A New Proof of Torellis Theorem
A New Proof of Torellis Theorem
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In our eReader you can find the full English version of the book. Read A New Proof of Torellis Theorem Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book A New Proof of Torellis Theorem
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By virtue of Riemann's theorem it follows that the image in C^ under the map " --^ of (g-l)-tuples of points on X, is determined up to a trans- lation by the period matrix (TriE; A), since it is the set of zeros of 0(u; A) . Thus the sets V/^~ are determined as the zero manifolds of a first-order theta-functions. We shall now drav; an interesting conclusion from the irreduc- Ibility of W^' . Assiome that, in the period matrix, (7riE, A) A decomposes into a direct sum \0 A^J where A^ is an nXn m...atrix. It is readily seen from the definition that in this case 29 e(u; A) = . ^(v; A^)b(w; A^ ), ^^Jhere v is an n-vector, w is a (g-n)-vector^ and u = (v, . .. ;V^, v;, . .. , \^^~^) . Nov/, since W^~ is the zero a manifold of ^{u; A) for some a, it follows that &(v; A, ) vanishes identically on a full neighborhood of W^", but not everywhere, since the zeros of :(v; A ) and 6;(w; A^) do not coincide. This contradicts the irreducibility of W^~, and we a conclude Theorem : If {vlE; A) is a period matrix v/ith respect to a canonical homology basis on a closed Riemann surface, then A is not symplectically equivalent to a direct sum .
What to read after A New Proof of Torellis Theorem?
You can find similar books in the "Read Also" column, or choose other free books by Henrik H Martens to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
By virtue of Riemann's theorem it follows that the image in C^ under the map " --^ of (g-l)-tuples of points on X, is determined up to a trans- lation by the period matrix (TriE; A), since it is the set of zeros of 0(u; A) . Thus the sets V/^~ are determined as the zero manifolds of a first-order theta-functions. We shall now drav; an interesting conclusion from the irreduc- Ibility of W^' . Assiome that, in the period matrix, (7riE, A) A decomposes into a direct sum \0 A^J where A^ is an nXn m...atrix. It is readily seen from the definition that in this case 29 e(u; A) = . ^(v; A^)b(w; A^ ), ^^Jhere v is an n-vector, w is a (g-n)-vector^ and u = (v, . .. ;V^, v;, . .. , \^^~^) . Nov/, since W^~ is the zero a manifold of ^{u; A) for some a, it follows that &(v; A, ) vanishes identically on a full neighborhood of W^", but not everywhere, since the zeros of :(v; A ) and 6;(w; A^) do not coincide. This contradicts the irreducibility of W^~, and we a conclude Theorem : If {vlE; A) is a period matrix v/ith respect to a canonical homology basis on a closed Riemann surface, then A is not symplectically equivalent to a direct sum .
What to read after A New Proof of Torellis Theorem?
You can find similar books in the "Read Also" column, or choose other free books by Henrik H Martens to read onlineMoreLess
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