Holomorphic Differentials Research Articles

On the Dimension of Varieties of Special Divisors

Let ${T_g}$ denote the Teichmüller space and let $V$ denote the universal family of Teichmüller surfaces of genus $g$ Let $V_{{T_g}}^{(n)}$ denote the $n$th symmetric product of $V$ over ${T_g}$ and let $J$ denote the family of Jacobians over ${T_g}$. Let $f:V_{{T_g}}^{(n)} \to \text {J}$ be the natural relativization over ${T_g}$ of the classical map defined by integrating holomorphic differentials. Let \[ u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1\] be the map induced by $f$. We define $G_n^r$ to be the analytic subspace of $V_{{T_g}}^{(n)}$ defined by the vanishing of ${ \wedge ^{n - r + 1}}u$. Put $\tau = (r + 1)(n - r) - rg$. We show that $G_n^1 - G_n^2$, if nonempty, is smooth of pure dimension $3g - 3 + \tau + 1$. From this result, we may conclude that, for a generic curve $X$, the fiber of $G_n^1 - G_n^2$ over the module point of $X$, if nonempty, is smooth of pure dimension $\tau + 1$, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of $G_n^r$.

On the dimension of varieties of special divisors

Let T g {T_g} denote the Teichmüller space and let V V denote the universal family of Teichmüller surfaces of genus g g Let V T g ( n ) V_{{T_g}}^{(n)} denote the n n th symmetric product of V V over T g {T_g} and let J J denote the family of Jacobians over T g {T_g} . Let f : V T g ( n ) → J f:V_{{T_g}}^{(n)} \to \text {J} be the natural relativization over T g {T_g} of the classical map defined by integrating holomorphic differentials. Let \[ u : f ∗ Ω J / T g 1 → Ω V T g / T g ( n ) 1 u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1 \] be the map induced by f f . We define G n r G_n^r to be the analytic subspace of V T g ( n ) V_{{T_g}}^{(n)} defined by the vanishing of ∧ n − r + 1 u { \wedge ^{n - r + 1}}u . Put τ = ( r + 1 ) ( n − r ) − r g \tau = (r + 1)(n - r) - rg . We show that G n 1 − G n 2 G_n^1 - G_n^2 , if nonempty, is smooth of pure dimension 3 g − 3 + τ + 1 3g - 3 + \tau + 1 . From this result, we may conclude that, for a generic curve X X , the fiber of G n 1 − G n 2 G_n^1 - G_n^2 over the module point of X X , if nonempty, is smooth of pure dimension τ + 1 \tau + 1 , a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of G n r G_n^r .

Holomorphic differentials on open Riemann surfaces

Holomorphic differentials on open Riemann surfaces

Positive holomorphic differentials on Klein surfaces

Positive holomorphic differentials on Klein surfaces

On the Periods of Certain Rational Integrals: I

In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how Macaulay's theorem 4.11 is essentially equivalent to a suitable vanishing theorem for sheaf cohomology. We shall also give a proof of the de Rham algebraic theorem used in the proof of Theorem 5.3. However, our principal motivation is to be able to discuss rational integrals in case our hypersurface V c P, has rather simple singularities, and the localization technique of sheaf theory seems to be the best method for doing this (cf. ?? 15, 16 below). Let then V c P, be a non-singular hypersurface. We want to give a sheaf-theoretic version of the Hodge filtration of Hq(V, C) (cf. ? 8). For this we let Qv be the sheaf on V of holomorphic q-forms and & c QV the subsheaf of closed forms. The Poincare' lemma for holomorphic differentials gives an exact sequence

Holomorphic quadratic differentials on surfaces inE3

Holomorphic quadratic differentials on surfaces in<i>E</i>³

Holomorphic differentials as functions of moduli

Holomorphic differentials as functions of moduli

Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces.