Helge Øystein Maakestad
Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the paper a proof of the existence of a Chern-character from complex K-theory to any cohomology theory with values in graded Q-algebras is given. The Chern-character respects the Adams and Steenrod operations.
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This note outlines some first steps in the classification of Fano manifolds for which $c_1^2-2c_2$ is positive or nef.
This is a note on MacPherson's Chern class for algebraic stacks, based on a previous paper of the author [arXiv:math/0407348]. We also discuss other additive characteristic classes in the same manner.
In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.
In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our cons…
In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf F on a complex manifold X is the product of the Chern character of F with the Todd class of X. As a corollary, we obtain a functorial proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology for complex manifolds.
We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface.
Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. In this paper, we prove the positivity of the characteristic differential form $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form $ c_2(E,h) $ in any rank, starting …
We show that the second Chern character of any projective toric manifold of Picard number three is not ample. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric manifolds.
Following Bloch-Esnault-Kerz and Green-Griffiths' recent works on deformation of algebraic cycle classes, we use Chern character from K-theory to negative cyclic homology to show how to eliminate obstructions to deforming cycles.
We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supports on the finite fibers. Next, we compute these classes in the situation encountered by the second author when proving a "Kodaira vanishing theorem" for arithmetic surfaces.
Towards small quantum Chern character
本文在射影空间和关联簇情形下证明了从小量子K理论到小量子上同调的量子Chern特征同态，并给出了Milnor超曲面小量子K理论的环表示。
Fano manifolds whose Chern characters satisfy some positivity conditions
研究Chern特征满足正性条件的Fano流形，证明其具有高阶极小有理曲线族长链并被高阶有理簇覆盖。
We give a formula for the Chern character on the DG-category of global matrix factorizations on a smooth scheme $X$ with superpotential $w\in Γ(\mathcal{O}_X)$. Our formula takes values in a Cech model for Hochschild homology. Our methods may also be adapted to get an explicit formula for the Chern character for perfect complexes of sheaves on $X$ taking values in right derived global sections of the De-Rham algebra. Along the way we prove that the DG version of the Chern character coincides with the classical one for perfect complexes.
Multiplicative property of localized Chern characters for 2-periodic complexes
本文证明了局部化Chern特征的乘法性质，由此得到从周期复形K-群到双变Chow上同调群的环同态，并应用于证明Kiem-Li余截面局部化相交同态的函子性。
In this note we study the moduli space of rank two semistable sheaves on a smooth cubic hypersurface $X\subset\mathbb{P}^{4}$ with $c_{1}=0$, $c_{2}=2$ and $c_{3}=0$. We show that it is isomorphic to the blow-up of the intermediate jacobian $J(X)$ of $X$ along the Fano surface of $X$. We complete previous results of Iliev-Markushevich AG/9910058 and Markushevich-Tikhomirov AG/9910063.
We show that any toric Fano manifold of dimension at most eight with the positive second Chern character is isomorphic to the projective space by using polymake.
Characterising Ball Quotients through their (higher) Chern Numbers
本文通过特征数特征化球商，扩展了Miyaoka等人之前的研究。
Chern character of Schur bundles
本文提出一种方法，通过E的Chern类显式计算Schur丛S^αE的Chern类。
In this paper, we investigate higher order minimal families $H_i$ of rational curves associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold if the Chern characters of $X$ satisfy some positivity conditions. We also provide a sufficient condition for Fano manifolds to be covered by higher rational manifolds.
We calculate the parabolic Chern character of a bundle with locally abelian parabolic structure on a smooth strict normal crossings divisor, using the definition in terms of Deligne-Mumford stacks. We obtain explicit formulas for $ch_1$, $ch_2$ and $ch_3$, and verify that these correspond to the formulas given by Borne for $ch_1$ and Mochizuki for $ch_2$.
We prove an explicit formula for the total Chern character of the Verlinde bundle over the moduli space of pointed stable curves in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman's classification of semisimple CohFTs, there exists an element of Givental's group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the moduli space of nonsingular curves and the proje…
Generalised Lelong-Poincaré formula in complex Bott-Chern cohomology
本文用拓扑方法证明广义 Lelong–Poincaré 公式，表明在零点轨迹纯余维等于全纯向量丛秩时，最高陈类等于其概形零点轨迹在复 Bott-Chern 上同调中的闭链类。
The aim of this note is to point out that Chern characters can be computed using curvatures o ``super-connections up to homotopy'. We also present an application to the vanishing theorem for Lie algebroids which is at the origin of new secondary classes of algebroids (Fernandes), hence, in particular, of Poisson manifolds.
In this short note we give an explicit presentation of the algebra A_n generated by the curvature 2-forms of the standard Hermitiam line bundles over SL_n/B as the quotient of the polynomial ring. The difference between A_n and H^*(SL_n/B) reflects the fact that SL_n/B is not a symmetric space. Possible applications of A_n lie in the field of arithmetic intersection theory on flag varieties.
Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular characteristic class, the Chern-Schwartz-MacPherson class, of a complete simplicial toric variety X defined by a fan from the combinatorial data contained in the fan. Specifically, we give an effective combinatorial algorithm to compute the Chern-Schwartz-MacPherson class of X, in the Chow ring (or rational Chow ring) of X. This method is formulated by combining, a…
Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN_0^{II}(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.
Superconnection and Orbifold Chern character
该研究采用Bismut等人提出的方法，利用平坦反全纯超联络研究轨形陈特征，并通过证明轨形嵌入的黎曼-罗赫-格罗滕迪克定理，揭示了轨形陈特征的唯一性。
In this paper, we introduce six axioms for relative Bott-Chern secondary characteristic classes and prove the uniqueness and existence theorem for them. Such a work provides us a natural way to understand and hence to prove the arithmetic Grothendieck-Riemann-Roch theorem.
We construct a quasi-inverse of the cochain map on the negative cyclic complexes of the second kind induced from the quasi-Yoneda embedding on a curved dg algebra. This gives an explicit formula for the Chern character of a perfect module.
Refined Chern characteristic classes of blow-ups
该研究针对紧复流形的精细化陈特征类，证明其爆破公式，建立对应无分母黎曼-罗赫定理，并以岩泽流形为例分析其精细化陈特征类在爆破变换下的行为。
Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a ring homomorphism from the Grothendieck ring of locally free finite rank A-modules with an L-connection to the Lie-Rinehart cohomology of L, where L is a Lie-Rinehart algebra. As a corollary we get the classical Chern character.
For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of epsilon-stable quasimaps and moduli of the corresponding LG epsilon-stable quasimaps, in full generality.
Equivariant Chern character for coherent sheaves and Riemann-Roch-Grothendieck
本文在有限群作用的紧致复流形上，为相干层建立等变陈特征理论，并证明相应的Riemann-Roch-Grothendieck定理。
Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham cohomology. The main result of this paper shows that such characteristic classes are independent of the Hermitian metric and only dependent on the homotopy class of the cohesive module in the homotopy category. This extends the Bott-Chern cohomology to the bounded derived catego…
In this thesis we construct a modified version of Karoubi's relative Chern character for smooth varieties over the complex numbers or the ring of integers in a p-adic number field. Comparison results with the Deligne-Beilinson Chern character and the p-adic Borel regulator constructed by Huber and Kings are proven. As a corollary we obtain a new proof of Burgos' theorem that Borel's regulator is twice Beilinson's regulator.
Products, multiplicative Chern characters, and finite coefficients, are unarguably among the most important tools in algebraic K-theory. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this article, making use of the theory of non-commutative motives, we change this state of affairs by characterizing these constructions in terms of simple, elegant, and precise universal properties. We illustrate the potential of our results by developing two of its manyfold consequences: (1) the multiplicativity of the negative Chern character…
A note of characteristic class for singular varieties
本文研究奇异簇的示性类，用对数与多重对数微分形式层的μ类和陈类表示Schwartz-MacPherson类，并给出奇异簇补集的欧拉示性数表达式。
We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (inf…
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