Naoki Koseki
We investigate the stronger form of the Bogomolov-Gieseker inequality on smooth hypersurfaces in the projective space of any degree and dimension. The main technical tool is the theory of tilt-stability conditions in the derived category.
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The Bogomolov-Gieseker-Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
本文将Bogomolov-Gieseker不等式从特征零光滑射影曲面推广到任意特征、具典范奇点的射影曲面，并推广其若干经典应用。
We give a counterexample to the generalized Bogomolov-Gieseker inequality for threefolds conjectured by Bayer, Macrì and Toda using the blow up of a point over three dimensional projective space.
The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.
We prove a generalized Bogomolov-Gieseker inequality as conjectured by Bayer, Macrì and Toda for the smooth quadric threefold. This implies the existence of a family of Bridgeland stability conditions.
A generalized Bogomolov-Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general, if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.
We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.
We prove a version of the Bogomolov-Gieseker inequality on smooth projective surfaces of general type in positive characteristic, which is stronger than the result by Langer when the ranks of vector bundles are sufficiently large. Our inequality enables us to construct Bridgeland stability conditions with full support property on all smooth projective surfaces in positive characteristic.
Stronger Bogomolov--Gieseker type inequality on quintic threefold
本文通过结合细化限制定理与Clifford型界限，为五次三次曲面的斜半稳定sheaf建立了更强的Bogomolov-Gieseker不等式，并暗示Toda的猜想不等式。
We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint a…
We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples. Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.
We develop a framework to modify the Bogomolov-Gieseker type inequality conjecture introduced by Bayer-Macri-Toda, in order to construct a family of geometric Bridgeland stability conditions on any smooth projective 3-fold. We show that it is enough to check these modified inequalities on a small class of tilt stable objects. We extend some of the techniques in the works by Li and Bernardara-Macri-Schmidt-Zhao to formulate a strong form of Bogomolov-Gieseker inequality for tilt stable objects on Fano 3-folds. Consequently, we establish our modified Bogomolov-Gieseker type inequality conjecture…
We strengthen a conjecture by the author. This conjecture is a Bogomolov-Gieseker type inequality involving the third Chern character of mixed tilt-stable complexes on fibred threefolds. We extend it from complexes of mixed tilt-slope zero to arbitrary relative tilt-slope. We show that this stronger conjecture implies the support property of Bridgeland stability conditions, and the existence of explicit stability conditions. We prove our conjecture for ruled threefolds, hence improving a previous result by the author.
We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland. For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tilt-stable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on $\mathbb{P}^3$.
Tilt-stability on singular schemes and Bogomolov-Gieseker-type inequalities
本文将倾斜稳定性框架推广到奇异方案，并提出针对奇异三种折的广义博戈莫洛夫-吉斯克尔不等式猜想，同时发展了相对版本的构造，证明了半稳定sheaves在任意射影方案上的博戈莫洛夫-吉斯克尔型不等式。
Orbifold Chern classes and Bogomolov-Gieseker inequalities
本文研究了具有商奇点的复流形上反射sheaf的叠映Chern类，并推导了Bogomolov-Gieseker不等式。
We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langer's estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.
Orbifold Bogomolov-Gieseker inequalities on compact Kähler varieties
本文研究了具klt奇点的紧Kähler簇上稳定反演层的Orbifold Bogomolov-Gieseker不等式，并给出了当等式成立时该层的特征刻画。
Subspace concentration condition and Bogomolov--Gieseker inequality
利用Bogomolov--Gieseker不等式的凸几何解释，给出了整多胞体满足子空间集中型不等式的必要条件。
We propose a conjectural stronger version of Bogomolov-Gieseker inequality for stable sheaves on quintic 3-folds. Our conjecture is derived from an attempt to construct a Bridgeland stability condition on graded matrix factorizations, which should correspond to the Gepner point via mirror symmetry and Orlov equivalence. We prove our conjecture in the rank two case.
Generalized Bogomolov--Gieseker inequalities via Frobenius
该研究针对维数≥3的光滑射影簇上的倾斜半稳定对象，建立了广义博戈莫洛夫-吉泽克不等式，利用弗罗贝尼乌斯态射的作用分析证明了卡斯泰尔诺沃界猜想，并结合相关工作得出卡拉比-丘三维簇上半稳定层的唐纳森-托马斯不变量为辛不变量的结论。
Let K be an algebraic number field, O_K the ring of integers of K, and f : X --> Spec(O_K) an arithmetic surface. Let (E, h) be a rank r Hermitian vector bundle on X such that $E$ is semistable on the geometric generic fiber of f. In this paper, we will prove an arithmetic analogy of Bogomolov-Gieseker's inequality: c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 >= 0.
We study a conjectural relationship among Donaldson-Thomas type invariants on Calabi-Yau 3-folds counting torsion sheaves supported on ample divisors, ideal sheaves of curves and Pandharipande-Thomas's stable pairs. The conjecture is a mathematical formulation of Denef-Moore's formula derived in the study of Ooguri-Strominger-Vafa's conjecture relating black hole entropy and topological string. The main result of this paper is to prove our conjecture assuming a conjectural Bogomolov-Gieseker type inequality proposed by Bayer, Macri and the author.
We study the structure of a log smooth pair when the equality holds in the Bogomolov-Gieseker inequality for the logarithmic tangent bundle and this bundle is semistable with respect to some ample divisor. We also study the case of the canonical extension sheaf.
Bogomolov-Gieseker inequality for log terminal Kähler threefolds
本文研究具有对数终端奇性的三维紧 Kähler 空间，为稳定 $\mathbb Q$-层建立了 Bogomolov-Gieseker 不等式的 orbifold 版本。
Let f : X --> Spec(Z) be an arithmetic variety of dimension d >= 2 and (H, k) an arithmetically ample Hermitian line bundle on X. Let (E, h) be a rank r vector bundle on X. In this paper, we will prove that if E is semistable with respect to H on each connected component of the infinite fiber of X, then { c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 } c_1(H, k)^{d-2} >= 0. Moreover, if the equality of the above inequality holds, then E is projectively flat and h is a weakly Einstein-Hermitian metric.
We give a conjectural construction of Bridgeland stability conditions on the derived category of fibred threefolds. The construction depends on a conjectural Bogomolov-Gieseker type inequality for certain stable complexes. It can be considered as a relative version of the construction of Bayer, Macrì and Toda. We prove the conjectural Bogomolov-Gieseker type inequality in the case of relative projective planes over curves. This gives the the existence of Bridgeland stability conditions on such threefolds.
Let (X,H) be a polarized, smooth, complex projective surface, and let v be a Chern character on X with positive rank and sufficiently large discriminant. In this paper, we compute the Gieseker wall for v in a slice of the stability manifold of X. We construct explicit curves parameterizing non-isomorphic Gieseker stable sheaves that become S-equivalent along the wall. As a corollary, we conclude that if there are no strictly semistable sheaves of character v, the Bayer-Macri divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous resu…
Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville-Bogomolov-Fujiki square.
We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri and Toda for some products of three curves. This gives the first examples of Bridgeland stability conditions on some threefolds of general type. The key ingredients are the spreading out technique, Frobenius morphism and Bogomolov's inequality for product type varieties in positive characteristic, proved by the author recently.
In this paper, we prove the Bogomolov-Gieseker type inequality conjecture for threefolds with nef tangent bundles. As a corollary, there exist Bridgeland stability conditions on these threefolds.
Admissible Hermitian-Yang-Mills connections over normal varieties
本文在正规簇上证明完整版 Donaldson-Uhlenbeck-Yau 定理，建立 Hitchin-Kobayashi 对应，并由此得到 Bogomolov-Gieseker 不等式及其等号刻画，还给出平凡第一 Chern 类正规 Kaehler 簇为环面有限商的新判据。
We prove the conjectural Bogomolov-Gieseker type inequality for tilt slope stable objects on each Fano threefold X of Picard number one. Based on the previous works on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D^b(X). We also get a new stronger bound for the Chern characters of slope semistable sheaves on X.
We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macrì, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda (such as $\mathbb P^3$, the quintic threefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class then we show it has very negative square.
In this paper, we prove a stronger form of the Bogomolov-Gieseker (BG) inequality for stable sheaves on two classes of Calabi-Yau threefolds, namely, weighted hypersurfaces inside the weighted projective spaces $\mathbb{P}(1, 1, 1, 1, 2)$ and $\mathbb{P}(1, 1, 1, 1, 4)$. Using the stronger BG inequality as a main technical tool, we construct open subsets in the spaces of Bridgeland stability conditions on these Calabi-Yau threefolds.
We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem.
Movable curve classes and slope stability on Deligne-Mumford stacks
将光滑射影簇上关于可动曲线类和相干层的斜率稳定性的结果推广到具有射影粗模空间的光滑真DM栈上，并建立了Bogomolov-Gieseker不等式。
We show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are proper algebraic stacks of finite type, if they satisfy the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson-Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi-Yau 3-folds satisfying the BG inequality conjecture, for example on…
In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb{P}^5$. We thus prove a stronger Bogomolov-Gieseker inequality for characters of stable vector bundles and stable objects on $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.
Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt semistable objects conjectured by Bayer-Macrì-Stellari, then the minimum degree of a dominant rational map $S\dashrightarrow\mathbb{P}^2$ is either relatively large or determined by a net of curves of low degree on $S$. As one application, we prove that the complete intersection of three very general quadrics in $\mathbb{P}^5$ has degree of irrationality 4.
We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasi projective variety.
A complex analytic approach to orbifold Chern classes on singular varieties and its applications
本文提出了一种复分析方法，用于研究奇异变种上的环面 Chern 类，并应用于稳定有理 sheaf 的 Bogomolov-Gieseker 不等式及复杂 torus 商的特征化。
We consider hypersurfaces in $(\mathbb{P}^1)^n$ that contain a generic sequence of small dynamical height with respect to a split map and project onto $n-1$ coordinates. We show that these hypersurfaces satisfy strong coincidence relations between their points with zero height coordinates. More precisely, it holds that in a Zariski-open dense subset of such a hypersurface $n-1$ coordinates have height zero if and only if all coordinates have height zero. This is a key step in the resolution of the dynamical Bogomolov conjecture for split maps.
We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample cotangent bundle. We discuss the conjecture that the analogous statement should hold in the projective space. Finally, we present a construction due to Bogomolov of varieties with ample cotangent bundle as linear sections of a product of varieties with big cotangent bundle.
The Miyaoka-Yau inequality for minimal Kähler klt spaces
本文通过在紧Kähler klt空间正则轨迹上得到自反Higgs层的广义Bogomolov不等式，建立所有极小Kähler klt空间的宫冈-丘不等式，并通过Gauduchon orbifolds上的分析结果推导奇异情形下无挠层的相关性质。
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