Yongbin Ruan
This is an expository article about the recent results on the relation between the naturality problem of quantum cohomology and birational geometry. We give a short survey on Li-Ruan's results. Some conjecture-proposals are discussed.
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We study the formation of finite time singularities of the Kahler-Ricci flow in relation to high codimensional birational surgery in algebraic geometry. We show that the Kahler-Ricci flow on an n-dimensionl Kahler manifold contracts a complex submanifold $\mathbb{P}^m$ with normal bundle $\oplus_{j=1}^{n-m}\mathcal{O}_{\mathbb{P}^m}(-a_j)$ for $a_j\in\mathbb{Z}^+$ and $\sum_{j=1}^{n-m} a_j \leq m$ in Gromov-Hausdorff topology with suitable initial Kahler class. We also show that the Kahler-Ricci flow resolves a family of isolated singularities uniquely in Gromov-Hausdorff topology. In particul…
We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.
Quantum cohomology and birational geometry of Verra fourfolds
本文通过计算 Verra 四重体的小量子上同调环，利用原子理论证明了 Verra 四重体不与非常一般的三次四重体或 Gushel-Mukai 四重体双有理等价，并刻画了与之双有理等价的那些四重体的原始上同调结构。
I show that the cohomology of the generic points of algebraic complex varieties becomes {\sl stable} birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'.
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable.
We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".
Torsion birational motives of surfaces and unramified cohomology
该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。
This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray--Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi--Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.
Birational and $\mathbf{A}^1$-invariant lattices in the cohomology of the structure sheaf over non-archimedean fields
本文研究了非阿基米德域上光滑和正则方案的结构层上同调，并将其细化为光滑（不一定正则）方案上的$\mathbf{A}^1$-不变上同调理论，同时在正特征情况下，维度不超过3时也成立。作为应用之一，得到了正则光滑变种的函数域的自同构群在结构层上同调上的准单性作用。
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.
The main goal of this paper is to break up motivic cohomology into smaller pieces as suggested by the conjectural Bloch-Beilinson filtrations for the Chow groups.
Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture
本文引入广义 Hessenberg 簇，证明对称群作用的扩展性，通过双有理几何分析给出 Shareshian-Wachs 猜想及其推广的初等证明，并提出两种计算上同调表示的算法。
Gamma classes and quantum cohomology
本文介绍复流形的Gamma类如何定义量子上同调的积分结构，提出关于该结构及Riemann-Hilbert问题的Gamma-猜想，并探讨量子上同调在双有理变换下的猜想函子性。
Four paths from birational geometry to the elliptic genus
本文通过四条原因说明椭圆亏格是可推广至奇异空间的最一般特征类，证明其特征类（加额外因子）在特定修改下唯一不变，证实并扩展Totaro的结论，方法基于等变上同调局部演算。
The Grassmannian Gr(2,5) is embedded in $\Bbb{P}^9$ via the Plücker embedding. The intersection of two general PGL(10)-translates of Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y, deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain "birational" Torelli statement for…
Rigid currents in birational geometry
本文系统研究紧致复流形上的刚性流，展示其在最小模型纲领中的应用，并提供大量新例子。
An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.
We establish two exact sequences for the lattice cohomology associated with non-degenerate plumbing graphs. The first is the analogue of the surgery exact triangle proved by Ozsvath and Szabo for the Heegaard-Floer invariant HF^+; for the lattice cohomology over Z_2-coefficients it was proved by J. Greene. Here we prove it over the integers, and we supplement it by some additional properties valid for negative definite graphs. The second exact sequence is an adapted version which does not mix the classes of the characteristic elements (spin^c-structures); it was partially motivated by the surg…
Birational Geometry of sextic del Pezzo surfaces
该研究针对任意完美域上皮卡数为1的6次del Pezzo曲面，运用Galois上同调技术完成双正则分类与自同构群计算，解答了相关公开问题，证明其为唯一具无限柔韧性的立体曲面，并构造了双有理变换群的非平凡商。
Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers.…
Let π: Y -> X be a crepant projective resolution of an affine symplectic variety X with a good C^*-action. We interpret the second cohomology H^2(Y, C) in two ways. First, H^2(Y, C) is the Picard group of Y tensorised with C. By the ample cones of different crepant resolutions of X, there is a natural chamber structure in H^2(Y, C). The second interpretation of H^2(Y, C) is the base space of the universal Poisson deformation $\mathcal Y$ of Y. Let D \subset H^2(Y, C) be the locus where the corresponding Poisson varieties are not affine. Then D is the union of finite number of hyperplanes, whic…
This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology. We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology. Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured. An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.
Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of Ganatra-Pardon-Shende t…
Quantum cohomology of variations of GIT quotients and flips
本文证明 GIT 商簇变异穿墙过程中量子 D-模的直和分解定理，并将其应用于双有理几何中标准翻折局部模型的量子上同调分解。
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.
Functions on Nilpotent Orbit Covers and Birational Geometry
本文利用Springer分解的类似物构造簇$\tilde{\mcM}$，通过双有理几何证明其有理奇点，从而将幂零轨道万有覆盖上的函数环描述为Levi子群的诱导表示，并给出其分次$G$-模结构。
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
该研究证明了任意光滑三次四fold的中等雅可比纤维化具有超凯勒紧化，并计算了其双有理几何中的锥，同时确定了Mordell-Weil群与代数上同调的同构关系。
On the algebraizability of formal deformations in $K$-cohomology
本文证明了光滑本征簇上两个K-上同调函子及其étale层化的可代数化性是稳定双有理不变量，并为此引入了相对K-同调的新概念。
For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.
Decomposition and framing of F-bundles and applications to quantum cohomology
本文针对F-丛建立谱分解定理，并证明对数F-丛框架扩展的唯一性，进而应用于射影丛和爆破的量子上同调，补充了已有存在性结果。
The quantum D-module of product varieties
本文研究乘积簇的量子D-模，证明其量子谱为因子谱的成对和，并建立量子D-模的形式同构，进而得到原子双有理不变量的乘法性及motivic测度的存在。
An atomic criterion for irrationality without quantum computations
该研究提出一种无需量子计算、基于原子理论的无理性判据，结合量子乘法单值群等变性与单值群表示不可约性，可判定三类特殊代数簇的无理性。
In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism from Y to X where the geometry of Y is "nicer" than that of X. One version of this problem, first studied by Faltings, requires Y to be Cohen-Macaulay; in this case Y is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition S_r. In this paper, the authors introduce generalized Serre conditions--these are local cohomology conditions which include S_r and the Cohen-Macaulay condition as special cases. To any generalized Serre condition…
Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projective variety over $\mathbb{K}$ and $f:X\rightarrow X$ a dominant regular morphism. Let $N^i(X)$ be the group of algebraic cycles modulo numerical equivalence. Let $χ(f)$ be the spectral radius of the pullback $f^*:H^*(X,\mathbb{Q}_l)\rightarrow H^*(X,\mathbb{Q}_l)$ on $l$-adic cohomology groups, and $λ(f)$ the spectral radius of the pullback $f^*:N^*(X)\rightarrow N^*(X)$. We prove in this paper, by using consequences of Deligne's proof of Weil's Riemann hypothesis, that $χ(f)=λ(f)$. This answers affirmatively a question pos…
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