledge of their quantum cohomology.
Teleman's Theorem: The structure of 2D semi-simple field theories. arXiv

Kontsevich's HMS, [KPP], should give a far-reaching adaptation of Givental’s reconstruction conjecture without the semi-simplicity assumption. It is a chain-level, open-closed FT that generalizes CohFT. Applying this programme to Gromov-Witten theory requires the construction a good Fukaya category for compact symplectic manifolds (A-model in HMS) and is generally open. The B-model of HMS, that is D^b(-), is much better understood. See [C] for implementation of these ideas.

Other activities

Regular birational geometry research meeting with Kenji Matsuki.