Kefeng Liu, Hao Xu
We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of $n$-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of Gromov-Witten invariants.
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A new proof of Faber's intersection number conjecture
该研究提出了曲线模空间$\bar{M}_g$的tautological环中Faber相交数猜想的新证明，证明基于双分歧循环的直接几何与组合计算，为该猜想提供了新的推导路径。
In this note, we prove that the free energies F_g constructed from the Eynard-Orantin topological recursion applied to the curve mirror to C^3 reproduce the Faber-Pandharipande formula for genus g Gromov-Witten invariants of C^3. This completes the proof of the remodeling conjecture for C^3.
Faber's socle intersection numbers via Gromov--Witten theory of elliptic curve
本文通过改进Oberdieck-Pixton的工作，重新证明了Faber在模空间tautological环中关于socle交数的公式，并揭示了新的tautological关系。
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by ``top intersections'' of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a ``combinatorialization'' of top intersections of psi-classes. As genus 0 double Hur…
We prove that a curious generating series identity implies Faber's intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Faber's conjecture by directly proving this identity.
We present certain new properties about the intersection numbers on moduli spaces of curves $\bar{\sM}_{g,n}$, including a simple explicit formula of $n$-point functions and several new identities of intersection numbers. In particular we prove a new identity, which together with a conjectural identity implies the famous Faber's conjecture about relations in $\mathcal R^{g-2}(\sM_g)$. These new identities clarify the mysterious constant in Faber's conjecture and uncover novel combinatorial structures of intersection numbers. We also discuss some numerical properties of Hodge integrals which ha…
Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.
We give a new method of producing relations in the tautological ring R(M_{g, 1}), using the sl_2-action on the Chow ring of the universal Jacobian. With these relations, we prove that R(M_{g, 1}) is generated by κ_i for i no greater than g/3, together with ψ. Our computation shows that Faber's conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also give an algebraic proof of an identity of Morita.
Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
该研究在射影空间光滑法诺完全交的ambient状态空间上证明了亏格一Virasoro猜想，推广了拟映射不变量的跨壁公式，为完全交与射影空间的Gromov-Witten理论建立了关键桥梁。
The tautological ring of $M_{25}$ is not Gorenstein
本文证明了模空间 $M_{25}$ 的 tautological 环非 Gorenstein，推翻 Faber 猜想，并确认 Faber-Zagier 关系为完整关系列表。
We discuss analogs of Faber's conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincare duality.
In this short note, we expose some of the works on Serre intersection multiplicity conjecture. I provide a proof of the vanishing of Serre intersection multiplicity in non-proper intersection over a regular ring based on the intersection theory in W. Fulton.
We prove that Fano complete intersections in projective spaces satisfy Conjecture $\mathcal O$ proposed by Galkin-Golyshev-Iritani.
We state and check the analogue of Faber's conjectures for the tautological ring of the moduli spaces $M_{g,n}$ of $n$-pointed smooth curves of genus $g$.
The contents is changed.
We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein toric Fano varieties.
Variational Hodge conjecture for complete intersections on hypersurfaces in projective space
该研究针对射影空间中超曲面上完全交闭链的变分Hodge猜想，提出了一种全新的简化证明方法，为相关代数几何问题的研究提供了新的论证思路。
The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove that for a Fano complete intersection $X\subset \mathbb{P}^n$ of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on $X$ has the expected dimension.
We explain intersection multiplicity defined by J. P. Serre, in terms of the Poincare product in Hodge theory by a modification of the chern character map. We also discuss a formulation of the Euler characteristic via the action of correspondences on the Chow groups of projective varieties, assuming the Grothendieck Standard conjectures over $Q_l$.
We translate the results of Yansong Xu into the language of~\cite{GGV1}, obtaining nearly the same formulas for the intersection number of Jacobian pairs, but with an inequality instead of an equality.
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.
A Deformation Theoretic Reduction of the Hodge Conjecture via Derived Categories and Complete Intersections
本文提出几何与范畴方法，通过嵌入平坦族并利用导出范畴证明完全交情形的霍奇猜想，并将一般情形约化为极限代数性假设，统一了霍奇理论、形变与导出技术。
We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent $\ell$-blocks of finite reductive groups.
We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotica…
Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
We prove that the intersection homology Poincare' polynomial P(X) of an affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is the closure of any orbit in X and X/Y is a slice transverse to the orbit. This proves a combinatorial conjecture of Kalai for rational polytopes.
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.
We survey recent joint work with M. Rapoport and W. Zhang related to the arithmetic Gan-Gross-Prasad conjecture for Shimura varieties attached to unitary groups.
Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and $I$-functions of the mirror Calabi-Yau family and those of the two mirror Landau-Ginzburg models. Our proof makes explicit the "gluing/splitting" of fibrations in the Doran-Harder-Thompson mirror conjecture. Our gluing formula implies an identity, obtained by composition with their respective mirror maps, that relates the absolute Gromov-Witten invariants for the Calabi-Yaus and relative Gromov-Witten invariants for the quas…
The Poincaré torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincaré torsor, but other, more geometric, approache…
Strong Watanabe-Yoshida conjecture for Complete Intersections
本文证明了各正特征下完全相交奇点的强Watanabe-Yoshida猜想，并在特征2和3中显式计算了A1、A2奇点的Hilbert-Kunz函数。
For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.
Toric degenerations of Calabi--Yau complete intersections and metric SYZ conjecture
本研究在Gross--Siebert纲领下，以热带几何刻画对应非阿基米德Calabi--Yau度量的实Monge--Ampère方程，在解存在的假设下证明了Batyrev--Borisov型Calabi--Yau完全交环面退化的度量SYZ猜想。
We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.
On the p-adic weight-monodromy conjecture for complete intersections in toric varieties
本文通过借鉴 Scholze 的 $\ell$-adic 证明并运用刚性解析动机中的同伦论方法，证明了射影光滑环面簇内概型论完全交的 $p$-adic 权重–单值性猜想。
We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.
In 1991 Sørensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case when $d=2$. In this paper, we prove that the conjecture is true for $d=3$ and $q \ge 8$. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degene…
We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.
The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber's intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of m…
J. Wahl conjectured that every quasihomogeneous isolated normal singularity admits a positive grading for which there are no derivations of negative weighted degree. We confirm his conjecture for quasihomogeneous isolated complete intersection singularities of either order at least 3 or embedding dimension at most 5. For each embedding dimension larger than 5 (and each dimension larger than 3), we give a counter-example to Wahl's conjecture.
The tangential ramification locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of points in the ramification locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \emptyset$ implies $\operatorname {codim}_V B_{V/k}\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying g…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let $\mathcal{P}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname{codim} B}χ(A\cap B, \mathcal{P}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersection…
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--Némethi's conjecture.
Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier-Mukai transformation between the K-groups of X and Y, via an equivariant version of the …
Effective nonvanishing for weighted complete intersections of codimension two
本研究证明Kawamata有效非零猜想对余维数为2的拟光滑加权完全交成立，验证了满足条件的丰沛Cartier除子对应的线性系非空，推进了该猜想在特定代数簇类中的验证。
We address the conjecture of [Durfee1978], bounding the singularity genus, p_g, by a multiple of the Milnor number, μ, for an n-dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely (n+1)!p_g\leq μ, fails whenever the codimension r is greater than one. Moreover, we propose a new inequality, and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a `combinatorial inequality', that might have an independent interest.
We show a method in constructing algebraic cycles via intersection theory. It leads to a proof of the Lefschetz standard conjecture.
In this paper we examine different problems regarding complete intersection varieties of high degree in a complex projective space. First we show how one can deduce hyperbolicity for generic complete intersection of high multidegree and high codimension from the known results on hypersurfaces. Then we prove a existence theorem for jet differentials that generalizes a theorem of S. Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove th…
We show that the cone construction extends the Lefschetz standard conjecture to the coniveau filtration.
We prove two conjectures on weighted complete intersections and give the complete classification of threefold weighted complete intersections in weighted projective space that are canonically or anticanonically embedded.
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
$p$-adic monodromy and mod $p$ unlikely intersections, I
本文提出特征p下的Mumford-Tate和André-Oort猜想的类比，并通过Ax-Schanuel类型的猜想证明了这两个猜想，同时解决了Chai的Tate线性猜想。
Deligne's weight-monodromy conjecture gives control over the poles of local factors of L-functions of varieties at places of bad reduction. His proof in characteristic p was a step in his proof of the generalized Weil conjectures. Scholze developed the theory of perfectoid spaces to transfer Deligne's proof to characteristic 0, proving the conjecture for complete intersections in toric varieties. Building on Scholze's techniques, we prove the weight-monodromy conjecture for complete intersections in abelian varieties.
We prove that generic complete intersections associated to multiple mirror nef-partitions are all birational. This result solves a conjecture by Batyrev and Nill under some mild assumptions.
We are interested in the intersection cohomology of the minimal compactification of Siegel modular varieties at some places of bad reduction. We compute the semi-simple trace of the Frobenius morphism on the fibers of the nearby cycles of the intersection complex. We obtain a common generalization of results of Morel and Haines and Ngo.
On decompositions for Fano schemes of intersections of two quadrics
本文研究两个二次曲面交上线性子空间的 Fano 概形，提出以相伴超椭圆或叠曲线的对称幂表示的半正交分解猜想，并通过 Hodge 数公式与组合恒等式提供证据。
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