- Positive form
In

complex geometry , the term "positive form"refers to several classes of real differential formsof Hodge type "(p, p)".**(1,1)-forms**Real ("p","p")-forms on a complex manifold "M"are forms which are of type ("p","p") and real,that is, lie in the intersection :$Lambda^\{p,p\}(M)cap\; Lambda^\{2p\}(M,\{Bbb\; R\}).$A real (1,1)-form $omega$ is called

**positive**if any of thefollowing equivalent conditions hold#$sqrt\{-1\}omega$ is an imaginary part of a positive (not necessarily positive definite)

Hermitian form .

#For some basis $dz\_1,\; ...\; dz\_n$ in the space $Lambda^\{1,0\}M$ of (1,0)-forms,$sqrt\{-1\}omega$ can be written diagonally, as $sqrt\{-1\}omega\; =\; sum\_i\; alpha\_i\; dz\_iwedge\; dar\; z\_i,$ with $alpha\_i$ real and non-negative.

#For any (1,0)-tangent vector $vin\; T^\{1,0\}M$, $-sqrt\{-1\}omega(v,\; ar\; v)\; geq\; 0$

#For any real tangent vector $vin\; TM$, $omega(v,\; I(v))\; geq\; 0$, where $I:;\; TMmapsto\; TM$ is thecomplex structure operator.**Positive line bundles**In algebraic geometry, positive (1,1)-forms arise as curvatureforms of

ample line bundle s (also known as "positive line bundles"). Let "L" be a holomorphic Hermitian linebundle on a complex manifold,:$arpartial:;\; Lmapsto\; Lotimes\; Lambda^\{0,1\}(M)$

its complex structure operator. Then "L" is equipped with a unique connection preserving the Hermitian structure and satisfying

:$abla^\{0,1\}=arpartial$.

This connection is called "the

Chern connection ".The curvature $Theta$ of a Chern connection is always apurely imaginary (1,1)-form. A line bundle "L" is called "positive" if

:$sqrt\{-1\}Theta$

is a positive (1,1)-form. The

Kodaira vanishing theorem claims that a positive line bundle is ample, and conversely, anyample line bundle admits a Hermitian metric with $sqrt\{-1\}Theta$ positive.**Positivity for "(p, p)"-forms**Positive (1,1)-forms on "M" form a

convex cone .When "M" is a compactcomplex surface , $dim\_\{Bbb\; C\}M=2$, this cone is

self-dual, with respectto the Poincaré pairing:$eta,\; zeta\; mapsto\; int\_M\; etawedgezeta$For "(p, p)"-forms, where $2leq\; p\; leq\; dim\_\{Bbb\; C\}M-2$,there are two different notions of positivity. A form is called

**strongly positive**if it is a linear combination ofproducts of positive forms, with positive real coefficients.A real "(p, p)"-form $eta$ on an "n"-dimensionalcomplex manifold "M" is called**weakly positive**if for all strongly positive "(n-p, n-p)"-forms ζ with compact support, we have$int\_M\; etawedgezetageq\; 0$.Weakly positive and strongly positive formsform convex cones. On compact manifoldsthese cones are dualwith respect to the Poincaré pairing.

**References***Phillip Griffiths and Joseph Harris (1978), "Principles of Algebraic Geometry", Wiley. ISBN 0471327921

*J.-P. Demailly, " [

*http://arxiv.org/abs/alg-geom/9410022 L*] ".^{2}vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)

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**positive form**— noun : either of a pair of congruent crystal forms that together correspond to a single form in a class of higher symmetry … Useful english dictionary**positive**— I. adjective Etymology: Middle English, from Anglo French, from Latin positivus, from positus, past participle of ponere Date: 14th century 1. a. formally laid down or imposed ; prescribed < positive laws > b. expressed clearly or peremptorily <… … New Collegiate Dictionary**positive**— positiveness, n. /poz i tiv/, adj. 1. explicitly stated, stipulated, or expressed: a positive acceptance of the agreement. 2. admitting of no question: positive proof. 3. stated; express; emphatic: a positive denial. 4. confident in opinion or… … Universalium**positive**— pos•i•tive [[t]ˈpɒz ɪ tɪv[/t]] adj. 1) confident in opinion or assertion; sure: He is positive that he ll win[/ex] 2) cvb showing or expressing approval or agreement; favorable: a positive reaction to the speech[/ex] 3) cv gram. expressing or… … From formal English to slang**Positive Psychotherapie**— (seit 1968) wird die Ausgestaltung des psychotherapeutischen Vorgehens von Nossrat Peseschkian und Mitarbeitern genannt, das dieser aus dem Iran stammende Wiesbadener Facharzt für Neurologie, Psychiatrie und Psychotherapeutische Medizin… … Deutsch Wikipedia**positive**— [päz′ə tiv] adj. [ME positif < OFr < L positivus < positus: see POSITION] 1. formally or arbitrarily set; conventional; artificial [a positive law] 2. definitely set; explicitly laid down; admitting of no question or modification;… … English World dictionary**Positive psychotherapy**— is a psychodynamic method of psychotherapy founded by Dr. Nossrat Peseschkian in 1968 in Germany. It is based on a positive conception of humanity, and has an integral and holistic approach. It is today spread in many countries. The main center… … Wikipedia**Positive Discipline**— (or PD) is a discipline system used by schools that focuses on the positive points of behaviour. Some practitioners believe that educators should act with a philosophy that there are no bad children, just good and bad behaviors. You can teach and … Wikipedia**Positive**— is a property of positivity and may refer to: Mathematics and science * Positive number, a number that is greater than 0 * Positive operator, in functional analysis, a bounded linear operator whose spectrum consists of positive real numbers *… … Wikipedia**positive definite**— positive definiteness. Math. 1. (of a quadratic form) positive for all real values of the variables, where the values are not all zero. 2. (of a matrix) displaying the coefficients of a positive definite quadratic form. [1905 10] * * * … Universalium