New PDF release: Applied Picard--Lefschetz Theory

Algebraic Geometry

By V. A. Vassiliev

Many very important features of mathematical physics are outlined as integrals reckoning on parameters. The Picard-Lefschetz concept reviews how analytic and qualitative homes of such integrals (regularity, algebraicity, ramification, singular issues, etc.) rely on the monodromy of corresponding integration cycles. during this e-book, V. A. Vassiliev provides a number of types of the Picard-Lefschetz thought, together with the classical neighborhood monodromy thought of singularities and whole intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz thought, and in addition twisted models of these kind of theories with purposes to integrals of multivalued kinds. the writer additionally indicates how those types of the Picard-Lefschetz idea are utilized in learning numerous difficulties coming up in lots of parts of arithmetic and mathematical physics. specifically, he discusses the next periods of capabilities: quantity features bobbing up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; primary suggestions of hyperbolic partial differential equations; multidimensional hypergeometric services generalizing the classical Gauss hypergeometric crucial. The publication is aimed at a huge viewers of graduate scholars, study mathematicians and mathematical physicists attracted to algebraic geometry, complicated research, singularity conception, asymptotic tools, strength thought, and hyperbolic operators.

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Then P (8(Y)xF)(El"'" Ep) = YXF(E 1 , ... , Ep) - L XF(E 1 , ... , 8(Y)Ei"'" Ep) i=1 = - Lg(8(Y)Ei ,Ei )XF(E 1 , ... ,Ep) = - L g([Y, E i ], Ei). On the other hand, p p TrLA y = Lg(AyEi,E;) = - Lg(\l~y,Ei) i=1 i=1 p p i=1 i=1 = - Lg(\l~Ei + [Ei,Y],Ei) = - Lg([Ei,Y],Ei ), where we have used g(\l~ E i , E i ) = ~ Y g(Ei, E i ) = O. Comparison proves the desired identity. D We consider two special cases. 20 COROLLARY. 18, and let Y E fL1.. 21) where Ii is the mean curvature one-form of:F (RummIer [Ru 1]).

It follows that H~(F) -I- o. 0 Next we calculate the value of the integral of divE Y for Y E V (F) in case of a Riemannian foliation. 24 THEOREM. Let F be a transversally oriented Riemannian foliation on a closed oriented Riemannian manifold (M,g). Let Y E V(F). Then (the global scalar product of the sections T and Y of Q). 26). Since 'Po E F 2 n p+ 1, and i(Y)v E Fq-1nq-1, it follows that the term i(Y)v /\ 'Po is of filtration degree q + 1, and hence vanishes. Thus diVE YJ1 = d(i(Y)v /\ XF) + '" /\ i(Y)v /\ XF· 5 Since K 1\ v TRANSVERSAL RIEMANNIAN GEOMETRY 49 ErA q+ 1 Q*, and hence vanishes, we have further 0= i(Y)KV - K 1\ i(Y)v.

The transverse orient defined invariant, by the orientability of Q. 2) holonomy invariant transversal metric gQ there is a corresponding transversal 8(V)v = 0 for all V E rL, volume form v E rAqQ* c ~q(M). e. 2) 8(V)v = 0 for all V E rL, q L vh,···, 8(V)so;,"" Sq) (8(V)V)(Sl"'" vh,···,and 8(V)so;,"" for Sl, ... ,Sq rQ. ForSq) q ==1VV(Sl,"" the conceptSq)of-a L transversal holonomySq) invariant where (8(V)V)(Sl"'" Sq) = VV(Sl,"" Sq) - 0;=1 q E 0;=1 invariant transversal metric. volume coincides with the concept of a holonomy simple a Riemannian is given by nonsingular Killing for Sl,A...

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