Stefan Schroeer
I construct some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.
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Rigid and obstructed tangent bundles on Calabi--Yau threefolds
本文在不假设单连通性的条件下，通过构造自由商和Igusa商例子，回答了Huybrechts关于Calabi-Yau三维流形切丛刚性与奇异形变空间的两个问题，并给出了三维反例及形变芽的显式结构。
In this article we prove a fundamental inequality between Artin-Mazur heights and Yobuko heights of certain proper log smooth schemes of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field $κ$ of characteristic $p>0$. We also prove that the cohomologies of Witt-sheaves of them are finitely generated ${\cal W}(κ)$-modules if the Yobuko heights of them are finite. As an application, we prove that the $p$-primary torsion parts of the Chow groups of codimension $2$ of proper smooth threefolds over $κ$ is of finite cotype if the Yobuko heights of them are …
We compute certain open Gromov-Witten invariants for toric Calabi-Yau threefolds. The proof relies on a relation for ordinary Gromov-Witten invariants for threefolds under certain birational transformation, and a recent result of Kwokwai Chan.
Higher genus Gromov-Witten theory of one-parameter Calabi-Yau threefolds II: Feynman rule and anomaly equations
本研究证明了两个针对三类单参数卡拉比-丘三维流形Gromov-Witten理论的猜想，给出了高亏格不变量生成级数的递归确定方法，仅含有限未知参数。
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_q$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.
We solve Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equation to determine the higher genus Gromov-Witten invariants ($g \leq 5$) of the derived equivalent Calabi-Yau threefolds, which are of the appropriate codimensions in the Grassmannian Gr(2,7) and the Pfaffian Pf(7).
The moduli stack of Mixed Spin P-fields (MSP) provides an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. This paper is to apply the algorithm in genus one case. We use the localization formula, the proposed algorithm in [CLLL1, CLLL2], and Zinger's packaging technique to compute the genus one Gromov-Witten invariants of quintic Calabi-Yau threefolds. New hypergeometric series identities are also discovered in the process.
We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $\F_3$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $\F_5$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $\F_p$ that do not lift to algebraic spaces in characteristic zero.
Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds are compared. In certain situations, the Donaldson-Thomas invariants are very easy to handle, sometimes easier than the other invariants. This point is illustrated in several ways, especially by revisiting computations of Gopakumar-Vafa invariants by Katz, Klemm, and Vafa in a rigorous mathematical framework. This note is based on my talk at the 2004 Snowbird Conference on String Geometry.
We derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures emerging in the Gromov-Witten theory of the quintic, such as the Frobenius manifold structure and the special Kähler structure.
We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are Calabi-Yau threefolds. We will discuss especially the generating functions of Gromov-Witten and Gopakumar-Vafa invariants.
Based on the large N duality relating topological string theory on Calabi-Yau 3-folds and Chern-Simons theory on 3-manifolds, M. Aganagic, A. Klemm, M. Marino and C. Vafa proposed the topological vertex (hep-th/0305132), an algorithm on computing Gromov-Witten invariants in all genera of any non-singular toric Calabi-Yau 3-fold. In this expository article, we describe the mathematical theory of the topological vertex developed by J. Li, K. Liu, J. Zhou, and the author (math/0408426).
Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds
本文发展了亏格二拟Siegel模形式与拟Siegel Jacobi形式的微分环理论，并将其应用于带膜的局部Calabi-Yau三流形的拓扑递归，证明了开闭Gromov-Witten势本质上是亏格二拟Siegel模形式。
If X is a nonsingular curve in a Calabi--Yau threefold Y whose normal bundle N_{X/Y} is a generic semistable bundle, are the local Gromov-Witten invariants of X well defined? For X of genus two or higher, the issues are subtle. We will formulate a precise line of inquiry and present some results, some positive and some negative.
Higher-genus Gromov-Witten theory of one-parameter Calabi-Yau threefolds I: Polynomiality
该研究利用MSP场理论证明了三类单参数卡拉比-丘三维超曲面的Gromov-Witten势满足有限生成猜想，并给出了其亏格一Gromov-Witten势的公式，推进了高亏格Gromov-Witten理论的研究。
We investigate Gromov-Witten invariants associated to exceptional classes for primitive birational contractions on a Calabi-Yau threefold X. It was observed in a previous paper that these invariants are locally defined, in that they can be calculated from knowledge of an open neighbourhood of the exceptional locus of the contraction; in this paper, we make this explicit. For Type I contractions (i.e. only finitely many exceptional curves), a method is given for calculating the Gromov-Witten invariants, and these in turn yield explicit expressions for the changes in the cubic form $D^3$ and the…
For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
The Gopakumar-Vafa invariants are numbers defined as certain linear combinations of the Gromov-Witten invariants. We prove that the GV invariants of a toric Calabi-Yau threefold are integers and that the invariants for high genera vanish. The proof of the integrality is based on elementary number theory and that of the vanishing uses the operator formalism and the exponential formula.
We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme $\operatorname{Hilb}^{sc} V_4$ of smooth con…
In the present paper the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $P^3$ or the quadric $Q^3$ is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov-Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold $X$ with $b_3(X)=0$ admits a complete exceptional set of th…
Liftability and vanishing theorems for Fano threefolds in positive characteristic I
本文证明正特征光滑Fano三维流形可提升到Witt向量环，并满足Akizuki-Nakano消失、Hodge到de Rham谱序列E1退化及晶体上同调无挠性，针对$|-K_X||$非常丰沛且Picard群由$\omega_X$生成的情形。
Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N=(2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -- recently computed via localization by Benini et al. and Doroud et al. -- yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More g…
We study BPS states of 5d $\mathcal{N}=1$ $SU(2)$ Yang-Mills theory on $S^1\times \mathbb{R}^4$. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface $\mathbb{F}_0$. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle inter…
Two-Dimensional Torus Action on the Quadric Threefold and GW-DT-GV Correspondence
该研究针对光滑二次三维簇Q上的局部CY 4-流形，计算了次数至4的GW与DT不变量，证实了零亏格下DT与GV不变量的对应关系，并预测了Q上的一个亏格1的BPS不变量。
Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.
This manuscript from August 1995 (revised February 1996) studies the Kaehler cone of Calabi-Yau threefolds via symplectic methods. For instance, it is shown that if two Calabi-Yau threefolds are general in complex moduli and are symplectic deformations of each other, then their Kaehler cones are the same. The results are generalizations of those in the author's previous paper "The Kaehler cone on Calabi-Yau threefolds" (Inventiones math. 107 (1992), 561-583; Erratum: Inventiones math. 114 (1993), 231-233), where the behaviour of the Kaehler cone under deformations of the complex structure was …
We use a topological framework to study descendent Gromov-Witten theory in higher genus, non-toric settings. Two geometries are considered: surfaces of general type and the Enriques Calabi-Yau threefold. We conjecture closed formulas for surfaces of general type in classes K and 2K. For the Enriques Calabi-Yau, Gromov-Witten invariants are calculated in genus 0, 1, and 2. In genus 2, the holomorphic anomaly equation is found.
We study the web of dualities relating various enumerative invariants, notably Gromov-Witten invariants and invariants that arise in topological gauge theory. In particular, we study Donaldson-Thomas gauge theory and its reductions to D=4 and D=2 which are relevant to the local theory of surfaces in Calabi-Yau threefolds.
This is the second part of the project toward an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, the localization formula is derived, and algorithms toward evaluating these Gromov-Witten invariants are derived.
Liftability and vanishing theorems for Fano threefolds in positive characteristic II
本文研究正特征域上光滑 Fano 三维簇的提升性与上同调性质，证明当 Picard 数或 Fano 指数大于 1 时其具有拟 F-分裂性，从而在除特定极丰沛情形外建立 Akizuki–Nakano 消失、Hodge–de Rham 谱序列 E1-退化及晶体上同调无挠性。
Displays can be thought of as relative versions of Fontaine's notion of strongly divisible lattice from integral $p$-adic Hodge theory. In favourable circumstances, the crystalline cohomology of a smooth projective $R$-scheme $X$ is endowed with a display-structure coming from complexes associated with the relative de Rham-Witt complex $WΩ_{X/R}^{\bullet}$ of [LZ04]. In this article, we use the crystal of relative displays of [GL21] to prove a Grothendieck-Messing type result for the deformation theory of Calabi-Yau threefolds.
We propose Picard-Fuchs equations for periods of nonabelian mirrors in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which is eventually reduced to the actual number of Kähler parameters. These Picard-Fuchs equations are concise and novel. We justify our proposal by reproducing existing mathematical results, namely Picard-Fuchs equations of Grassmannians and Calabi-Yau manifolds as complete intersections in Grassmannians. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute Picard-Fuchs equ…
Quadratic Donaldson-Thomas invariants for $(\mathbb{P}^1)^3$ and some other smooth proper toric threefolds
本文利用Witt层上同调的虚拟局部化方法，证明了取值于实Witt环的(P¹)³二次Donaldson-Thomas不变量生成级数为MacMahon函数M(q²)⁻⁸，证实修正的Viergever猜想，并证明其局部化版本对(P¹)³某些迭代胀开及相关光滑proper托里拆利簇成立。
The relative Clemens Conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds
本文提出了1/2-log Calabi-Yau三维流形的相对Clemens猜想，并通过验证素Fano三维流形的框架，证明了相对Gromov-Witten理论的虚拟复杂性被抑制，从而实现了经典计数。
We prove transformation formulae for generating functions of Gromov-Witten invariants on general toric Calabi-Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi-Yau threefolds.
In this note we show that Frobenius split smooth, projective threefolds are Hodge-Witt (using a criterion for the degeneration of the slope spectral sequence of smooth projective threefolds which we prove), and that smooth, projective Frobenius split varieties do not have exotic torsion in their slope spectral sequence. We also record a few simple observations on Frobenius split threefolds. This paper complements authors paper with C. S. Rajan ``On Frobenius splitting and ordinarity'' which is also available on this archive.
We study geometric transitions for topological strings on compact Calabi-Yau hypersurfaces in toric varieties. Large N duality predicts an equivalence between topological open and closed string theories connected by an extremal transition. We develop new open string enumerative techniques and perform a high precision genus zero test of this conjecture for a certain class of toric extremal transitions. Our approach is based on a) an open string version of Gromov-Witten theory with convex obstruction bundle and b) an extension of Chern-Simons theory treating the framing as a formal variable.
In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the gauged linear sigma model, which in their case encom- passes Gromov-Witten theory and its three companions (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, we calculate all four genus zero theories explicitly. Furthermore, we relate the I-functions of these theories by analytic continuation and symplectic transfor- mation…
In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version modified by Ph. Ellia) concerning maximal families of space curves lying on a smooth cubic surface, assuming the quadratic normality of its general members. We also give a sufficient condition for curves lying on a cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as …
In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -- even though in most of these cases the Picard group is infinite.
We study the contribution of multiple covers of an irreducible rational curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten invariants in the following cases. (1) If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by the sum of all 1/n^3 where n divides d. (2) For a smoothly embedded contractable curve C in Y we define schemes C_i for i=1,...,l where C_i is supported on C and has multiplicity i, and the integer l (0<l<7) is Kollar's invariant ``length''. We prove that the contribution of …
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Göttsche-Shende. A formula for hyp…
We calculate the D-brane superpotentials for two Calabi-Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials $\mathcal{W}_{GVW}$ of the dual F-theory compactification on the relevant Calabi-Yau fourfolds in the weak decoupling limit. We also compute the Ooguri-Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov-Witten invariants.
Universally counting curves in Calabi--Yau threefolds
本文证明了具有nef反典范丛的复三重折叠的曲线计数不变量由局部曲线的值决定，从而在初级插入下为所有此类三重折叠（含Calabi--Yau三重折叠）确立了MNOP猜想，关键技术是复流形中全纯曲线的一般横截性结果。
Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local Calabi-Yau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincaré polynomials of wild character varieties.
We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface.
The Hirokado variety is a Calabi-Yau threefold in characteristic 3 that is not liftable either to characteristic~0 or the ring $W_2$ of the second Witt vectors. Although Deligne-Illusie-Raynaud type Kodaira vanishing cannot be applied, we show that $H^1(X, L^{-1})=0$, for an ample line bundle such that $L^3$ has a non-trivial global section, holds for this variety.
The log-open correspondence for two-component Looijenga pairs
本文建立了双分量Looijenga对的log Gromov-Witten理论与环面Calabi-Yau三维流形的开Gromov-Witten理论间的全亏格对应，解决了相关猜想，并基于拓扑顶点方法提供了计算对数不变量及证明其BPS完整性的新途径。
We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.
A. Zinger defined reduced Gromov-Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). H. -L. Chang and J. Li provided a proof of the comparison theorem for quintic Calabi-Yau 3-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. T. Coates and C. Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger and proved the comparison theorem for every Calabi-Yau threefold. In this paper,…
An explicit description of the spectral data of stable U(n) vector bundles on elliptically fibered Calabi-Yau threefolds is given, extending previous work of Friedman, Morgan and Witten. The characteristic classes are computed and it is shown that part of the bundle cohomology vanishes. The stability and the dimension of the moduli space of the U(n) bundles are discussed. As an application, it is shown that the U(n) bundles are capable to solve the basic topological constraints imposed by heterotic string theory. Various explicit solutions of the Donaldson-Uhlenbeck-Yau equation are given. The…
We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande--Thomas invariants of an elliptic Calabi--Yau 3-fold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov--Witten invariants and universal Jacobi forms. As applications we prove new formulas and recover several known formulas for the PT invariants of $\mathrm{K3} \times E$, abelian 3…
Vector bundles over certain Koras-Russell threefolds of the third kind
该研究证明了特定Koras-Russell三维流形上的Chow群和Chow-Witt群为平凡，表明其上所有向量丛均为平凡。
In the first part of this paper, we obtain mirror formulas for twisted genus 0 two-point Gromov-Witten (GW) invariants of projective spaces and for the genus 0 two-point GW-invariants of Fano and Calabi-Yau complete intersections. This extends previous results for projective hypersurfaces, following the same approach, but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper. In the second and third parts of this paper, we confirm Walcher's mirror symme…
This is a modest attempt to study, in a systematic manner, the structure of low dimensional varieties in positive characteristics using $p$-adic invariants. The main objects of interest in this paper are surfaces and threefolds. It is known that there are many (counter) examples of `pathological' or unexpected behavior in surfaces and even of threefolds. Our focus has been on, obtaining systematically, general positive results instead of counter examples. There are many results we prove in this paper and not all can be listed in this abstract. Here are some of the results we prove: We show tha…
We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drasticall…
The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be solved in terms of the Noether-Lefschetz intersection numbers of the fibration and the reduced invariants of a K3 surface. Toward a generalization of this result to families with singular fibers, we introduce completed Noether-Lefschetz numbers using toroidal compactifications of the period space of elliptic K3 surfaces. As an application, we prove quasi-modularity for some genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and show that they satisfy a holomorphic anomaly equation.
Curve classes on conic bundle threefolds and applications to rationality
研究几何有理曲面上圆锥丛三维流形的曲线类结构，构造Galois等变同态推广Beauville结果，并应用于精细中间Jacobian挠子(IJT)有理性阻碍，证明在低算术复杂度基域下IJT可刻画有理性。
We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.
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