This is the basic theory of virtual fundamental class which is the fundation of the enumerative geometry, including the construction of virtual fundamental class and the localization formula over it.
This is the notes about dualizing complexes in commutative algebra which is the reading report of the course “advanced commutative algebra” in AMSS, CAS. This is a very interesting course about Cohen-Macaulay rings and local cohomology.
The theory of Varieties of Minimal Rational Tangents (VMRT) motivated from the proof of Hartshorne conjecture due to S. Mori. Then developed by Ngaiming Mok and Junmuk Hwang in 80s to 00s. Now this is a important area to discover the geometric properties of Fano varieties of Picard number $1$. Here is our main topics:
(a) First we introduce the basic theory of rational curves, including Hilbert schemes of morphisms, the Chow scheme of rational components $\mathrm{RatCurves}^n_d(X)$ and their useful properties. Then we will give some applications, such as Bend-and-Break, boundedness of smooth Fano varieties and the proof of Hartshorne’s conjecture using our theory.
(b) Then we will introduce some more general facts about Fano manifolds. Moreover we will also introduce some special Fano manifolds such as Gushel-Mukai varieties, rational homogeneous varieties, Hermitian symmetric space and Del-Pezzo manifolds.
(c) Then we will construct the theory of VMRT over smooth uniruled varieties and mainly focused on the Fano varieties of Picard number $1$. We will dicuss the properties of distributions $\mathcal{D}\subset T_X$ and Cartan-Fubini type extension theorem.
(d) Then we will discuss some basic applications of VMRT, such as stability of the tangent bundles and rigidity of generically finite morphisms.
(e) We will consider the VMRT of rational homogeneous varieties which is very important We will consider the properties of minimal sections and dual VMRT.
(g) We will talk about Campana-Peternell conjecture and prove the case $T_X$ is nef, big and $1$-ample.
(h) Consider the most recent work of the conjecture of Fano manifolds with non-isomorphic surjective endomorphism and the most recent work of the conjecture of Fano manifolds with big automorphism group.
(i) Remmert-Van de Ven/Lazarsfeld problem which is also fundamental.
(j) We will again talk about Campana-Peternell conjecture and prove the case $G/B$ and some special cases, such as lower dimension case and large Picard number case.
We will first introduce the theory of good moduli space of Prof. Jarod Alper and his cooperators, more precisely, the first 6 sections in their paper1 (the more earier version since the results in the paper are too general which we will not use).
Then we will introduce a very general moduli space in the 7th section in paper1, the good moduli space of objects in abelian categories which will give our a very general idea to construct the moduli space of semistable sheaves and Bridgeland stable complexes.
Next we will see the detailed theory of stability and moduli theory of stable sheaves.
Fianlly We will introduce the general theory of Bridgeland stablility on the triangulated category and construct the good moduli space of Bridgeland semistable objects in heart. We will also prove the Wall-crossing structure and give some examples. Moreover we will construct the Bridgeland stablility on the surface in detail and give the second tilt about threefolds with some conjectures.
Here is my notes about this part: ModernModuliTheory, finished at 2023-10-14.
Jarod Alper, Daniel Halpern-Leistner, and Jochen Heinloth. Existence of moduli spaces for algebraic stacks. Invent. Math., pages 1–90, 2023. ↩↩2
This is one of my plan (also the requirement of the supervisor) after reading the two books about intersection theory (Fulton’s and 3264). Here is my notes about this part: Hodge-Cycles, update at 2023-08-18.
Here is my notes: BirGeo23, update at 2023/03/07. Our main is to learn some basic theory in birational geometry, including Kollar/Mori’s book and BCHM and the basic theory of moduli of higher dimensional varieties. Here is our plane: Birational Geometry Reading Seminar.
I began to study the basic results about the moduli space of curves, aiming to discover the constructions and geometrical properties of moduli stacks and spaces of smooth curves $M_g$ and stable curves $\overline{M}_g$. The detailed information we refer Reading program on moduli space of curves. Here is my notes: Moduli space of algebraic curves, finished at 2023/09/07.
Here consist ofsome informal notes about math and more things.
[2024/08/30] Lecture note about the symplectic resolutions, symplectic duality and coulomb branches after reading the survey paper Kamnitzer22. Here is my notes: Informal-Symp-dual, updated at 2024-09-02.
[2022/01/01] Here including some naive notes during undergraduate. See: Other Notes.
