Algebraic Manifolds Research Articles

Superposition interpolation process in [formula omitted

In this paper, we deeply research Lagrange interpolation by polynomial in several variables and give an application of Cayley–Bacharach theorem for it. Particularly on the sufficiently intersected algebraic manifold (or SIAM, for short), we introduce a general method of constructing properly posed set of nodes (or PPSN, for short) for Lagrange interpolation, namely the superposition interpolation process. Then we give an equivalent condition about a PPSN along a SIAM. Further we introduce a relation between the sufficiently intersected algebraic hypersurfaces and H-basis. At the end of this paper, we use the extended Cayley–Bacharach theorem to resolve some problems of Lagrange interpolation along the zero-dimensional and one-dimensional algebraic manifolds.

Building blocks of étale endomorphisms of complex projective manifolds

Etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak Calabi-Yau varieties.

An update on semisimple quantum cohomology and F-manifolds

To the memory of V.A. Iskovskikh Abstract—In the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative OM -bilinear multiplication on its tangent sheaf TM is an F -manifold in the sense of Hertling- Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle T ∗ M , is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.

Equivariant Genera of Complex Algebraic Varieties

For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain Atiyah-Meyer type formulae for equivariant Hodge-theoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera y(X;g) of a quasi-projective variety X acted upon by a nite group of algebraic automorphisms are dened by combining the group action with the information encoded by the Hodge ltration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold y(X;g) can by computed by the Atiyah- Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer type formula for y(X;g) in the case when X is not necessarily smooth or compact, but just bers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodge-theoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodge-theoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities.

Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: (1) There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and $$\varphi\,:\, C\,\longrightarrow\, X$$ a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable. (2) The variety X admits an etale covering by an abelian variety. (3) The dimension dim X ≤ 1.

PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

Maximal rationally connected fibrations and movable curves

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.

Affine Hermitian-Einstein Metrics

Here c1(E, h) is the first Chern form of E with respect to a Hermitian metric h. The famous theorem of Donaldson [7, 8] (for algebraic manifolds only) and Uhlenbeck-Yau [24, 25] says that an irreducible vector bundle E → N is ω-stable if and only if it admits a HermitianEinstein metric (i.e. a metric whose curvature, when the 2-form part is contracted with the metric on N , is a constant times the identity endomorphism on E). This correspondence between stable bundles and Hermitian-Einstein metrics is often called the Kobayashi-Hitchin correspondence. An important generalization of this theorem is provided by Li-Yau [15] for complex manifolds (and subsequently due to Buchdahl by a different method for surfaces [3]). The major insight for this extension is the fact that the degree is well-defined as long as the Hermitian form ω on N satisfies only ∂∂ωn−1 = 0. This is because

Symplectic approach to quantum constraints

A general prescription for the treatment of constrained quantum motion is outlined. We consider in particular constraints defined by algebraic submanifolds of the quantum state space. The resulting formalism is applied to obtain solutions to the constrained dynamics of systems of multiple spin particles. When the motion is constrained to a certain product space containing all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in the sense that the equations of motion take a form identical to that of unitary motion, but with different boundary conditions. When the constrained subspace is a product space of disentangled states, the associated motion is more intricate. Nevertheless, the equations of motion satisfied by the dynamical variables are obtained in closed form.

Low rank perturbation of regular matrix polynomials

Let A ( λ ) be a complex regular matrix polynomial of degree ℓ with g elementary divisors corresponding to the finite eigenvalue λ 0 . We show that for most complex matrix polynomials B ( λ ) with degree at most ℓ satisfying rank B ( λ 0 ) < g the perturbed polynomial ( A + B ) ( λ ) has exactly g - rank B ( λ 0 ) elementary divisors corresponding to λ 0 , and we determine their degrees. If rank B ( λ 0 ) + rank ( B ( λ ) - B ( λ 0 ) ) does not exceed the number of λ 0 -elementary divisors of A ( λ ) with degree greater than 1, then the λ 0 -elementary divisors of ( A + B ) ( λ ) are the g - rank B ( λ 0 ) - rank ( B ( λ ) - B ( λ 0 ) ) elementary divisors of A ( λ ) corresponding to λ 0 with smallest degree, together with rank ( B ( λ ) - B ( λ 0 ) ) linear λ 0 -elementary divisors. Otherwise, the degree of all the λ 0 -elementary divisors of ( A + B ) ( λ ) is one. This behavior happens for any matrix polynomial B ( λ ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ℓ . If A ( λ ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.

Linear manifolds in the moduli space of one-forms

We study closures of GL2+(R)-orbits in the total space ΩMg of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus, or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to ΩMg. For nongeneric strata, similar results can be shown by a case-by-case inspection. We also propose to study a notion of linear manifold that comprises Teichmuller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow [DM]. Moreover, we give an explanation for the difference between Hilbert modular surfaces and Hilbert modular threefolds with respect to this notion of linearity

A Second Main Theorem of Nevanlinna Theory for Meromorphic Functions on Complex Submanifolds in C n

We show a second main theorem of Nevalinna theory for meromorphic functions on complex submanifolds in Cn. This has a similar form to the classical one and has a remainder term including Ricci curvature. We also give a concrete computation of the remainder term in the case of nonsingular algebraic submanifolds.

Schematic homotopy types and non-abelian Hodge theory

AbstractWe use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$ on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

On classification and construction of algebraic Frobenius manifolds

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld–Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

Finite generation of canonical ring by analytic method

In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425).

Ample subvarieties and rationally connected fibrations

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.

Division theorems and twisted complexes

We prove new Skoda-type division, or ideal membership, theorems. We work in a geometric setting of line bundles over Kahler manifolds that are Stein away from an analytic subvariety. (This includes complex projective manifolds.) Our approach is to combine the twisted Bochner-Kodaira Identity, used in the Ohsawa-Takegoshi Theorem, with Skoda’s basic estimate for the division problem. Techniques developed by McNeal and the author are then used to provide many examples of new division theorems. Among other applications, we give a modification of a recent result of Siu regarding effective finite generation of certain section rings.

On Semistable Principal Bundles over a Complex Projective Manifold

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P. We prove that a holomorphic principal G-bundle E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with if and only if for every pair of the form (Y, ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y, and for every holomorphic reduction of structure group E P ⊂ ψ*E G to the subgroup P, the line bundle over Y associated with the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with if and only if for each pair (Y, ψ) of the above type the G-bundle ψ*E G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Miyaoka [12], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations, one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

On period maps that are open embeddings

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.

Coupling Nonlinear Sigma-Matter to Yang-Mills Fields: Symmetry Breaking Patterns

We extend the traditional formulation of Gauge Field Theory by incorporating the (non-Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinear-sigma-model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling prescription, which resembles the so-called Stueckelberg transformation [1], but for the non-Abelian case. Here we study the case of internal gauge symmetry groups, in particular, unitary groups U(N). We show how to couple standard Yang-Mills Theory to Nonlinear-Sigma Models on cosets of U(N): complex projective, Grassman and flag manifolds. These different couplings lead to distinct (chiral) symmetry breaking patterns and Higgs-less mass-generating mechanisms for Yang-Mills fields.