1,720,987 research outputs found
Quasi-negative holomorphic sectional curvature and ampleness of the canonical class
Full text linkThis note is an extended version of a 50 min talk given at the INdAM Meeting “Complex and Symplectic Geometry”, held in Cortona from June 12th to June 18th, 2016. What follows was the abstract of our talk.
Let X be a compact Kähler manifold with a Kähler metric whose holomorphic sectional curvature is strictly negative. Very recent results by Wu–Yau and Tosatti– Yang confirmed an old conjecture by S.-T. Yau which claimed that under this curvature assumption X should be projective and canonically polarized. We will explain how one can relax the assumption on the holomorphic sectional curvature to the weakest possible, i.e. non positive and strictly negative in at least one point, in order to have the same conclusions. We shall also try to motivate this generalization by arguments coming from birational geometry, such as the abundance conjecture.
The results presented here were originally contained in the joint work with Diverio and Trapani (Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, 2016, ArXiv e-prints 1606.01381v3)
Existence of global invariant jet differentials on projective hypersurfaces of high degree
No full textLet X ⊂ Pn+1 be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X , vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair (Pn, D), where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n
Segre forms and Kobayashi–Lübke inequality
Full text linkStarting from the description of Segre forms as directimages of (powers of) the first Chern form of the (anti) tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi–Lübke inequality
Differential equations on complex projective hypersurfaces of low dimension
Get PDFLet n = 2, 3, 4, 5 and let X be a smooth complex projective hypersurface of Pn+1. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k = n = dim X , and also similar bounds for order k > n. Moreover, for every integer n 2, we show that there are no such algebraic differential equations of order k < n for a smooth hypersurface in Pn+1
Smooth metrics on jet bundles and applications
No full textFollowing a suggestion made by J.-P. Demailly, for each k≥1, we endow, by an induction process, the k-th (anti)tautological line bundle O_X_k(1) of an arbitrary complex directed manifold (X,V) with a natural smooth Hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically---in a sense to be specified later) only on the curvature of V and on the structure of the fibration Xk→X. When X is a surface and V=TX, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for X a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem [1] of Bogomolov
On a conjecture of Oguiso about rational curves on Calabi-Yau threefolds
No full textLet X be a Calabi–Yau threefold. We show that if there exists on X a non-zero nef non-ample divisor then X contains a rational curve, provided its second Betti number is greater than 4
About the hyperbolicity of complete intersections
Full text linkThis note is an extended version of a thirty minutes talk given at the “XIX Congresso dell’Unione Matematica Italiana”, held in Bologna from September 12th to September 17th, 2011. This was essentially a survey talk about connections between Kobayashi hyperbolicity properties and positivity properties of the canonical bundle of projective algebraic varieties
Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture
Full text linkLet be a complex manifold, be a rank holomorphic
hermitian vector bundle, and be a sequence of dimensions . Let , , be the
tautological line bundles over the (possibly incomplete) flag bundle
associated to , endowed with the natural
metrics induced by that of , with Chern curvatures . We show
that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the
push-forward of a homogeneous polynomial in the Chern classes of the
's also hold pointwise at the level of the Chern forms
in this hermitianized situation. As an application, we show the
positivity of several polynomials in the Chern forms of a Griffiths
(semi)positive vector bundle not previously known, thus giving some new
evidences towards a conjecture by Griffiths, which in turn can be seen as a
pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on
numerically positive polynomials for ample vector bundles.Comment: 24 pages, no figures, comments are very welcome! v3: several minor
corrections, the main application is now stated for strongly positive form
Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle
Full text linkWe show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non-positive and strictly negative in at least one point, then its canonical bundle is positive. This answers in the affirmative to a question first asked by S.-T. Yau
Hyperbolicity of projective hypersurfaces
Full text linkThis book presents recent advances on Kobayashi hyperbolicity in complex geometry, especially in connection with projective hypersurfaces. This is a very active field, not least because of the fascinating relations with complex algebraic and arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta, among others, resulted in precise conjectures regarding the interplay of these research fields (e.g. existence of Zariski dense entire curves should correspond to the (potential) density of rational points). Perhaps one of the conjectures which generated most activity in Kobayashi hyperbolicity theory is the one formed by Kobayashi himself in 1970 which predicts that a very general projective hypersurface of degree large enough does not contain any (non-constant) entire curves. Since the seminal work of Green and Griffiths in 1979, later refined by J.-P. Demailly, J. Noguchi, Y.-T. Siu and others, it became clear that a possible general strategy to attack this problem was to look at particular algebraic differential equations (jet differentials) that every entire curve must satisfy. This has led to some several spectacular results. Describing the state of the art around this conjecture is the main goal of this work
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