Zhanjun Ran

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For a semistable reflexive sheaf E of rank r and c 1 = a on P n and an integer d such that r|ad, we give sufficient conditions so that the restriction of E on a generic rational curve of degree d is balanced, i.e. a twist of the trivial bundle (for instance, if E has balanced restriction on a generic line, or r = 2 or E is an exterior power of the tangent(More)
Recall that a Q-Fano variety is by definition a normal projective variety X such that the anticanonical divisor class −K = −K X is Q-Cartier and ample. For such X we define the (Weil) index i = i(X) to be the largest integer such that K X /i exists as a Weil divisor (see [R] for a discussion of Weil divisors and reflexive sheaves, and also Lemma 2 below; NB(More)
We describe a kind of deformation of the anti-DeRham algebra on a Calabi-Yau manifold X. These are in 1-1 correspondence with the total cohomology ⊕H i (X, C). In his article [W] in this volume's precursor, Witten proposed, as a possible approach to constructing a mirror map, a certain extended moduli space N , a thickening of the usual moduli space M of(More)
We show that the formal moduli space of a Calabi-Yau manifold X n carries a linear structure, as predicted by mirror symmetry. This linear structure is canonically associated to a splitting of the Hodge filtration on H n (X). We begin by establishing terminology. In this paper a Calabi-Yau n-manifold means a compact complex manifold X such that
One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to CP 2 and a projection of the image curve from an appropriate point p ∈ CP 2 to the pencil of lines through p. We introduce a natural stratification of Hurwitz spaces according to the minimal degree(More)
  • Z Ran, By Riemann-Roch, Lemma, Professors P Burchard, J Kollár, R K Lazarsfeld
  • 2008
Recall that a Q-Fano variety is by definition a normal projective variety X such that the anticanonical divisor class −K = −K X is Q-Cartier and ample. For such X we define the (Weil) index i = i(X) to be the largest integer such that K X /i exists as a Weil divisor (see [R] for a discussion of Weil divisors and reflexive sheaves, and also Lemma 2 below; NB(More)
The purpose of this paper is establish and apply an enumerative formula or 'method' dealing with a family C = { ¯ C y : y ∈ Y } of rational curves on a variety S, e.g. a rational surface. Significantly, the family C is not assumed to be the family of 'all' rational curves of given homology class: rather, we require only that it be sufficiently large (n =(More)
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