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In our previous work, we introduced a special type of Hermitian metrics called torsion-critical, which are non-Kähler critical points of the L^2-norm of Chern torsion over the space of all Hermitian metrics with unit volume on a compact complex manifold. In this short note, we restrict our attention to the class of compact Chern flat manifolds, which are compact quotients of complex Lie groups equipped with compatible left-invariant metrics. Our main result states that, if a Chern flat metric is torsion-critical, then the complex Lie group must be semi-simple, and conversely, any semi-simple complex Lie group admits a compatible left-invariant metric that is torsion-critical. [ABSTRACT FROM AUTHOR]

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran. [ABSTRACT FROM AUTHOR]

In the setting of the non-archimedean SYZ mirror construction (Yuan in Family Floer program and non-archimedean SYZ mirror construction, Diss. State University of New York at Stony Brook, 2021), we prove the folklore conjecture that the critical values of the mirror superpotential are the eigenvalues of the quantum multiplication by the first Chern class. Our result relies on a weak unobstructed assumption, but it is usually ensured in practice by Solomon's results (Solomon in Adv Math 367:107107, 2020) on anti-symmetric Lagrangians. Lastly, we note that some explicit examples are presented in the recent work (Yuan in Family Floer mirror space for local SYZ singularities, Forum Mathe Sigma 12:e119, 2024). [ABSTRACT FROM AUTHOR]

Studying the Gieseker–Maruyama moduli space of normalized semistable coherent rank 3 sheaves of positive second Chern class and nonnegative third Chern class on the projective space , we find the first example of an irreducible component of this moduli space with small values of Chern classes in which the generic sheaf has singularities of mixed dimension: zero- and one-dimensional singularities simultaneously. Previously, some examples of components of moduli of semistable sheaves with singularities of mixed dimension were constructed only for rank 2 sheaves by Ivanov and Tikhomirov in 2018 as well as by Almeida, Jardim, and Tikhomirov in 2022. [ABSTRACT FROM AUTHOR]

We construct two infinite series of irreducible components of the moduli space of semistable reflexive rank 2 sheaves on the three-dimensional complex projective space with even and odd first Chern class. In both cases the second and third Chern classes are representable as polynomials in three integer variables. We establish the uniqueness of components in the series and describe the relations among these series and previous series of irreducible components. In the series we constructed by the authors in 2024, we find infinite subseries of rational components; these subseries are included into those constructed by Jardim, Markushevich, and Tikhomirov in 2017, as well as by Almeida, Jardim, and Tikhomirov in 2022 with the use of other constructions of series of components, for which Vassiliev established rationality in 2023. We give an example of moduli space with two irreducible components, one of which belongs to a series of components constructed in this article; while the other, to one previously known. We find the spectra of sheaves whose equivalence classes constitute these components. [ABSTRACT FROM AUTHOR]

We classify nef vector bundles on a smooth hyperquadric of dimension ≥ 4 with first Chern class two over an algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

In this paper, we provide computer-based construction of Ulrich bundles of small ranks over some special cubic fourfolds of small discriminants via deformation theory. First, we construct a simple sheaf whose Chern classes are the same as an Ulrich bundle of the same rank (if exists) as the syzygy sheaf of an Ulrich sheaf on a surface contained in a special cubic fourfold. We observe that a general deformation of this simple sheaf becomes Ulrich in several cases. We also provide some experimental constructions of Ulrich bundles on special cubic fourfolds of small discriminants. [ABSTRACT FROM AUTHOR]

Let X$X$ be a compact Kähler manifold and (E,∂¯E,θ)$(E,\overline{\partial }_E,\theta)$ be a Higgs bundle over it. We study the structure of the Kuranishi space for the pair (X,E,θ)$(X, E,\theta)$ when the Higgs bundle admits a harmonic metric or equivalently when the Higgs bundle is polystable and the Chern classes are 0. Under such assumptions, we show that the Kuranishi space of the pair (X,E,θ)$(X,E,\theta)$ is isomorphic to the direct product of the Kuranishi space of (E,θ)$(E,\theta)$ and the Kuranishi space of X$X$. Moreover, when X$X$ is a Riemann surface and (E,∂¯E,θ)$(E,\overline{\partial }_E,\theta)$ is stable and the degree is 0, we show that the deformation of the pair (X,E,θ)$(X,E,\theta)$ is unobstructed and calculate the dimension of the Kuranishi space. [ABSTRACT FROM AUTHOR]

We consider a holomorphic vector field on the complex Grassmannian constructed from a nilpotent matrix. We show that this vector field vanishes only at a single point. Using the Baum-Bott localization theorem we give a Grothendieck residue formula for the intersection numbers of the Grassmannian. Knowing that Chern classes of the tautological bundle generate the cohomology ring of the Grassmannian we can compute the ideal of relations explicitly from the residue formula. This shows that the cohomology ring of the Grassmannian is determined by holomorphic vector field around its only zero. [ABSTRACT FROM AUTHOR]

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over ℂ). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space. [ABSTRACT FROM AUTHOR]

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η -Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. [ABSTRACT FROM AUTHOR]

We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [Lazarev, Geom. Funct. Anal. 30 (2020), no. 1, 188–254]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact. [ABSTRACT FROM AUTHOR]

We investigate rank 3 instanton vector bundles on P3$\mathbb {P}^3$ of charge n$n$ and its correspondence with rational curves of degree n+3$n+3$. For n=2$n=2$, we present a correspondence between stable rank 3 instanton bundles and stable rank 2 reflexive linear sheaves of Chern classes (c1,c2,c3)=(−1,3,3)$(c_1,c_2,c_3)=(-1,3,3)$ and we use this correspondence to compute the dimension of the family of stable rank 3 instanton bundles of charge 2. Finally, we use the results above to prove that the moduli space of rank 3 instanton bundles on P3$\mathbb {P}^3$ of charge 2 coincides with the moduli space of rank 3 stable locally free sheaves on P3$\mathbb {P}^3$ of Chern classes (c1,c2,c3)=(0,2,0)$(c_1,c_2,c_3)=(0,2,0)$. This moduli space is irreducible, has dimension 16 and its generic point corresponds to a generalized't Hooft instanton bundle. [ABSTRACT FROM AUTHOR]

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$ -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$ -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$ -line bundles. As an application, we study a major open question in $p$ -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$ -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$ , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$. [ABSTRACT FROM AUTHOR]

The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in [3] that R ζ = c ζ 2 where R ζ is their map on K 3 defined using the cyclic quantum dilogarithm and c ζ is the Chern class map on K 3. [ABSTRACT FROM AUTHOR]

We consider morphisms of smooth projective varieties over . We show that if π has at most one singular fibre, then X is uniruled and π admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if π has at most two singular fibres, and the first Chern class of X is supported in a single fibre of π. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections. [ABSTRACT FROM AUTHOR]

Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in $H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence , if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence $\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$ , which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When $X$ and $Y$ are projective, the correspondences $f$ and $\tilde {f}$ are algebraic. [ABSTRACT FROM AUTHOR]

We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (nonsymmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke–Santaló inequality and Radon's theorem to prove nontrivial facts about the normalized volume in the toric setting. For example, we prove that for every ε>0$\epsilon > 0$ there are only finitely many Q$\mathbb {Q}$‐Gorenstein toric singularities with normalized volume at least ε$\epsilon$. From this result it directly follows that there are also only finitely many toric Sasaki–Einstein manifolds of volume at least ε$\epsilon$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the nonsymmetric Mahler conjecture in convex geometry. [ABSTRACT FROM AUTHOR]

The diffeomorphism class of simply connected smooth Calabi‐Yau threefolds with torsion‐free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi‐Yau threefolds constructed from the Kreuzer‐Skarke (KS) list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis‐independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favorable entries of the KS list of reflexive polytopes lead to some 10400$10^{400}$ diffeomorphically distinct Calabi‐Yau threefolds. The diffeomorphism class of simply connected smooth Calabi‐Yau threefolds with torsion‐free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi‐Yau threefolds constructed from the Kreuzer‐Skarke (KS) list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis‐independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favorable entries of the KS list of reflexive polytopes lead to some 10400 diffeomorphically distinct Calabi‐Yau threefolds. [ABSTRACT FROM AUTHOR]

We study moduli stacks of principal C ∗ -bundles over nodal complex algebraic curves and determine their rational cohomology algebras in terms of Chern classes. [ABSTRACT FROM AUTHOR]

This paper considers the (negative) cyclic open–closed map O C - , which maps the cyclic homology of the Fukaya category of a symplectic manifold to its S 1 -equivariant quantum cohomology. We prove (under simplifying technical hypotheses) that this map respects the respective natural connections in the direction of the equivariant parameter. In the monotone setting this allows us to conclude that O C - intertwines the decomposition of the Fukaya category by eigenvalues of quantum cup product with the first Chern class, with the Hukuhara–Levelt–Turrittin decomposition of the quantum cohomology. We also explain how our results relate to the Givental–Teleman classification of semisimple cohomological field theories: in particular, how the R-matrix is related to O C - in the semisimple case; we also consider the non-semisimple case. [ABSTRACT FROM AUTHOR]

We survey recent developments in the study of torus equivariant motivic Chern and Hirzebruch characteristic classes of projective toric varieties, with applications to calculating equivariant Hirzebruch genera of torus-invariant Cartier divisors in terms of torus characters, as well as to general Euler-Maclaurin type formulae for full-dimensional simple lattice polytopes. We present recent results by the authors, emphasizing the main ideas and some key examples. This includes global formulae for equivariant Hirzebruch classes in the simplicial context proved by localization at the torus fixed points, weighted versions of a classical formula of Brion, as well as of the Molien formula of Brion-Vergne. Our Euler-Maclaurin type formulae provide generalizations to arbitrary coherent sheaf coefficients of the Euler-Maclaurin formulae of Cappell-Shaneson, Brion-Vergne, Guillemin, etc., via the equivariant Hirzebruch-Riemann-Roch formalism. Our approach, based on motivic characteristic classes, allows us, e.g., to obtain such Euler-Maclaurin formulae also for (the interior of) a face. We obtain such results also in the weighted context, and for Minkovski summands of the given full-dimensional lattice polytope. [ABSTRACT FROM AUTHOR]

Let G ⊆ GL (n , F) be a finite subgroup G of the general linear group over a field F. Assume that F is algebraically closed. We show that the associated ring of polynomial invariants is characterized by its orbit Chern classes and vice versa. Moreover, the associated ring of polynomial invariants is characterized by its orbits on the dual space V * = (F n) * and vice versa. [ABSTRACT FROM AUTHOR]

A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires, is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its equivariant Chern classes can be described in terms of the corresponding Delzant polytope and the stabilization of its tangent bundle splits as a direct sum of complex line bundles. But in general a toric origami manifold is not simply connected, so the algebraic topology of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper they give an explicit formula of the equivariant Chern classes of an oriented toric origami manifold in terms of the corresponding origami template. Furthermore, they prove the stabilization of the tangent bundle of an oriented toric origami manifold also splits as a direct sum of complex line bundles. [ABSTRACT FROM AUTHOR]

Let Y be an integral nodal curve. We show that the connected component of the moduli space of torsion free sheaves of rank 1 on the compactified Jacobian J ¯ (Y) of Y, which contains Pic 0 J ¯ (Y) , is isomorphic to J ¯ (Y) under the map induced by the Abel–Jacobi embedding of Y in J ¯ (Y) . We determine the Chern classes (in Chow group) of the Picard bundles on the desingularisation of the compactified Jacobian over a nodal curve Y. We study the relation between the singular cohomology of J ¯ (Y) , J ~ (Y) and J(X) and use it to determine the singular cohomology of the compactified Jacobian of an integral nodal curve. We prove that the compactified Jacobian of an integral nodal curve with k nodes is homeomorphic to the product of the Jacobian of the normalisation X 0 and k rational nodal curves of arithmetic genus 1. [ABSTRACT FROM AUTHOR]

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu‐Zhou. The first application of our formula is a geometric description of Chern‐Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013. [ABSTRACT FROM AUTHOR]

Every compact Kähler manifold with negative first Chern class admits a unique metric g$g$ such that Ric(g)=−g$\text{Ric}(g) = -g$. Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler–Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang in the Calabi–Yau case [2019, arXiv:1906.03368], I construct a Kähler–Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family. [ABSTRACT FROM AUTHOR]

We classify globally generated vector bundles on a general hyperplane section of P² × P² embedded by the Segre embedding, considering small first Chern classes c1 = (1, 1) and c1 = (2, 1). [ABSTRACT FROM AUTHOR]

Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka--Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projectivemanifold X is home-omorphic to a variety of this type, then X is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces. [ABSTRACT FROM AUTHOR]

We construct two new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the three-dimensional complex projective space. In the first series the sheaves have an even first Chern class, and in the second series they have an odd one, while the second and third Chern classes can be expressed as polynomials of a special form in three integer variables. We prove the uniqueness of components in these series for the Chern classes given by those polynomials. [ABSTRACT FROM AUTHOR]

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi's formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor. [ABSTRACT FROM AUTHOR]

We introduce Chern classes in $U(m)$ -equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$ -cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$ -completion theorem of Greenlees–May and La Vecchia. [ABSTRACT FROM AUTHOR]

Let M be a smooth algebraic variety of dimension 2(p + q) with an algebraic symplectic form and a compatible deformation quantization of the structure sheaf. Consider a smooth coisotropic subvariety Y of codimension q and a vector bundle E on Y .We show that if the pushforward of E admits a deformation quantization (as a module), then its "trace density" characteristic class lifts to a cohomology group associated to the null foliation of Y . Moreover, it can only be nonzero in degrees 2q, ..., 2(p + q). For Lagrangian Y, this reduces to a single degree 2q. Similar results hold in the holomorphic category. [ABSTRACT FROM AUTHOR]

An existence of invariant hypercomplex structure on compact homogeneous spaces implies strong restrictions on their root structure and consequently on their characteristic Pontrjagin classes and the corresponding Chern classes. We describe these constraints by making use of Lie theory. [ABSTRACT FROM AUTHOR]

We introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups GL$\operatorname{GL}$, SLc$\operatorname{SL}^c$, SL$\operatorname{SL}$, Sp$\operatorname{Sp}$). We show that hyperbolic orientations of η$\eta$‐periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that GL$\operatorname{GL}$‐orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that η$\eta$‐periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space BGLn$\operatorname{BGL}_n$. Finally, we construct the universal hyperbolically oriented η$\eta$‐periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum MGL$\operatorname{MGL}$. [ABSTRACT FROM AUTHOR]

In this note we present a solution to the question of whether or not, in the presence of torsion, the topological Nieh–Yan term contributes to chiral anomaly. The integral of Nieh–Yan term is non-zero if topology is non-trivial; the manifold has a boundary or vierbeins have singularities. Noting that singular Nieh–Yan term could be written as a sum of delta functions, we argue that the heat kernel expansion cannot end at finite steps. This leads to a sinusoidal dependence on the Nieh–Yan term and the UV cut-off of the theory (or alternatively the minimum length of spacetime). We show this ill-behaved dependence can be removed if a quantization condition on length scales is applied. It is expected as the Nieh–Yan term can be derived as the difference of two Chern class integrals (i.e. Pontryagin terms). On the other hand, in the presence of a cosmological constant, we find that indeed the Nieh–Yan term contributes to the index with a dimensionful anomaly coefficient that depends on the de Sitter length or equivalently inverse Hubble rate. We find similar result in thermal field theory where the anomaly coefficient depends on temperature. In both examples, the anomaly coefficient depends on IR cut-off of the theory. Without singularities, the Nieh–Yan term can be smoothly rotated away, does not contribute to topological structure and consequently does not contribute to chiral anomaly. [ABSTRACT FROM AUTHOR]

It is known that projective minimal models satisfy the celebrated Miyaoka–Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective, and nonuniruled varieties. [ABSTRACT FROM AUTHOR]

We apply Weiss calculus to enumerate rank r topological complex vector bundles over complex projective spaces \mathbb {C}P^l with vanishing Chern classes when r=l-1 and r=l-2, which are the first two cases in the metastable range. [ABSTRACT FROM AUTHOR]

We study the long time behavior of the Hesse–Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse–Einstein metric. We also derive a convergence result for a twisted Hesse–Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse–Einstein metric by Cheng–Yau and Caffarelli–Viaclovsky as well as the real Calabi theorem by Cheng–Yau, Delanoë and Caffarelli–Viaclovsky. [ABSTRACT FROM AUTHOR]

Let E be a holomophic vector bundle over a compact Astheno-Kähler manifold (M, ω). The authors would prove that E is a numerically flat vector bundle if E is pseudo-effective and the first Chern class c 1 B C (E) is zero. [ABSTRACT FROM AUTHOR]

Let G be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes c²1(S)/c2(S) of minimal nonsingular surfaces of general type S with S with π1(S)≃G are dense in the interval [1,3]. [ABSTRACT FROM AUTHOR]

It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities. [ABSTRACT FROM AUTHOR]

In this article, we give numerical restrictions on the Chern classes of Ulrich bundles on higher-dimensional manifolds, which are inspired by the results of Casnati in the case of surfaces. As a by-product, we prove that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface. Moreover, we prove that the cotangent bundle is never Ulrich. [ABSTRACT FROM AUTHOR]

We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin. These techniques seem also suitable to deal with three orthogonal generalisations: (1) the extension to the r -hyperelliptic locus; (2) the extension to an arbitrary number of non-Weierstrass pairs of points; (3) the extension to multiple descendants. [ABSTRACT FROM AUTHOR]

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ ≥ 0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces. [ABSTRACT FROM AUTHOR]